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ESTUDIO DE ALGUNOS PROBLEMAS INVERSOS Y DE

CONTROLABILIDAD: TRANSMISION DE ONDAS Y

TRANSPORTE-DIFUSION

TESIS PARA OPTAR AL GRADO DE DOCTOR EN

CIENCIAS DE LA INGENIERIA

MENCION MODELAMIENTO MATEMATICO

EN COTUTELA CON LA UNIVERSIDAD DE

VERSAILLES SAINT-QUENTIN-EN-YVELINES

ALBERTO CARLOS MERCADO SAUCEDO

PROFESOR GUIA:

AXEL OSSES ALVARADO

PROFESOR CO-GUIA:

JEAN-PIERRE PUEL

MIEMBROS DE LA COMISION:

CARLOS CONCA ROSENDE

JEAN-MICHEL CORON

OTARED KAVIAN

MICHAL KOWALCZYK

MAURICIO SEPULVEDA CORTES

SANTIAGO DE CHILE

ENERO 2007

MIS AGRADECIMIENTOS + SINCEROS

Al gobierno del estado de Jalisco

Al ayuntamiento de Guadalajara

Al Fondo de Cultura Econmica

A PEMEX a PIPSA a BANAMEX a BANCOMER

Y muy muy en particular

A la Loterıa Nacional para la Asistencia Publica

Sin loterıa yo no estoy aquı

Se comprueba la teorıa de Leonardo:

1 % de inspiracion

2 de transpiracion

Y el resto suerte

Nicanor Parra

“No importa quien lo dijo: La hora

de sentar cabeza no llegara jamas”.

Roberto Bolano

i

Muchas gracias

Agradezco enormemente a los profesores Carlos Conca, Jean-Michel Coron, Otared Ka-

vian, Michal Kowalczyk y Mauricio Sepulveda el haber aceptado ser parte de mi jurado

de tesis. En especial gracias para Enrique Fernandez-Cara y Jean-Michel Coron por haber

aceptado ser rapporteurs de este trabajo.

El apoyo constante de mis directores de tesis ha sido muy importante, no solo para la

elaboracion de esta tesis, sino para mi desarrollo como matematico. Por toda la paciencia,

la preocupacion y por proponerme interesantes temas de trabajo, gracias Jean-Pierre. Por la

guıa, el invaluable apoyo desde el principio de este camino, y por la amistad, muchas gracias

Axel.

Ha sido muy grato y provechoso para mı trabajar en colaboracion con Lucie Baudouin

y con Sergio Guerrero. A cada uno de ellos les agradezco su disponibilidad para trabajar en

conjunto.

Un reconocimiento a las instituciones donde este trabajo se ha elaborado, el DIM y el

CMM de la U de Chile, y el Laboratorio de Matematicas de la U de Versailles. En particular

gracias a Luis Mella y Regina por parte del DIM, y a Marie-France en Versailles, por su ayuda.

Agradezco el apoyo financiero de MECESUP, del CMM y del proyecto ECOS-CONICYT

CO4E08 2006. Tambien, gracias a Pablo Dartnell por su labor en el doctorado del DIM.

Muchas gracias para quienes han compartido conmigo estos ultimos anos, sin cuya amis-

tad todo esto tendrıa poco sentido: Entre ellos Loredana, Cabezon, Adriana, Cristopher,

Flavio, Guille, Joe, Jaliste, Nathalie, Muga. De la misma forma, recuerdo a mis companeros

versallescos Baba, Yousri, Mic, Eric, Jean Maxime; y a la banda chilena en Paris. Va un

especial agradecimiento para Chincho, companero de ruta en esta aventura.

Un sentido agradecimiento para mis hermanos Juan Pablo y Dora, y para mi madre Dora

Maria, sin cuyo apoyo, ası sea a la distancia, todo seria distinto.

Gracias a Isabel por su amor y su apoyo.

ii

Indice general

Introduccion 1

1. Motivacion y antecedentes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Primer Problema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Introduccion: Teorıa de Control de EDP’s. . . . . . . . . . . . . . . . . 3

2.2. Planteamiento del primer problema. . . . . . . . . . . . . . . . . . . . 4

3. Segundo Problema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1. Introduccion: Problemas inversos y desigualdades de Carleman. . . . . 5

3.2. Planteamiento del segundo problema. . . . . . . . . . . . . . . . . . . 6

4. Resultados principales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1. Una desigualdad inversa en transporte-difusion. . . . . . . . . . . . . . 7

4.2. Desigualdades de Carleman: un problema de transmision de ondas. . . 10

I. Transport-difussion equation. 15

1. Introduction and contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2. Basic definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1. Dual characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Transport equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1. Controllability properties. . . . . . . . . . . . . . . . . . . . . . . . . . 28

iii

iv

4. Transport-Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1. Results on the literature. . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2. Convergence of localized solutions. . . . . . . . . . . . . . . . . . . . . 36

4.3. The ε-estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4. An ε-inverse inequality for the right hand side. . . . . . . . . . . . . . 45

4.5. The cost of the -regional-approximate controllability . . . . . . . . . . 45

5. Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1. Transport equation (µ = 0) . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2. Fokker-Planck equation (µ > 0) . . . . . . . . . . . . . . . . . . . . . . 53

II. Transmission wave equation. 55

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2. The transmission wave equation. . . . . . . . . . . . . . . . . . . . . . . . . . 59

3. The Carleman estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1. The weight function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2. Development of the L2-product . . . . . . . . . . . . . . . . . . . . . . 71

3.3. The inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4. Proof of the Carleman estimate. . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1. The interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2. The traces at the boundary . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3. Proof of Theorem 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5. Application to the inverse problem . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1. The perturbed inverse problem. . . . . . . . . . . . . . . . . . . . . . . 85

5.2. Proof of Theorem 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Conclusiones. 92

Bibliografıa 95

v

RESUMEN DEL INFORME FINAL

PARA OPTAR AL GRADO DE

DOCTOR EN CIENCIAS DE LA INGENIERIA

MENCION MODELACION MATEMATICA

POR: ALBERTO CARLOS MERCADO SAUCEDO

FECHA: 23 DE ENERO DE 2007

PROFESORES GUIAS: DR. A. OSSES,

DR. J-P. PUEL

Estudio de algunos problemas inversos y de controlabilidad:

Transmision de ondas y transporte-difusion

Esta tesis tiene por objetivo estudiar algunas desigualdades inversas y su aplicacion a

algunos problemas de controlabilidad de ciertas ecuaciones en derivadas parciales lineales.

En el primer capıtulo estudiamos una ecuacion de transporte-difusion lineal con un coe-

ficiente de difusion igual a un parametro positivo pequeno. Estudiamos la controlabilidad

de la ecuacion de transporte puro, y la convergencia de las soluciones de la ecuacion de

transporte-difusion a aquellas de la ecuacion de transporte. Como consecuencia de estos dos

puntos demostramos, bajo ciertas propiedades geometricas, una estimacion de la solucion

en terminos de la condicion inicial, cuando esta posee regularidad H1, mas un termino de

error. Demostramos que esta desigualdad inversa proporciona una cota del costo del control

aproximado cuando el coeficiente de difusion converge a cero. Estudiamos algunas propiedades

de controlabilidad de la ecuacion de Fokker-Planck: las analogas al caso del transporte puro

cuando la difusion es cero, y la controlabilidad aproximada para el caso general.

En el segundo capıtulo consideramos un problema inverso para la ecuacion de ondas

en un dominio acotado con coeficiente principal discontinuo: el problema de recuperar el

potencial de la ecuacion por medio de la observacion en el borde de la derivada normal de

la solucion. Consideramos el caso de un coeficiente principal constante por pedazos, cuyo

vi

conjunto de discontinuidades es precisamente la frontera comun de los dominios interior y

exterior. Demostramos que bajo las hipotesis de convexidad fuerte del dominio interior, y de

que la velocidad en el dominio exterior es menor que en el interior, se tiene una desigualdad

de Carleman para las soluciones de este problema, con una funcion peso adaptada a nuestro

caso. Utilizando esta desigualdad de Carleman y siguiendo el metodo de Bukhgeim-Klibanov

resolvemos el problema inverso planteado. Cabe destacar que la desigualdad de Carleman

obtenida es de interes por sı misma y deja abierta la posibilidad de su aplicacion al estudio

de otros problemas inversos en transmision de ondas y de la controlabilidad de la ecuacion

semilineal.

vii

Abstract

The object of this thesis is the study of inverse inequalities for some linear partial differential

equations. In the first chapter we study the linear transport-diffusion equation with a diffusion

coefficient given by a small positive parameter. We study the controllability for the transport

equation and the convergence of the solutions of the transport-diffusion equation when the

diffusion goes to zero. We deduce some bound for the cost of the approximate controllability

for the transport diffusion equation when the diffusion goes to zero. In the second chapter we

consider a transmission wave equation in two embedded domains in Rn, where the speed is

a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman ine-

quality for this problem under some convexity hypothesis for the inner domain, and assuming

a2 < a1. As a consequence of this inequality, we obtain uniqueness and Lipschitz stability for

the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet

data and discontinuous principal coefficient from a single time-dependent Neumann boundary

measurement.

Key-words: approximate controllability, transport, Carleman estimate, transmission wa-

ve equation, strong convexity.

viii

Quelques problemes inverses et de controlabilite: transmission

des ondes et transport-diffusion

Resume

L’objet de cette these est l’etude de certaines inegalites inverses pour les solutions de

quelques equations aux derivees partielles lineaires. Dans le premier chapitre nous etudierons

les proprietes de controlabilite pour l’equation de transport pure, aussi que des proprietes de

convergence des solutions de l’equation de transport diffusion vers des solutions de l’equation

du transport lorsque le coefficient de diffusion tend vers zero. Nous en deduirons une borne

pour la norme du controle approche lorsque la diffusion tend vers zero. Dans le deuxieme

chapitre nous considererons une equation de transmission des ondes dans deux domaines

emboıtes de Rn, avec vitesse a1 dans le domaine interieur et a2 dans le domaine exterieur. En

supposant que a2 < a1 et que le domaine interieur est fortement convexe, nous montrerons

une inegalite de Carleman pour les solutions de ce probleme. En utilisant cette inegalite,

nous obtiendrons la stabilite Lipschitz pour le probleme inverse qui consiste a determiner le

potentiel stationnaire pour l’equation des ondes avec des donnees Dirichlet et un coefficient

principal discontinu, avec la connaissance de la derivee normale dans la frontiere.

Mots-cles: Controlabilite approche, transport, inegalite de Carleman, transmission des

ondes, convexite forte.

Introduccion

1

INTRODUCCION

1 Motivacion y antecedentes.

Los problemas inversos y la teorıa de control aparecen frecuentemente en varios dominios

de las ciencias y de la ingenierıa, y al mismo tiempo constituyen una fuente de interesantes

problemas matematicos. Sin intentar dar una definicion precisa, podrıamos decir que en gene-

ral un problema inverso consiste en invertir algun proceso, proceso que es comprendido mas o

menos cabalmente, pero cuya inversion constituye un nuevo problema y presenta dificultades

propias, segun el caso.

Por ejemplo, sabiendo que el angulo con el cual un rayo luminoso se refleja es igual al

angulo con el cual incide, es facil determinar donde se reflejara una imagen determinada en un

espejo dado. Pero no es tan directo el problema de determinar la posicion y la altura mınima

que debe tener un espejo plano para que, colgado de manera vertical frente a alguien, este se

observe a sı mismo -exactamente- de cuerpo entero (y probar que la respuesta no depende de

la distancia!). Este es, en cierto modo, un -sencillo- problema inverso.

La clase de problemas inversos en los que estamos interesados consisten en, dada una

ecuacion diferencial parcial (EDP), recuperar algun dato de la ecuacion (por ejemplo algun

coeficiente) suponiendo que se conoce alguna medicion u observacion de su solucion, (la traza

de la solucion en una parte del borde del dominio, por ejemplo).

Por otra parte, la teorıa de control estudia -grosso modo- problemas consistentes en ac-

tuar o intervenir en un sistema dado, de manera de obtener una respuesta determinada. En

particular, en la teorıa de control de EDP’s se estudian las propiedades de controlabilidad

de sistemas gobernados por este tipo de ecuaciones. Basicamete, el problema consiste en,

dada una EDP en evolucion en el dominio Ω e intervalo de tiempo (0, T ), encontrar el control

(tıpicamente, lado derecho de la ecuacion, o valor en la frontera), para que la solucion de la

EDP evaluada en el tiempo final T , satisfaga una condicion dada de antemano (sea igual o

este cerca de cierta funcion dada -control exacto o aproximado- o sea igual a cero -control a

cero-, etcetera).

En esta tesis estamos interesados, por una parte, en el estudio de algunas propiedades

de controlabilidad de una ecuacion de transporte-difusion y de la ecuacion de Fokker Planck

2

INTRODUCCION

frente al efecto del transporte. Por otra parte, estudiamos el problema inverso de determinar

un potencial desconocido en la ecuacion de ondas con coeficientes discontinuos en Rn.

A continuacion, presentamos una sucinta introduccion bibliografica, para exponer el con-

texto en el que se situan los problemas que nos planteamos. Despues, presentamos de manera

precisa los resultados alcanzados en esta tesis con respecto a cada uno de los temas abordados.

2 Primer Problema

2.1 Introduccion: Teorıa de Control de EDP’s.

Dos referencias introductorias clasicas dentro de la teorıa moderna del control de EDP

son los trabajos de D.-L. Russell [61] y J.-L. Lions [48], en este ultimo es donde se introduce

el Metodo de Unicidad Hilbertiana (HUM), pero en ambos trabajos se establece claramente

la dualidad entre controlabilidad y observabilidad, en el contexto de espacios L2. Utilizando

el metodo de los multiplicadores, en tal trabajo (ver tambien la idea original en [33]), se

prueba que para un tiempo T suficientemente grande, y bajo ciertas condiciones geometricas

sobre la zona de control, la ecuacion de ondas es controlable exactamente. En [44] y [52] se

estudian problemas mas generales con variantes de este metodo como por ejemplo la idea

de multiplicadores rotados. Por medio del analisis microlocal, en [3], [16] se dan condiciones

necesarias y suficientes para tener control exacto en la ecuacion de ondas.

Con respecto a la controlabilidad interior de la ecuacion del calor, por una parte una

propiedad basica de esta ecuacion -el efecto regularizante- implica que no se tiene control

exacto. En [60] se probo que la controlabilidad de la ecuacion de ondas en algun tiempo T

con controles soportados en ω ⊂ Ω implica la controlabilidad interior del calor con controles

en ω, para cualquier tiempo T > 0. Sin embargo, la hipotesis impuesta por la ecuacion de

ondas en ω no resulta natural en el marco de la ecuacion del calor (donde se tiene propagacion

de la informacion a velocidad infinita); posteriormente en [47] se probo que la ecuacion del

calor es controlable a cero para todo T > 0 y para todo abierto ω ⊂ Ω no vacıo.

Una herramienta muy importante en la teorıa del control de EDP (y en el estudio de los

problemas inversos, como veremos mas adelante) la constituyen las desigualdades de Carle-

3

INTRODUCCION

man, introducidas en [18] y usadas sistematicamente en [29], [64] para probar desigualdades

de observabilidad en ecuaciones en derivadas parciales. Esta herramienta tiene la enorme

ventaja de dar caracter dominante al coeficiente principal (que debe ser suficientemente sua-

ve) frente a los terminos de orden inferior en la EDP. Esto en particular implica, para todo

abierto no vacıo ω ⊂ Ω, la controlabilidad a cero de la ecuacion parabolica de segundo orden

con terminos de orden menor (y coeficiente principal regular), con control en ω y en tiempo

T , para todo T > 0.

2.2 Planteamiento del primer problema.

Si Ω es un abierto acotado regular, con subconjuntos abiertos Ω0 y ω, estamos interesados

en el estudio de la ecuacion

yt + v · ∇y − ε∆y = 0 in Ω (1)

con condiciones de borde Dirichlet y condicion inicial y(0) = y01Ω0 , donde 1Ω0 representa la

funcion indicatriz del conjunto Ω0. Por los resultados mencionados anteriormente, se tiene

que para cada ε > 0, la ecuacion (1) es controlable a cero, (y aproximadamente). Sin embargo,

las estimaciones obtenidas a partir de las desigualdades de Carleman (ver [26]) muestran que

la norma del control (tanto del control a cero como del control aproximado) diverge cuando

ε converge a cero.

Estamos interesados en encontrar una desigualdad inversa que proporcione la estabilidad

de la condicion inicial y0 en terminos de la traza de la solucion de 1 en un subconjunto ω.

Esto, bajo ciertas condiciones para los conjuntos Ω0 y ω. La idea es que, al disminuir el efecto

de la difusion, primara el efecto del transporte, el cual debera ayudar a tener las estimaciones

inversas deseadas.

Cabe mencionar que en los trabajos [21] y [31] se probo la convergencia a cero de la norma

del control nulo de (1), bajo ciertas condiciones para la zona de control y el tiempo T .

Nuestro enfoque consiste en estudiar dos puntos:

La controlabilidad de la ecuacion de transporte puro (la ecuacion resultante de tomar

4

INTRODUCCION

ε = 0 en (1), con condiciones de borde adecuadas).

La convergencia uniforme de las soluciones de la ecuacion de transporte-difusion hacia

las del transporte puro, cuando ε→ 0.

Despues de estudiar los dos puntos anteriores, en esta tesis se demuestra una desigualdad

inversa para la condicion inicial de (1), soportada en Ω0, en terminos de la restriccion de

la solucion en ω × (0, T ). Despues mostramos que esta desigualdad inversa proporciona una

estimacion de la norma del control aproximado, en terminos del error y del coeficiente de

difusion.

Como problema anexo, se estudio tambien en esta tesis las propiedades de controlabilidad

aproximada de la ecuacion de Fokker-Planck frente al efecto del transporte.

3 Segundo Problema

3.1 Introduccion: Problemas inversos y desigualdades de Carleman.

El uso de desigualdades de Carleman para resolver problemas inversos en EDP’s fue

introducido por Bukhgeim y Klibanov [14], (ver tambien [13]). Este metodo permite resolver

problemas inversos que consisten en determinar elementos de una EDP por medio de una

observacion de la solucion en algun subconjunto del dominio o su frontera. Usualmente se

obtiene estabilidad Lipschitz local alrededor de una solucion regular, (ver [42]).

Existen numerosos trabajos que utilizan estos metodos para resolver el problema inverso

de determinar el potencial de una ecuacion de ondas por medio de observaciones en la frontera

del dominio. Entre ellos estan [56] y [66], donde se consideran condiciones de borde Dirichlet

y observaciones Neumann, y [38], donde se estudian condiciones Neumann y observaciones

Dirichlet.

El caso de determinar el coeficiente principal (que se asume variable, pero regular) ha

sido abordado tambien por medio de las desigualdades de Carleman. En [39] y [37], donde se

estudia en caso isotropico, y se usan observaciones interiores. En [46] y [5] se estudia el caso

anisotropico por medio de observaciones en la frontera.

5

INTRODUCCION

Nosotros estamos interesados en el caso de determinar el potencial por medio de la obser-

vacion de la derivada normal en la frontera, en una ecuacion de ondas con coeficiente principal

discontinuo, caso que, a nuestro conocimiento, no ha sido estudiado previamente.

Cabe mencionar que en los trabajos mencionados (y en el caso que nos interesa) para

resolver el problema inverso planteado se dispone de una sola observacion. Existe tambien

el enfoque que consiste en determinar un elemento de una EDP, por medio de multiples

observaciones. Es el caso de la teorıa matematica de la Tomografıa de Impedancia Electrica

(EIT), usada sobre todo en reconstruccion de imagenes medicas. Una buena referencia sobre

este tema es [65]. Para la aplicacion de desigualdades de Carleman en este enfoque vease [15].

Trabajos que estudian problemas inversos en sistemas de transmision por medio de multiples

observaciones son por ejemplo [57, 32, 58].

3.2 Planteamiento del segundo problema.

Como hemos mencionado, estamos interesados en el problema

utt − div(a(x)∇u) + p(x)u = g(x, t), (2)

donde a(x) es constante en cada uno de los dominios encajados Ω1 y Ω2.

Nuestro enfoque consiste en trabajar con una formulacion equivalente a la ecuacion (2), el

sistema de transmision, que consiste en un sistema de dos ecuaciones de ondas (con coeficiente

principal constante a1 y a2) acopladas por una condicion de transmision (que involucra las

trazas de la soluciones y de su derivadas normales en la interfaz Γ1 = ∂Ω1). Desarrollamos

entonces una desigualdad de Carleman para cada ecuacion por separado, y despues reunimos

los terminos para obtener una desigualdad para el sistema completo.

La principal dificultad proviene entonces del hecho que las soluciones de las ecuaciones to-

man valores no nulos en la frontera comun Γ1, que afectan el producto escalar en la obtencion

de la desigualdad de Carleman. Para controlar estos terminos, se tiene la informacion de la

condicion de transmision, pero la dificultad surge de la derivada tangencial de las soluciones

(de la cual no se tiene informacion). La funcion peso que usemos debe tener cuenta de esta

6

INTRODUCCION

dificultad.

La funcion φ que proponemos es de la forma c(x)|x−x0|2 −βt2 +M , con x0 ∈ Ω1, donde

c(x) es tal que φ cumple las siguientes condiciones:

1. Cumple las condiciones de transmision.

2. Es constante en la interfaz.

3. Cada una de las restricciones de φ a los dominios Ω1 y Ω2 pueda ser usada como funcion

de peso para una desigualdad de Carleman en la ecuacion correspondiente.

Por una parte, demostraremos que bajo la hipotesis a2 < a1, la propiedad 2. implica

que podemos controlar las trazas en la interfaz. Por otra parte, se mostrara que bajo cierta

hipotesis de convexidad del dominio interior, se cumple la condicion 3.

Demostramos entonces, una desigualdad de Carleman con funcion de peso φ. Por ultimo,

siguiendo el metodo de Bukhgeim-Klibanov resolvemos, por medio de nuestra desigualdad de

Carleman, el problema inverso consistente en determinar el potencial por medio de la traza

en el borde de la derivada normal de la solucion.

4 Resultados principales.

A continuacion, enunciaremos los resultados que obtenemos para cada uno de los proble-

mas que hemos planteado en las secciones anteriores.

4.1 Una desigualdad inversa en transporte-difusion.

En esta parte de la tesis, comenzamos por establecer las definiciones de los distintos

tipos de controlabilidad de una EDP en evolucion del tipo ut + Au = 0, con A igual a un

operador diferencial parcial, concentrandonos en la controlabilidad interior. Establecemos

las caracterizaciones duales de cada tipo de control, presentando una demostracion de tales

caracterizaciones.

En la segunda seccion, estudiamos las distintas propiedades de controlabilidad de la ecua-

cion de transporte puro, dada por

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INTRODUCCION

yt + v · ∇y = 0 Ω × (0, T )

y = 0 Γ− × (0, T )

y(0) = y0 Ω

(3)

donde Ω ⊂ Rn es un abierto con frontera de clase C1, v ∈ C1

b (Ω) con div(v) = 0, y ademas:

Γ− = x ∈ Γ : v · ν(x) < 0, y

Γ+ = x ∈ Γ : v · ν(x) > 0.

Definimos las curvas caracterısticas de la equacion (3), dadas por la funcion X(x, t) defi-

nida por

d

dtX(t, x) = v(X(t, x)), t ∈ R

X(0, x) = x

para todo x ∈ Rn.

En terminos del comportamiento de estas trayectorias, damos condiciones para tener los

distintos tipos de control para la ecuacion (3).

En la seccion 3, estudiamos la ecuacion de transporte-difusion dada por

yεt + v · ∇yε − ε∆yε = 0 in Ω × (0, T ),

yε = 0 on Γ × (0, T ),

yε(0) = y0 in Ω.

(4)

donde ε > 0 es un parametro dado (que eventualmente tendera a cero).

Dado A ⊂ Ω definimos el conjunto de trayectorias que parten de A como:

Ψv,T (A) = X(t, x) : x ∈ A, X(s, x) ∈ Ω, ∀ 0 ≤ s ≤ t ≤ T (5)

Consideramos el caso en el que la condicion inicial y0 esta soportada en el subconjunto

8

INTRODUCCION

abierto Ω0, el cual asumiremos que cumple la siguiente propiedad:

v(x) · ν(x) 6= 0 ∀x ∈ Ψv,T (Ω0) ∩ Γ (6)

Siguiendo ideas de [49], y gracias a algunas propiedades de la ecuacion de transporte,

probamos un resultado de convergencia de las soluciones de (4) hacia las de (3), siempre que

la condicion inicial sea regular, y que se cumpla (6).

Proposicion 1. Existe C > 0 tal que

|yε − y|L2(Ω×(0,T )) ≤ Cε1/2|y0|H1(Ω0)

para todo y0 ∈ H10 (Ω0), ε ∈ (0, 1), donde yε, y son las soluciones de (4), (3) respectivamente,

con yε(0) = y(0) = y01Ω0 .

Entonces, gracias a esta convergencia y a propiedades de observabilidad demostradas para

la ecuacion del transporte, se prueba la siguiente desigualdad inversa para la ecuacion (4):

Teorema 1. Supongase que Ω0 ⊂⊂ Ψ−v,T (ω). Entonces, existen C > 0, ε0 > 0 tales que

|y0|L2(Ω0) ≤ C(|yε|L2(ω×(0,T )) + ε1/2|y0|H1(Ω0)) (7)

para todo y0 ∈ H10 (Ω0), para todo ε < ε0.

Por ultimo, aplicamos la desigualdad (7) a la ecuacion adjunta de (4), y probamos que por

dualidad nos proporciona una estimacion del costo del control aproximado de la restriccion

al conjunto Ω0 de las soluciones de (4), con control soportado en ω × (0, T ) actuando en el

lado derecho.

De manera precisa, si definimos

Definicion 1. Para cada ε, γ > 0 y f ∈ L2(ω), denotamos yε a la solucion de (4) con lado

derecho igual al control v ∈ L2(ω × (0, T )). Definimos

Cλε (f, γ) = ınf|v|L2(ω×(0,T )) : |qε(T ) − f |H−1(Ω0) ≤ γ.

9

INTRODUCCION

Entonces, hemos probado el

Teorema 2. Existe C > 0 tal que para cada f ∈ L2(ω) y γ > 0 existe ε0 > 0 tal que

Cε(f, γ) ≤ C|f |L2(Ω0) ∀ε ≤ ε0.

4.2 Desigualdades de Carleman: un problema de transmision de ondas.

Sea n ≥ 2, y sean Ω y Ω1 dos subconjuntos abiertos y acotados de Rn con fronteras Γ y

Γ1 regulares, tales que Ω1 es simplemente conexo y Ω1 ⊂ Ω. Sea Ω2 = Ω \Ω1, de manera que

∂Ω2 = Γ ∪ Γ1. Definimos:

a(x) =

a1 x ∈ Ω1

a2 x ∈ Ω2

con aj > 0 para j = 1, 2.

En esta parte de la tesis, estudiamos la ecuacion dada por

utt − div(a(x)∇u) + p(x)u = g(x, t) Ω × (0, T )

u = h Γ × (0, T )

u(0) = u0 Ω

ut(0) = u1 Ω.

(8)

con p ∈ L∞(Ω), g ∈ L1(0, T ;L2(Ω)), h ∈ L2(0, T ;L2(Γ)), u0 ∈ H10 (Ω) y u1 ∈ L2(Ω).

El enfoque que usamos en el trabajo es considerar la ecuacion (8) como dos ecuaciones

de ondas con coeficientes constantes, junto con la condicion de transmision

u1 = u2 on Σ1

a1∂u1

∂ν1+ a2

∂u2

∂ν2= 0 on Σ1.

(9)

Para establecer las hipotesis que usaremos, consideremos la

Definicion 2. Sea U ⊂ Rn un conjunto abierto, acotado y convexo cuya frontera ∂U es una

hipersuperficie de clase C2. Diremos que U is fuertemente convexo si para cada plano

10

INTRODUCCION

Π ⊂ Rn que intersecta U , la frontera del convexo (2-dimensional) U ∩ Π posee curvatura

estrictamente positiva en cada punto.

En esta parte de la tesis, se considera la hipotesis de que Ω1 es fuertemente convexo.

Siguiendo la motivacion planteada arriba, construimos la funcion de peso como sigue:

Sea x0 ∈ Ω1, y por cada x ∈ Ω \ x0 definamos ℓ(x0, x) = x0 + λ(x − x0) : λ ≥ 0.Como Ω1 is convexo y acotado existe un unico punto y(x) tal que

y(x) ∈ Γ1 ∩ ℓ(x0, x). (10)

Definimos ρ : Ω \ x0 −→ R+ por

ρ(x) = |x0 − y(x)|. (11)

Ahora, tomamos una funcion cut-off η ∈ C∞(R) tal que toma el valor de cero en una

vecindad cerrada de x0, y es constante igual a 1 fuera de una vecindad de x0.

Para cada j, k tales que j, k = 1, 2 definimos

φj(x, t) = η(x)ak

ρ(x)2|x− x0|2 − βt2 +Mj (x, t) ∈ Ω × R, (12)

donde β, M1 y M2 son numeros positivos que escogeremos posteriormente.

Entonces, nuestra funcion peso esta dada por:

φ(x, t) =

φ1(x, t) (x, t) ∈ Ω1 × R

φ2(x, t) (x, t) ∈ Ω2 × R.

(13)

Denotamos

L = ∂2t − a∆ y E(z) = |zt|2 − a|∇z|2.

Como es usual en las desigualdades de Carleman, hacemos el cambio de variables

ϕ = eλφ , λ > 0, w = esϕu , s > 0, P (w) = esϕL(e−sϕw) (14)

11

INTRODUCCION

y despues de algunas manipulaciones algebraicas, descomponemos P (w) como sigue:

P (w) = P1(w) + P2(w) +R(w),

donde se tiene que (γ es algun numero en (0, 1) a fijar posteriormente).

P1(w) = wtt − a∆w + s2λ2ϕ2E(φ)w,

P2(w) = (γ − 1)sλϕL(φ)w − sλ2ϕE(φ)w − 2sλϕ(φtwt − a∇φ · ∇w),

R(w) = −γsλϕL(φ)w.

De ser necesario escribiremos P φ, P φ1 , P φ

2 , etc. para hacer la dependencia en φ explıcita.

Dado el abierto U ⊂ Rn y T > 0, definimos la norma in H1(U × (−T, T )) dada por

‖g‖2U,ϕ

= sλ

∫ T

−T

∫

U(|gt|2 + |∇g|2)ϕ+ s3λ3

∫ T

−T

∫

U|g|2ϕ3, (15)

y el espacio

X = u ∈ L2(−T, T ;L2(Ω)) | Luj ∈ L2(−T, T ;L2(Ωj)), j = 1, 2; u|Σ = 0,

u(±T ) = ut(±T ) = 0, y u satisface (9).

Finalmente, definimos

Σφ+ = (x, t) ∈ Σ : ∇φ(x, t) · ν(x) > 0. (16)

Hacemos notar que un punto importante de nuestra funcion peso es que su gradiente

se anula en x0, por lo que sera necesario usar no una, sino dos funciones peso y usarlas

combinadamente.

Sean pues xk ∈ Ω1, k = 1, 2 y sea φk, ϕk, wk la correspondiente funcion definida como

antes, ahora para xk.

Se prueba la siguiente desigualdad de Carleman

Teorema 3. Sea Ω ⊂ Rn abierto y acotado. Asumamos que Ω1 es fuertemente convexo, de

12

INTRODUCCION

clase C3, y a1 > a2 > 0. Entonces existen C > 0, s0 > 0, λ0 > 0 tales que

2∑

k=1

(∥∥∥P φk

1 (wk)∥∥∥

2

L2(Q)+∥∥∥P φk

2 (wk)∥∥∥

2

L2(Q)+∥∥∥wk

∥∥∥2

Q,ϕk

)

≤ C

2∑

k=1

(∫∫

Qe2sϕk |Lpu|2 + sλ

∫∫

Σφk

+

ϕk

∣∣∣∣a2∂wk

∂ν

∣∣∣∣2) (17)

para todo u ∈ X, λ ≥ λ0, s ≥ s0.

En la ultima seccion, estudiamos el problema inverso que consiste en determinar el po-

tencial p de la ecuacion (8), conociendo la derivada normal de la solucion u(p).

Mas precisamente, probaremos las siguientes propiedades:

Unicidad :

La igualdad ∂u(q)∂ν = ∂u(p)

∂ν en Γ × (0, T ) implica q = p en Ω.

Estabilidad :

Se prueba una estimacion del tipo

‖q − p‖X(Ω) ≤∥∥∥∥∂u(q)

∂ν− ∂u(p)

∂ν

∥∥∥∥Y (Γ×(0,T ))

para espacios adecuados X e Y .

Para establecer de manera precisa nuestro resultado acerca de estas preguntas, definimos,

para x1, x2 ∈ Ω1, las cantidades dadas por

αj = d(xj ,Γ1), Rj = sup|x− yj(x)| : x ∈ Ω2 (18)

para cada j = 1, 2, donde yj se define como en (10) con respecto a xj .

Tambien, hacemos

D0 = max

R1 + α1

α1,R2 + α2

α2

. (19)

Entonces, como aplicacion de la desigualdad de Carleman (3), probamos el siguiente

13

INTRODUCCION

resultado de estabilidad de nuestro problema inverso:

Teorema 4. Sea Ω acotado, Ω1 ⊂⊂ Ω fuertemente convexo con borde Γ1 de clase C3, y sean

a1 > a2 > 0. Si T > 0, p ∈ L∞(Ω), u0 ∈ H10 (Ω), u1 ∈ L2(Ω), r > 0 satisfacen

T > D0

√a1β

|u0(x)| ≥ r > 0 c.t.p. en Ω, y

u(p) ∈ H1(0, T ;L∞(Ω))

Entonces, dado U ⊂ L∞(Ω) acotado, existe una constante

C = C(a1, a2,Ω1,Ω2, T, ‖p‖L∞(Ω), ‖u(p)‖H1(L∞),U , r) > 0 tal que:

‖p − q‖L2(Ω) ≤ C

∥∥∥∥a2∂u(p)

∂ν− a2

∂u(q)

∂ν

∥∥∥∥H1(0,T ;L2(Γ))

para todo q ∈ U , donde u(p), u(q) son las soluciones de (8) con potencial p, q respectivamente.

14

CAPITULO I

Transport-difussion equation.

15

CAPITULO I. TRANSPORT-DIFUSSION EQUATION.

The results exposed in the third section of this chapter are contained in the paper An inverse inequality for some

transport-diffusion equation. Application to the regional approximate controllability submitted to Asymptotic Analysis,

in collaboration with Sergio Guerrero and Axel Osses.

1 Introduction and contents.

In this chapter we study the controllability properties of some linear transport-diffusion

equation when the diffusion is small compared with the effect of the transport.

In the first section we introduce the basic concepts on controllability of evolution equa-

tions, and we prove in detail the standard equivalences between the different properties of

controllability of a given equation and the corresponding observabiliy properties for the ad-

joint equation.

In the second section, we study the controllabily properties of the transport equation. We

find conditions for the velocity, the time and the control zone for each control property.

In the third section, our objetive is the study of the controllability properties of a transport

diffusion equation, wen the diffusion coefficient goes to zero. We are interested in obtain some

information from the observability properties of the transport when the effect of the diffusion

decreases. The usual method for proving the observability estimates for the parabolic equation

(the Carlman estimates) do not take into account the efect of the transport. Our approach is

elementary: we analyze the convergence of the transport-diffusion solutions to the transport

solutions when the diffusion goes to zero. We prove the uniformity on this convergence if the

initial condition has H1-regularity and their support is appropriately localized, and we show

that this convergence together with the observability inequality for the transport equation

-studied on the previous section- imply our two main results of this chapter of the thesis:

on one hand, we get an inverse inequality for the transport diffusion equation with right-

hand side, and in the other hand, applying this result to the adjoint equation and passing

by duality, we obtain that the H−1-norm of the regional approximate control for the original

transport diffusion equation (for any L2-function as target) remains bounded when the error

goes to zero if also the diffusion goes to zero.

16


In the last section we state some controllability properties of another equation wherein are

present both effects, the diffusion and the transport: Fokker-Planck equation. However, the

study is faraway to be complete. We mainly deduce some results for the transport equation

analogous to the these of the second section, and we prove the approximate controllability

for the Fokker-Planck equation.

2 Basic definitions.

Let Ω ⊂ Rn be an open bounded set, and T > 0. We consider the evolution system given

by

ut + Lu = f Ω × (0, T )

u(0) = u0 Ω(I.1)

where L : D(L) ⊂ L2(Ω) −→ L2(Ω) is a given partial differential operator.

We assume this problem is well posed: for each u0 ∈ L2(Ω) and f ∈ L2((0, T );L2(Ω))

there exist a unique weak solution u ∈ C([0, T ];L2(Ω)) of (I.1), which depends continuously

on u0 and f .

Given an open set ω ⊂⊂ Ω, here we are interested in the interior control of system

(I.1). That means that given the data u0 ∈ L2(Ω) and f ∈ L2((0, T );L2(Ω)), we want to find

a control h ∈ L2(ω × (0, T )) such that the solution of

ut + Lu = f + h1ω Ω × (0, T )

u(0) = u0 Ω(I.2)

will have a prescribed behavior.

More precisely, given T > 0, we define the set of reached targets of problem (I.1) in

time T as:

RT (u0, f) = u(T ) : h ∈ L2(ω × (0, T )) and u is the solution of (I.2).

Then we will say that:

17


Definition 1. Given a time T > 0, the system (I.2) is:

Approximately controllable if RT (u0, f) = L2(Ω) for all (u0, f) ∈ L2(Ω) ×L2(Ω × (0, T )).

Exactly controllable if RT (u0, f) = L2(Ω) for all (u0, f) ∈ L2(Ω)×L2(Ω×(0, T )).

Null controllable if 0 ∈ RT (u0, f) for all (u0, f) ∈ L2(Ω) × L2(Ω × (0, T )).

If u = u(f, u0) is the solution of the uncontrolled equation (I.1), thanks to the linearity

of the problem, we get:

RT (u0, f) = RT (0, 0) + u(T ) (I.3)

Hence we deduce directly the following characterization:

Lemma 1. Problem (I.2) is:

Approximately controllable iff RT (0, 0) = L2(Ω).

Exactly controllable iff RT (0, 0) = L2(Ω).

Null controllable iff u(T ) ∈ RT (0, 0) ∀(u0, f) ∈ L2(Ω) × L2(Ω × (0, T )).

Remark 1. Lemma 1 in particular means that, in linear systems, null-controllability is equi-

valent to controllability to the trajectories property.

Now, depending on the concrete equations we study, usually there is possible to control

to zero the system, but only for some data. This motivates the next definition.

Definition 2. Given T > 0, and a subspaces Z ⊂ L2(Ω) and X ⊂ L2(Ω) × L2(Ω × (0, T )),

the system (I.2) is:

Exactly controllable to Z if RT (0, 0) = Z.

Null controllable in X if u(T ) ∈ RT (0, 0) ∀ (u0, f) ∈ X.

18


Another particular case corresponds to controlling the restriction of u(T ) to some given

open subset Ω0 ⊂ Ω. We will say that problem (I.2) is regional exactly (approximately)

controllable in Ω0 if

RT |Ω0 = u(T )|Ω0 : h ∈ L2(ω × (0, T )) and u is the solution of (I.2)

is (dense in) the space L2(Ω0).

2.1 Dual characterizations

In this part of the work, we develop the classic dual characterizations controllability /

observability for the evolution system (I.1) which are very useful for proving that a given

system is controllable.

For each h ∈ L2(ω × (0, T )), consider the system

ut + Lu = h1ω Ω × (0, T )

u(0) = 0 Ω(I.4)

and define the operator:

A : L2(ω × (0, T )) → L2(Ω)

h 7→ u(T )(I.5)

where u is the solution of problem (I.4).

Then, we have that RT (0, 0) = R(A), and the (null or exact) control properties of equa-

tion (I.4) correspond exactly with the onto-ness properties of operator A. Moreover, we can

reformulate this properties by means of the adjoint operator of A.

Consider the backward problem given by:

−ϕt + L∗ϕ = 0 Ω × (0, T )

ϕ(T ) = ϕT Ω.(I.6)

where L∗ is the formal adjoint operator of L∗. We suppose problem (I.6) has a unique weak

19


solution ϕ ∈ C([0, T ];L2(Ω)) for each ϕT ∈ L2(Ω), with continuous dependence.

Multiplying equation (I.4) by ϕ and integrating by parts we get:

∫

Ωu(T )ϕT =

∫ T

0

∫

ωhϕ (I.7)

for all h ∈ L2(ω × (0, T )), ϕT ∈ L2(Ω).

Thus, from (I.7) we have that the adjoint operator of A is given by:

A∗ : L2(Ω) → L2(Ω)

ϕT 7→ ϕ|ω×(0,T )

(I.8)

where ϕ is the solution of equation (I.6) with ϕ(T ) = ϕT .

Using basics facts in Functional Analysis concerning the dual relations between operators

(see [12] for example), we get the following:

Proposition 1. Problem (I.2) is

Approximately controllable iff for all ϕT ∈ L2(Ω), the solution of (I.6) satisfies:

ϕ = 0 in ω × (0, T ) =⇒ ϕ ≡ 0 (I.9)

Exactly controllable iff:

∃C > 0

∫

Ω|ϕT |2 ≤ C

∫ T

0

∫

ω|ϕ|2 ∀ϕT ∈ L2(Ω) (I.10)

Let us mention that (I.9) is called a unique continuation principle, and (I.10) is called

observability inequality.

20


Weighted spaces.

Now, we introduce a class of subspaces of L2(Ω) which will be very useful in our study of

controllable data. Given a positive function ρ : Ω −→ R such that

0 < δ ≤ ρ <∞

we define the weighted L2-space by

X(ρ) = L2(Ω, ρdµ). (I.11)

We state at the following Lemma some facts about these spaces, whose proof follows direct

from the definition.

Lemma 2. If X(ρ) is defined as above, then:

1. X(ρ) ⊂ L2(Ω) with continuous and dense injection.

2. With the usual identification of L2 with its dual, we have that X(ρ)′ = X(ρ−1), with

X(ρ) ⊂ L2(Ω) ⊂ X(ρ−1).

3. In particular X(1) = L2(Ω).

We state the analogous result of Proposition 1 for weighted spaces:

Proposition 2. Problem (I.2) is Exactly controllable to the space X(ρ) if and only if

there exists C such that:

∫

Ωρ−1|ϕT |2 ≤ C

∫ T

0

∫

ω|ϕ|2 ∀ϕT (I.12)

Completely analogous with all the previous characterizations, concerning the regional

control, we have the following

Proposition 3. Problem (I.2) is

21


Approximately controllable in Ω0 iff for all ϕT ∈ L2(Ω0), the solution of (I.6)

satisfies:

ϕ = 0 in ω × (0, T ) =⇒ ϕ ≡ 0 (I.13)

Exactly controllable in Ω0 iff:

∃C > 0

∫

Ω0

|ϕT |2 ≤ C

∫ T

0

∫

ω|ϕ|2 ∀ϕT ∈ L2(Ω0) (I.14)

Null-controllability

In order to find a dual characterization for the null control, (I.3) is not really very helpful,

and we have to do a little extra work. Multiplying equation (I.2) by ϕ and integrating in

Ω × (0, T ) we get:

∫

Ωu(T )ϕT =

∫

Ωu0ϕ(0) +

∫ T

0

∫

Ωfϕ+

∫ T

0

∫

ωhϕ ∀u0, ϕT (I.15)

And then h ∈ L2(ω × (0, T )) is a null control for (u0, f) if and only if:

∫

Ωu0ϕ(0) +

∫ T

0

∫

Ωfϕ+

∫ T

0

∫

ωhϕ = 0 (I.16)

for all ϕT ∈ L2(Ω).

Firstly, we shall prove that inequality (I.16) give us not only the existence of the null-

control, but also the boundness of the operator that sends each pair (u0, f) to their null

control (of minimal norm).

Define the space

H0 = ϕ|ω×(0,T ) : ϕT ∈ L2(Ω), ϕ is solution of (I.6) (I.17)

and also

H = H0. (I.18)

22


where the closure is taken in the space L2(ω × (0, T )). Then H is a closed subespace of

L2(ω × (0, T )), and thus a Hilbert space by itself.

As we will see in the next result, the natural space where the null-control operator takes

its values is precisely H.

Proposition 4. Let X be a Banach space which is continuously injected in L2(Ω)×L2(Ω ×(0, T )). Suppose that (I.16) is valid for all (u0, f) ∈ X. Then there exists a a lineal bounded

operator

B : X −→ H

such that B(u0, f) is the minimal-norm control of (u0, f) ∈ X.

Proof: For each (u0, f) ∈ X, let Uad(u0, f) ⊂ L2(ω × (0, T )) be the set of null controls for

(u0, f).

If P : L2(ω × (0, T )) −→ H is the orthogonal projection, from (I.16) we can see that for

each pair (u0, f) ∈ X we have:

(i) P (h) ∈ Uad(u0, f) for all h ∈ Uad(u0, f),

(ii) P (h1) = P (h2) for all h1, h2 ∈ Uad(u0, f).

Indeed, let be h ∈ Uad(u0, f). Then h satisfies (I.16), and h = P (h) + Q(h), where Q

is the orthogonal projection onto H⊥. Thus, by definition we have

∫ T

0

∫

ωQ(h)ϕ = 0 for

each ϕ ∈ H, and then we conclude that P (h) also satisfies (I.16) and therefore it belongs to

Uad(u0, f). This proves (i).

To prove (ii), take h1, h2 ∈ Uad(u0, f). Then both functions satisfy (I.16), and this implies

that ∫ T

0

∫

ω(h1 − h2)ϕ = 0 ∀ϕT ∈ L2(Ω). (I.19)

By density, from equality (I.19) we have (h1 − h2) ∈ H⊥ and then P (h1 − h2) = 0.

Therefore P (h1) = P (h2) for each pair h1, h2 ∈ Uad(u0, f), and (ii) is proved.

Hence, for each (u0, f) ∈ X the set Uad(u0, f)∩H contains exactly one element. It makes

sense to define B(u0, f) as this element. From the linearity of equation (I.2) and the fact

23


that H is a subspace, we have that B is lineal. From the orthogonal projection properties,

we have also that ‖B(u0, f)‖ = ‖Ph‖ ≤ ‖h‖ for all h ∈ Uad(u0, f), and then B(u0, f) is the

minimal-norm control.

For the continuity of B, we can apply the closed graph theorem. Indeed, let (un, fn, hn) →(u, f, h) in X ×H with hn = B(un, fn). Then h ∈ H and (un, fn) → (u, f) in L2 × L2. On

the other hand, (un, fn, hn) fulfill equality (I.16) for each n, which we can pass to the limit

in n, resulting that (u, f, h) also fulfill (I.16). We deduce that h = B(u, f), and hence B is

bounded.

As we have done in (I.11), if ρ1 : Ω −→ R and ρ2 : Ω× (0, T ) −→ R are minored by some

δ > 0, we define for j = 1, 2 the space

X(ρ1, ρ2) = L2(Ω, ρ1dµ) × L2(Ω × (0, T ), ρ2dµdt).

where µ is the Lebesgue measure in Rn.

We can prove the next characterization for the null-controllability.

Proposition 5. System (I.2) is null-controllable in X(ρ1, ρ2) if and only if

∫

Ωρ−11 |ϕ(0)|2 +

∫ T

0

∫

Ωρ−12 |ϕ|2 ≤ C

∫ T

0

∫

ω|ϕ|2 (I.20)

for all ϕT ∈ L2(Ω), where ϕ is the solution of system (I.6).

Furthermore, the norm of the null-control operator defined in X is given by the minimal

constant C achieving (I.20).

Proof: (⇒) By proposition 4, the null-control operator B : X(ρ1, ρ2) −→ H is well defined

and bounded. Then B∗ is also a bounded operator, and by Lemma 2 we have B∗ : H −→X(ρ−1

1 , ρ−12 ).

Evaluating B∗ in the functions ϕ|ω ∈ H0, from the definition of adjoint operator and

(I.16) we have that B∗(ϕ|ω) = (−ϕ(0),−ϕ) for each ϕT ∈ L2(Ω). Hence (I.20) is true with

constant C = ‖B∗‖ = ‖B‖.

24


(⇐) We define the lineal operator A : H0 −→ X(ρ−11 , ρ−1

2 ) by

A(ϕ|ω) = (ϕ(0), ϕ).

From (I.20), we have that A is well defined and bounded (with ‖A‖ ≤ C). Thus we can

extend it at the whole space H, which is a Hilbert space. Then A∗ : X(ρ1, ρ2) −→ H is also

a bounded operator with

‖A∗‖ = ‖A‖ ≤ C. (I.21)

By definition of adjoint operator we have that, for each (u0, f) ∈ X(ρ1, ρ2), h = −A∗(u0, f)

achieve (I.16) and then is their null control.

Finally, it is clear that the minimal C attaining (I.20) is exactly ‖A‖, and thus is equal

to the null-control norm.

In particular, in the previous results is included the situation related to controlling an

equation with right hand side equal to zero, (take ρ2 = ∞ in the previous proposition). Hence

we have

Corollary 1. System (I.2) (with f = 0) is null-controllable in L2(Ω) if and only if there

exists C > 0 such that ∫

Ω|ϕ(0)|2 ≤ C

∫ T

0

∫

ω|ϕ|2

for all ϕT ∈ L2(Ω), where ϕ is the solution of (I.6) with ϕ(T ) = ϕT .

3 Transport equation.

Given an open set Ω ⊂ Rn with C1 boundary, let v ∈ C1

b (Ω) with div(v) = 0 and define

Γ− = x ∈ Γ : v · ν(x) < 0

Γ+ = x ∈ Γ : v · ν(x) > 0

25


Consider the transport equation given by:

ut + v · ∇u = f Ω × (0, T )

u = 0 Γ− × (0, T )

u(0) = u0 Ω

(I.22)

It is a known result that this problem is well posed in L2(Ω), (see [2], [22]):

Proposition 6. For each u0 ∈ L2(Ω) and f ∈ L2((0, T );L2(Ω)), there exists a unique weak

solution u ∈ C([0, T ], L2(Ω)) of problem I.22. There exists a constant C such that

|u|C([0,T ],L2(Ω)) ≤ C(|u0|L2(Ω) + |f |L2(Q)

)

In fact, we can write explicitly the solution of system (I.22) (in terms of u0 and f), with

the so-called method of characteristics, which is based on the existence of the trajectories

or characteristic curves defined in Ω by the vector field v. First of all, we assume that there

exists an extension of v to the whole Rn, which will be also called v. Given T > 0, we set:

Definition 3. • We denote by X(t, x) the trajectories defined by v in Rn. This means X is

such that:

d

dtX(t, x) = v(X(t, x)), t ∈ R

X(0, x) = x

for all x ∈ Rn.

• For each x ∈ Ω, we denote by (τ−(x), τ+(x)) the connected component of the set t ∈R : X(t, x) ∈ Ω ⊂ R which contains 0.

• For each A ⊂ Ω we denote:

Ψv,T (A) = X(t, x) : x ∈ A, and 0 ≤ t < mınτ+(x), T.

• Given the subsets U and O of Ω, we say that

U ⊂v,T O (I.23)

26


if

U ⊂⊂ Ψv,T (O).

The solution of system (I.22) is given by

u(x, t) =

u0(X(−t, x)) if 0 ≤ t < −τ−(x)

0 if t ≥ −τ−(x)

+

∫ t

max0,t+τ−(x)f(X(−(t− s), x)s)ds (I.24)

Remark 2. Property (I.23) means that the set U is contained in the union of all the trajec-

tories defined by v starting in O. Thanks to the continuity X(x, t), we have that U ⊂v,T O if

and only if there exists δ0 > 0 such that for any x ∈ U , we have |O(x)| ≥ δ0 > 0, where | · |designs the Lebesgue measure, and O(x) is defined as follows

O(x) = t : maxτ−(x),−T < t ≤ 0 , X(t, x) ∈ O.

Remark 3. The existence of the characteristic curves X(t, x) is guaranteed even if the vector

field v is only Lipschitz continuous. In this work we take v ∈ C1b (Ω) and ∂Ω of class C1 in

order to assure that the transport equation is well posed. Indeed, by the Sard’s Theorem, the

set of points belonging to some characteristic curve passing by Γ0 = x ∈ Γ : ν(x) ·v(x) = 0has zero measure (see [2]).

Extra-regularity of the transport equation.

Next we prove some extra regularity result for the solutions of the transport equation

which will be very useful later (see Lemma 4 below). In the next Proposition we show that

the solution of (I.22) has finite trace in the space L2(Σ+, |v · ν|dS) (see [20] for more general

results):

27


Proposition 7. Let u0 ∈ L2(Ω) and f ∈ L2(Ω× (0, T )). Then the solution of (I.22) satisfies

u ∈ L2(Σ+, |v · ν|dS) and there exists a constant C > 0 such that

|u|L2(Σ+,|v·ν|dS) ≤ C(|u0|L2(Ω) + |f |L2(Ω×(0,T ))

).

Proof: Multiplying the equation in (I.22) by u, integrating in Ω × (0, T ) and integrating by

parts (recall that div(v) = 0), it is not difficult to see that

∫

Ω|u(T )|2dx+

∫∫

Σ+

|u|2|v · ν|dσ dt =

∫

Ω|u0|2dx+ 2

∫ T

0

∫

Ωfu dx dt. (I.25)

Together with Proposition 6, (I.25) implies the desired result.

3.1 Controllability properties.

We are interested in the problem of interior control of the transport equation:

ut + v · ∇u = h1ω×(0,T ) Ω × (0, T )

u = 0 Γ− × (0, T )

u(0) = u0 Ω

(I.26)

with initial data u0 ∈ L2(Ω) and control h ∈ L2(ω × (0, T )).

In order to apply the characterizations developed on section 2.1, we consider the adjoint

equation of (I.26), given by:

−ϕt − v · ∇ϕ = 0 Ω × (0, T )

ϕ = 0 Γ+ × (0, T )

ϕ(T ) = ϕT Ω

(I.27)

where ϕT ∈ L2(Ω).

Under the change of variable given by t 7→ T − t, system (I.27) is a transport equation

like (I.22) with vector field of velocity −v.In order to get the unique continuation principle (I.9), all the trajectories defined by v

28


and starting in ω must fill out all the open set Ω after time T . Indeed, assume there exists

some nonempty open set U ⊂ Ω which is not intersected by any characteristic curve coming

from ω (before time T ), and take ϕT = 1U . Then the non-zero solution ϕ of equation (I.27)

has null trace in the set ω × (0, T ). Therefore, we have prove the

Proposition 8. System (I.26) is approximately controllable if and only if

Ψv,T (ω) = Ω a.e. (I.28)

Exact controllability for the transport equation.

It is clear that the geometric property (I.28) is a necessary condition for the exact con-

trollability (due to the fact that exact controllability obviously implies approximate contro-

llability). However, one can see with simple examples that it is not sufficient:

Example 1. An one-dimensional example where (I.28) is satisfied, but the transport equation

is not exactly controllable.

Let Ω = (0, 10), ω = (0, 2), v = 1, and T = 9.

Then it is clear that (I.28) is satisfied.

However, if we take

ϕnT = n1(0, 1

n)

then the solution of equation (I.27) is given by

ϕn = n1Dn

where Dn = (x, t) : x+ (9 − t) < 1/n, 0 ≤ x ≤ 10, 0 ≤ t ≤ 9.Hence we have:

∫

Ω|ϕn

T |2 = n,

∫ T

0

∫

ω|ϕn|2 = n2|Dn| =

1

2∀n

and then an inequality like (I.10) is not possible, and therefore the transport equation with

29


these parameters is not exactly controllable.

Although, if condition (I.28) holds, we can define a weight function in the following way.

Consider the equation:

yt + v · ∇y = 1ω Ω × (0, T )

y = 0 Γ− × (0, T )

y(0) = 0 Ω

(I.29)

If (I.28) holds, then we have y(T ) > 0 a.e. in Ω. Multiplying (I.27) by yϕ and integrating

by parts one obtains:

∫ T

0

∫

Ωϕy(ϕt + v · ∇ϕ) −

∫

Ω(y(T )ϕ2(T ) − y(0)ϕ2(T ))

−∫ T

0

∫

Γ(v · ν)yϕ+

∫ T

0

∫

Ωϕ2(yt + v · ∇y) = 0

and then:

∫

Ωy(T )ϕ2

T =

∫ T

0

∫

ωϕ2 (I.30)

From Proposition 2, equality (I.30) directly implies:

Proposition 9. Let ρ = y(T ), where y is the solution of equation (I.29). Then the space of

reached targets of equation (I.26) is L2ρ−1(Ω).

In other words, if∫Ω ρ

−1|uT |2 < ∞, then there exists a control h ∈ L2(ω × (0, T )) such

that the solution of (I.22) (with u0 = 0) satisfies u(T ) = uT

Example 2. In the example 1, if ρ = y(T ) is the function defined above, we have

ρ(x) =

x 0 ≤ x ≤ 2

2 2 ≤ x < 10.

(I.31)

Consequently, from Proposition 9 we have that the space of reached functions is (exactly)

L2((0, 10), x−1dµ).

30


In fact, is not difficult to show that, given a smooth open set Ω ⊂ Rn and v ∈ C1

b (Ω), if the

control region ω ⊂ Ω satisfies (I.28), then uT is an exactly controllable function in equation

(I.26) if and only if ∫

Ωρ−1|uT |dµ <∞

where ρ(x) = dist(x,Γ−) = ınf|x− y| : y ∈ Γ−.

On the other hand, even if the geometric condition (I.28) is not satisfied, if K ⊂ Ω is any

compact set such that K ⊂v,T ω (recall definition (I.23)) and y is defined by equation (I.29),

is not difficult to prove that

y(T ) ≥ δ > 0 in K

for some δ > 0.

Hence, we get:

Lemma 3. If Ω0 ⊂ Ω is a given open set such that Ω0 ⊂v,T ω, then exists C such that

∫

Ω0

ϕ2T ≤ C

∫ T

0

∫

ωϕ2 (I.32)

for all ϕT ∈ L2(Ω0), where ϕ is the solution of equation (I.27).

This Lemma together with Proposition 3 directly imply the following result about the

regional control:

Proposition 10. If Ω0 ⊂ Ω is a given open set such that Ω0 ⊂v,T ω, for each uT ∈ L2(Ω0)

there exists a control h ∈ L2(ω × (0, T )) such that the solution of equation (I.26) satisfies

u(T )|Ω0 = yT .

31


Null-Controllability of the transport equation.

Now we introduce a function which will be used as a weight function for the null-

controllability. Consider the following equation given by

−φt − v · ∇φ = 0 Ω × (0, T )

φ = 1 Γ+ × (0, T )

φ(T ) = y(T ) Ω

(I.33)

where y is the solution of equation (I.29). Then φ(0) ∈ L2(Ω) only depends on the geometry

of Ω and ω.

Multiplying (I.27) by ϕφ and integrating by parts we have:

∫ T

0

∫

Ωϕφ(ϕt + v · ∇ϕ) −

∫

Ω(φ(T )ϕ2(T ) − φ(0)ϕ2(T ))

−∫ T

0

∫

Γ(v · ν)ϕ2φ+

∫ T

0

∫

Ωϕ2(φt + v · ∇φ) = 0

and thus

∫

Ωφ(0)ϕ2(0) =

∫

Ωy(T )ϕ2

T +

∫ T

0

∫

Γ−

(v · ν)φϕ2

=

∫ T

0

∫

ωϕ2 +

∫ T

0

∫

Γ−

(v · ν)φϕ2 (I.34)

≤∫ T

0

∫

ωϕ2

In order to get the result on null-control given by inequality (I.34) via Proposition 5, we

have to find conditions to have φ(0) > 0 through Ω. By definition of equations (I.29) and

(I.33), it is not so difficult to see that a sufficient condition to have φ(0) ≥ δ > 0 on Ω is that

Ω ⊂ Ψ−v,T (ω) ∪ Ψ−v,T (Γ+) (I.35)

Therefore, we have the

32


Proposition 11. Assume that condition (I.35) is satisfied. Then, for all u0 ∈ L2(Ω) there

exists a control h ∈ L2(ω× (0, T )) such that the solution of equation (I.26) satisfies u(T ) = 0.

In some cases, taking T large enough is sufficient to drive any initial condition to zero,

even without control:

Example 3. Given a bounded open set Ω ⊂ Rn and a constant vector field v ∈ R

n, there

exists T > 0 such that the solution of the transport equation (I.26) satisfies u(T ) = 0 for any

u0 ∈ L2(Ω), with h = 0.

In fact, any T > T0 =diam(Ω)

|v| works.

4 Transport-Diffusion equation

The results in this section are contained in the paper An inverse inequality for some transport-diffusion equation.

Application to the regional approximate controllability, in collaboration with Sergio Guerrero and Axel Osses, submitted

to Asymptotic Analysis.

In this part of the work we study the following transport-difussion equation

yεt + v · ∇yε − ε∆yε = h1ω in Ω × (0, T ),

yε = 0 on Γ × (0, T ),

yε(0) = y0 in Ω.

(I.36)

where ε > 0 is a given parameter (intended to be small). The adjoint equation of (I.36) is

given by:

−ϕεt − v · ∇ϕε − ε∆ϕε = 0 in Ω × (0, T ),

ϕε = 0 on Γ × (0, T ),

ϕε(0) = ϕT in Ω.

(I.37)

We are interested in the controllability properties of equation (I.36) for a small ε > 0

derived from those of equation (I.22) which have been studied in the previous section.

33


We will begin enunciating the known controllability properties of parabolic equations

4.1 Results on the literature.

First of all, it is well known that equation (I.36) is not exactly controllable. For instance,

this can be seen as a consequence of the regularizing effect of the parabolic equations: for any

y0 ∈ L2(Ω), h ∈ L2(ω × (0, T )) we know that the solution of (I.36) satisfies u ∈ C∞(Ω \ ω),

and then we can not expect to reach any L2 function with the solutions of equation (I.36) (if

Ω \ ω 6= ∅).However, for any open non-empty set ω ⊂ Ω, equation (I.36) is approximately controllable:

The unique continuation property for equation (I.37):

ϕε = 0 in ω × (0, T ) ⇒ ϕε ≡ 0.

is a consequence of the Holmgren Uniqueness Theorem (see [35] for example) for C∞ coeffi-

cients.

The fact that this property is also true in the case of less regular coefficients can be proved

with the aid of Carleman estimates, which in fact imply some observability inequalities like

(I.20) (with weight ρ1 = 1 and with some ρ2 such that ρ2(T ) = ∞) for equation (I.37), see

[29], [64]. Therefore, for any nonempty open set ω ⊂ Ω, equation (I.36) is null-controllable.

We are interested in the cost of the approximate controllability, that is to say, in the

norm of the operator which assigns a fixed target f ∈ L2(Ω) to the control h with minimal

L2(ω × (0, T ))-norm such that the solution of (I.36) satisfies

|yε(T ) − f |L2(Ω) < γ. (I.38)

It is possible to show that the cost of this approximate controllability coincides with the

L2(ω)-norm of the element ϕT where the functional

Jf,γ(ϕT ) =1

2

∫ T

0

∫

O|ϕ|2 dx dt −

∫

ΩϕT f dx+ γ|ϕT |L2(Ω)

34


attains its minimum (see [25], [26]).

Using this approach for the heat equation, that is, taking ε = 1 and v = 0 in (I.36), and

with the help of an estimate of the observability constant (which is equal to the norm of the

null-control operator) in terms of the time T , in [26] (see also [53]) the authors prove that

given f ∈ H2(Ω)∩H10 (Ω), the norm of the control needed to have (I.38) can be estimated by

exp(C/γ1/2) as γ goes to zero.

It is clear that the fact that the norm of the control needed to approximate some given

function goes to infinity as the error goes to zero is related to the fact that the heat equation,

or equation (I.36), is not exactly controllable.

On the other hand, as we have seen in the previous section, under some geometrical

conditions on Ω, ω, v and T , the transport equation (I.26) is exactly controllable. In fact,

from Proposition 10 we have that in any case (if v 6= 0 and ω 6= ∅) there is an open set Ω0

such that (I.26) is exactly -regionally- controllable to Ω0.

Then it is natural to ask if, given a open set Ω0 ⊂ Ω satisfying the hypothesis of Proposi-

tion 10, for each f ∈ L2(Ω0), the norm of the control needed to have the regional version of

(I.38), that is,

|yε(T )|Ω0 − f |L2(Ω0) < γ. (I.39)

remains bounded when, in addition to γ, the coefficient ε goes to zero.

Related to this problem, in the works [21] and [31] the authors proved that, under some

natural conditions (which are equivalent to the fact that the corresponding transport equation

would be null-controllable), the cost of the null controllability of equation (I.36) goes to zero

as ε goes to zero. One of these conditions (which is in fact necessary) is to take the time T

large enough, so is not possible to follow the approach of [26] in order to prove the analogous

property for the approximate control.

Here we give a partial answer to this problem.

35


4.2 Convergence of localized solutions.

Let Ω0 be an open subset of Ω. We are interested here in controlling the solution u(T ) of

equation (I.36), restricted to Ω0. In order to do this, we will prove first some convergence of

the solutions of the transport-diffusion problem to those of the transport equation, when the

initial conditions are supported in Ω0 and has H1-regularity. Next, applying this result to

the adjoint equation (I.37), and using the observability inequality for the transport equation

proved in Lemma 3, we get some kind of estimate for the transport diffusion problem. Fina-

lly, we shall see that this estimate gives some information about the cost of some regional

approximate controllability property for (I.36).

Throughout this part of the work, we assume the following hypothesis:

ν(x) · v(x) 6= 0 ∀x ∈ Ψv,T (Ω0) ∩ Γ. (I.40)

were we have denoted ν = ν(x) the outward unit normal vector to Γ and we follow the

notation stated in Definition 3.

Consider, for each y0 ∈ L2(Ω0) and ε > 0, the equation given by

yεt + v · ∇yε − ε∆yε = 0 in Ω × (0, T ),

yε = 0 on Γ × (0, T ),

yε(0) = y01Ω0 in Ω.

(I.41)

and the corresponding transport equation

yt + v · ∇y = 0 Ω × (0, T )

y = 0 Γ− × (0, T )

y(0) = y01Ω0 Ω

(I.42)

36


Extra regularity for the transport equation

Now, we recall two regularity results for the solutions of (I.42), the first one concerning

some hidden regularity et the boundary, and the second one concerning its differentiability

in space.

Lemma 4. Assume hypothesis (I.40). Then, the solution of (I.42) satisfies y ∈ L2(Σ+) for

all y0 ∈ L2(Ω0), and there exists a constant C > 0 such that

|y|L2(Σ+) ≤ C|y0|L2(Ω0).

Proof:

Since we have that the initial condition of (I.42) has support in Ω0 ⊂⊂ Ω and Ω0 is

bounded, by (I.40) we deduce that

|v(x) · ν(x)| ≥ η > 0 ∀x ∈ supp(z) ∩ Γ (I.43)

for some η > 0.

Therefore, we have

|y|L2(Σ+) ≤ η−1/2|y|L2(Σ+,|v·ν|dS) (I.44)

Inequality (I.44) and Proposition 7 imply the desired result.

Lemma 5. If y0 ∈ H10 (Ω0) then y ∈ C([0, T ];H1(Ω)) and there exists a positive constant C

such that

|y|C([0,T ];H1(Ω)) ≤ C|y0|H10 (Ω0).

Proof: Let us denote by y0 the extension by zero of y0 to all Rn. Since y0 ∈ H1

0 (ω) we have

y0 ∈ H1(Rn), and y := y0 X(−t, ·) ∈ C(R;H1(Rn)).

In [2] it is proved that the only singular points of the mapping Υ : Γ × R −→ Rn given

by Υ(x, t) = X(t, x) are those pairs (x, t) such that x ∈ Γ0 = x ∈ Γ : ν(x) · v(x) = 0.Then, from to hypothesis (I.40) and the inverse mapping theorem, we get that the function

x 7→ X(τ+(x), x) (see definitions 3) is, in particular, continuous in ω.

37


Assuming without loss of generality that ω is connected, we have that Ψv,T (ω)∩Γ is also

connected, and hence Ψv,T (ω) is a component of the set X(t, x) : x ∈ ω, 0 ≤ t <∞ ∩ Ω.

Then we can take a cut-off function ϕ ∈ C∞(Rn) such that ϕ ≡ 1 in Ψv,T (ω) and ϕ ≡ 0

in Rn\V , where V is some neighborhood of Ψv,T (ω).

Therefore we have y = (ϕy)|Ω, which give us the desired result.

The corrector

Following classical ideas (see, for instance, [49]), for each y0 ∈ L2(Ω0) we define the

corrector θε as the solution of:

θεt + v · ∇θε − ε∆θε = 0 in Ω × (0, T ),

θε = y on Γ × (0, T ),

θε(0) = 0 in Ω,

(I.45)

where y is the solution of (I.42) with y(0) = y0.

We will work with system (I.45) by transposition. Thus, for each h ∈ L2(Ω × (0, T )), we

consider the following adjoint problem:

−uεt − v · ∇uε − ε∆uε = h in Ω × (0, T ),

uε = 0 on Γ × (0, T ),

uε(T ) = 0 in Ω.

(I.46)

It is well-known (see for instance [45]) that (I.46) possesses a unique solution belonging to

the space

X0 := L2(0, T ;H2(Ω)) ∩H1(0, T ;L2(Ω))

and which depends continuously on h, that is to say, there exists a positive constant Cε such

that

|uε|X0 ≤ Cε|h|L2(Ω).

38


We have also the next result:

Lemma 6. For each η > 0 there exists C > 0 independent of ε such that

ε1/2

∣∣∣∣∂uε

∂ν

∣∣∣∣L2(0,T ;L2(Γη))

≤ C |h|L2(Ω×(0,T )) (I.47)

for all h ∈ L2(Ω× (0, T )), where uε is the solution of (I.46), and Γη = x ∈ Γ : ν(x) · v ≥ η.

Proof:

We first see that from classical energy estimates we obtain

|uε|C0([0,T ];L2(Ω)) + ε1/2|uε|L2(0,T ;H1(Ω)) ≤ C|h|L2(Ω0), (I.48)

for some C > 0. On the other hand, we multiply the equation in (I.46) by ∆uε and we

integrate by parts in Ω × (0, T ). We have

− 1

2

∫

Ω|∇uε(0)|2 − 1

2

∫∫

Σ

∣∣∣∣∂uε

∂ν

∣∣∣∣2

(v · ν) − ε

∫∫

Q|∆uε|2

+

∫∫

Q(∂xj

vk)(∂xkuε)(∂xj

uε) =

∫∫

Qh∆uε. (I.49)

From (I.48) and (I.49) we get

∫∫

Σ

∣∣∣∣∂uε

∂ν

∣∣∣∣2

(v · ν) + 2ε|∆uε|2L2 ≤ 2|v|W 1,∞ |∇uε|2L2 + 2|h|L2 |∆uε|L2

≤ Cε−1|h|2L2 + 2|h|L2 |∆uε|L2

≤ Cε−1|h|2L2 + ε|∆uε|2L2 . (I.50)

Inequality (I.50) implies that

∫∫

Σ

∣∣∣∣∂uε

∂ν

∣∣∣∣2

(v · ν) ≤ Cε−1|h|2L2(Q). (I.51)

Now, we take a cut-off function ϕ ∈ C∞(Rn) such that

(i) ϕ ≡ 1 in a neighborhood of Γ2η/3,

39


(ii) ϕ ≡ 0 in a neighborhood of (Γη/3)c = x ∈ Γ : ν(x) · v < η/3.

Let ψε = ϕuε. It follows that ψε satisfies the system (I.46) with right hand side hε :=

ϕh− (∇ϕ · v)uε − 2ε∇ϕ · ∇uε − ε∆ϕuε instead of h.

By (I.48), we have

|hε|L2(Q) ≤ C|h|L2(Q) (I.52)

and by construction ψε ≡ 0 in a neighborhood of Γ− = x ∈ Γ : ν(x) · v < 0.Hence, from inequality (I.51) applied to ψε we get

∫ T

0

∫

Γη

∣∣∣∣∂ψε

∂ν

∣∣∣∣2

|v · ν| ≤ 2ε−1|hε|2L2(Q). (I.53)

By definition ∇ψε = ∇uε and |v ·ν| ≥ η on Γη. These facts together with (I.53) and (I.52)

yields (I.47). Lemma 6 is proved.

From Lemma 6 we can deduce the well-posedness of problem (I.45). Precisely, we have:

Proposition 12. For each ε ∈ (0, 1) and y0 ∈ L2(Ω0) we have θε ∈ L2(Ω× (0, T )) and there

exists a positive constant C > 0 independent of ε and y0 such that

|θε|L2(Ω×(0,T )) ≤ Cε1/2|y0|L2(Ω0). (I.54)

Proof: Multiplying the differential equation in (I.46) by θε, integrating in Ω × (0, T ), and

integrating by parts it is not difficult to deduce that

〈h, θε〉L2(Ω×(0,T )) = −ε∫ T

0

∫

Γη

z∂uε

∂νdσdt, (I.55)

where η > 0 is given by (I.43).

Then if we define the operator

Fε : L2(Ω × (0, T )) → L2(0, T ;L2(Γη))

h 7→ −ε1/2 ∂uε

∂ν1Γη

(I.56)

40


where uε is the solution of (I.46), then from (I.55) we have that its dual operator is given by

F ∗ε : L2(0, T ;L2(Γη)) → L2(Ω × (0, T ))

z 7→ ε−1/2θε.

From Lemma 6 we have ‖Fε‖ ≤ C and then ‖F ∗ε ‖ ≤ C, for all ε ∈ (0, 1) with C inde-

pendent of ε. This result together with Lemma 7 implies (I.54). Proposition 12 is proved.

Thanks to the result of the previous paragraph, one naturally introduces the function

wε = yε − (y − θε), expecting to have ‘nice’ estimates on it as ε → 0. Indeed, in our next

result we show that we can take y0 only a little more regular than L2(Ω0) in order to have a

convergence to zero for wε. Of course, the velocity of convergence depends on how regular y0

is.

We have the following:

Proposition 13. Let ε ∈ (0, 1). Then, for all δ ∈ [0, 1] and for all y0 ∈ Hδ0(Ω0), the function

wε defined above satisfies

|wε|L2(Ω×(0,T )) ≤ Cεδ/2|y0|Hδ(Ω0)

for some C > 0 independent of ε and y0.

Proof:

We check the two limit cases, and then apply interpolation:

δ = 1. Let us consider the function wε = yε − (y − θε). It fulfills:

wεt + v · ∇wε − ε∆wε = ε∆y in Ω × (0, T ),

wε = 0 on Γ × (0, T ),

wε(0) = 0 in Ω.

(I.57)

Now, if y0 ∈ H10 (Ω0), by Lemma 5 we have that y ∈ L2(0, T ;H1(Ω)) and then ∆y ∈

L2(0, T ;H−1(Ω)).

41


Multiplying the equation in (I.57) by wε and integrating by parts in Ω, we get

d

dt

(1

2

∫

Ω|wε|2dx

)+ ε

∫

Ω|∇wε|2dx = −ε

∫

Ω∇y · ∇wεdx

≤ ε

2

(∫

Ω|∇y|2dx+

∫

Ω|∇wε|2dx

)

Then, by Gronwall’s inequality and Lemma 5 we get:

|wε|2L∞(0,T ;L2(Ω)) ≤ ε|y|2L2(0,T ;H1(Ω))

≤ ε|y0|2H10 (ω),

which gives the desired inequality for δ = 1.

δ = 0. For all y0 ∈ L2(Ω0), the solutions y and yε of the equations (I.42) and (I.41)

respectively depend continuously on the L2-norm of y0. This is also true for θε by

Proposition 12, so we are done.

By the last two points, we have defined the operators

A0 : L2(Ω0) → L2(Ω × (0, T )),

A1 : H10 (Ω0) → L2(Ω × (0, T ))

which associates to each y0 the solution wε of problem (I.57) (recall that y denotes the

solution of (I.42) associated to y0). From the δ = 0 and δ = 1 cases, we know that

‖A0‖ ≤ C and ‖A1‖ ≤ Cε1/2, for a positive constant C independent of ε.

Applying a classical interpolation result (see, for instance, [50]), we have that Aδ :

Hδ0(Ω0) → L2(Ω × (0, T )) is well defined for δ ∈ [0, 1], and

‖Aδ‖ ≤ C‖A0‖1−δ‖A1‖δ ≤ Cεδ/2

for a constant C > 0 independent of ε and the proposition is proved.

42


In particular, from the two previous results we can establish the convergence of the solu-

tions of the transport-diffusion problem (I.41) to those of the transport problem (I.42):

Corollary 2. For each δ ∈ (0, 1], there exists C > 0 such that

|yε − y|L2(Ω×(0,T )) ≤ Cεδ/2|y0|Hδ(Ω0)

for all y0 ∈ Hδ0(Ω0) and all ε ∈ (0, 1), where yε and y are the solutions of (I.41) and (I.42)

respectively, with yε(0) = y(0) = y01Ω0 .

In particular, yε converges to y as εց 0+ in the space L2(Ω×(0, T )) as long as δ ∈ (0, 1].

Remark 4. In fact, is not difficult to see that

|yε − y|L2(Ω×(0,T )) ≤ γ(ε)|y0|L2(Ω0) ∀y0 ∈ L2(Ω0)

γ(ε) → 0 as ε→ 0

(I.58)

does not hold.

Indeed, take some open set ω ⊂ Ω such that Ω0 ⊂−v,T ω and Ω0 \ ω 6= ∅. From Lemma 3

there exists C > 0 such that

|y0|L2(Ω0) ≤ C|y|L2(ω×(0,T )) ≤ C(|yε|L2(ω×(0,T )) + |yε − y|L2(Ω×(0,T ))).

Assume for a moment that (I.58) holds. Then,

|y0|L2(Ω0) ≤ C(|yε|L2(ω×(0,T )) + γ(ε)|y0|L2(Ω0)).

Taking ε > 0 small enough (such that γ(ε) < 1/2), we deduce that

|y0|L2(Ω0) ≤ C|yε|L2(ω×(0,T )) ∀y0 ∈ L2(Ω).

which contradicts the fact that equation (I.36) is not exactly controllable.

43


4.3 The ε-estimate.

In this part of the work, from the convergence proved in the previous section and assuming

the geometrical hypothesis given by (I.28), we shall prove an estimate for the transport-

diffusion equation which will be interesting for ε small. The two key points in the proof of

this result are the convergence stated in Corollary 2 and the regional observability (I.32) for

the transport equation proved, under some geometrical assumptions, in Lemma 3.

Theorem 1. Assume that Ω0 ⊂−v,T ω. Then, there exist a positive constant C and ε0 > 0

such that for any δ ∈ (0, 1] we have:

|y0|L2(Ω0) ≤ C(|yε|L2(ω×(0,T )) + εδ/2|y0|Hδ(Ω0)) (I.59)

for all y0 ∈ Hδ0(Ω0) and ε < ε0, where yε is the solution of (I.41) associated to y0.

Proof: For each y0 ∈ Hδ0(Ω0) and ε > 0 let y and yε be the solutions of (I.42) and (I.41)

respectively, with y(0) = yε(0) = y01Ω0 .

We have:

|y0|Ω0 ≤ C|y|L2(ω×(0,T )) by Lemma 3

≤ C(|y − yε|L2(Ω×(0,T )) + |yε|L2(ω×(0,T ))

)

≤ C(εδ/2|y0|Hδ(Ω0) + |yε|L2(ω×(0,T ))

)by Corollary 2

and Theorem 1 is proved.

Remark 5. For each y0 ∈ L2(Ω0), we can take yn0 ⊂ Hδ(Ω0) such that yn

0 → y0 in L2(Ω0).

Then, applying the previous Theorem we get the following inequality for each yn0 and for each

ε < ε0:

|yn0 |L2(Ω0) ≤ C(|yε

n|L2(ω×(0,T )) + εδ/2|yn0 |Hδ(Ω0)),

where yεn is the solution of (I.41) associated to yn

0 .

Taking εn convergent to zero in such a way that εδ/2n |yn

0 |Hδ(Ω0) also converges to zero,

we get the inequality for the transport equation, proved in proposition 3.

44


4.4 An ε-inverse inequality for the right hand side.

Here we prove, as a direct consequence of the previous results, an inverse inequality for

the (time independent) right hand side of our equation.

To be precise, given the open subset Ω0 ⊂ Ω, for each g ∈ L2(Ω0) and ε > 0, consider the

equation given by

zεt + v · ∇zε − ε∆zε = g1Ω0 in Ω × (0, T ),

zε = 0 on Γ × (0, T ),

zε(0) = 0 in Ω.

(I.60)

We are interested in the following problem: given an open set ω ⊂⊂ Ω, find an estimate

for g in terms of the observation of the solution of (I.60) in ω × (0, T ), for small ε.

Taking time derivative, since g is time-independent we have that yε = ∂tzε solves equation

(I.41) with yε(0) = g. Then, from Theorem 1 we get directly the following result

Corollary 3. Assume that Ω0 ⊂−v,T ω. Then, there exist a positive constant C and ε0 > 0

such that for any δ ∈ (0, 1] we have:

|g|L2(Ω0) ≤ C(|zε|H1(0,T ;L2(ω)) + εδ/2|g|Hδ(Ω0)) (I.61)

for all g ∈ Hδ0(Ω0) and ε < ε0, where zε is the solution of (I.60) associated to g.

4.5 The cost of the -regional-approximate controllability

In this part of the work, we apply Theorem 1 to the adjoint transport-diffusion equation,

which we state again:

−ϕεt − v · ∇ϕε − ε∆ϕε = 0 in Ω × (0, T ),

ϕε = 0 on Γ × (0, T ),

ϕε(0) = ϕT in Ω.

(I.62)

45


and we prove that the resulting inequality (I.59) implies, by duality, the existence of some

kind of approximate regional controllability for the original transport-diffusion equation, with

control u ∈ L2(ω × (0, T )):

yεt + v · ∇yε − ε∆yε = u1ω in Ω × (0, T ),

yε = 0 on Γ × (0, T ),

yε(0) = y0 in Ω.

(I.63)

with error (measured in norm H−λ for λ ∈ (0, 1]) less or equal than some power of the

diffusion coefficient ε (precisely, ελ/2). Furthermore, the controls which lead to this property

are uniformly bounded with respect to ε.

In all the rest of this chapter, we assume that

Ω0 ⊂v,T ω (I.64)

Therefore, from Theorem 1 with δ = 1 applied to equation (I.62) (that is, with −v as

the vector field), we have that for all ϕT ∈ H10 (Ω0), the corresponding solution ϕ of equation

(I.62) satisfies inequality (I.59).

For each ε > 0 we define the space:

Zε = (ε1/2ϕT , ϕε|ω×(0,T )) : ϕT ∈ H1

0 (Ω0), (I.65)

where ϕε is the solution of (I.62) with ϕε(T ) = ϕT1Ω0 .

From the continuity of the solutions of (I.62) with respect to the initial condition, we

have that Zε is a closed subspace of H10 (Ω0) × L2(ω × (0, T )), and therefore, with the scalar

product induced by this space, is a Hilbert space. Let us remark that in fact Zε is a closed

subspace of H10 (Ω0) × Fε, where

Fε = ϕε|ω×(0,T ) : ϕT ∈ H10 (Ω0) ⊂ L2(ω × (0, T )).

46


Now, for each ε > 0 we define the linear operator

Bε : Zε → L2(Ω0)

(ε1/2ϕT , ϕε|ω×(0,T )) 7→ ϕT .

Thanks to inequality (I.59) for δ = 1, we directly have the following result:

Lemma 7. There exists C > 0 and ε0 > 0 such that

‖Bε‖ ≤ C

for all 0 < ε < ε0.

Therefore we also have that the adjoint operator of Bε, B∗ε : L2(Ω0) −→ Z ′

ε, is bounded

and

‖B∗ε‖ ≤ C for each 0 < ε < ε0. (I.66)

Since Fε is a closed subespace of L2(ω×(0, T )), it can be identified (using the L2-product)

with its dual. Thus, being Zε a closed subspace of H10 (Ω0)×Fε, we have (see for example [59],

section 4.8) that the dual space Z ′ε is isomorphic to the quotient space (H−1(Ω0) × Fε)/Z

⊥ε

where Z⊥ε is the set

(h, u) ∈ H−1(Ω0) × Fε :

⟨h, ε1/2ϕT

⟩−1,1

+

∫ T

0

∫

ωuϕε = 0 ∀ϕT ∈ H1

0 (Ω0)

,

called the annihilator of Zε.

This means that B∗εy is an equivalence class in H−1(Ω0) × Fε for each y ∈ L2(Ω0).

In fact, we can show that for each y ∈ L2(Ω0), B∗εy is exactly the equivalence class of all

the y-admissible pairs corrector-control in H−1(Ω0) × Fε:

Lemma 8. For all y ∈ L2(Ω0) we have that

y = yu(T )|Ω0 + ε1/2h

47


for each (h, u) ∈ B∗εy, where yu is the solution of equation (I.63) with control u.

Proof: Take ε > 0 and y ∈ L2(Ω0). For each (h, u) ∈ H−1(Ω0) × Fε such that B∗εy =

(h, u) + Z⊥ε , by the definition of dual operator and the construction of Bε, we have:

∫

Ω0

y ϕT dx =

∫

Ω0

y Bε(ε1/2ϕT , ϕ|ω×(0,T )) dx

=⟨B∗

εy, (ε1/2ϕT , ϕ

ε|ω×(0,T ))⟩

Z′

ε,Zε

=⟨h, ε1/2ϕT

⟩−1,1

+

∫ T

0

∫

ωuϕε dx dt . (I.67)

for all ϕT ∈ H10 (Ω0).

If we consider the equation (I.63) with control u and solution yu, multiplying it by ϕε and

integrating by parts results:

∫ T

0

∫

ωuϕε dx dt =

∫

Ω0

yu(T )ϕT dx ∀ϕT ∈ H10 (Ω0). (I.68)

From (I.67) and (I.68) we conclude that

⟨h, ε1/2ϕT

⟩−1,1

=

∫

Ω0

(y − yu(T ))ϕT dx ∀ϕT ∈ H10 (Ω0), (I.69)

that is, ε1/2h = y − yu(T ) in L2(Ω0).

Remark 6. In fact, we can describe explicitly the mentioned annihilator: It is not difficult

to see that

Z⊥ε = (−ε−1/2yu(T ), u) : u ∈ Fε.

Moreover, for each y ∈ L2(Ω), a trivial pair corrector-control is (ε−1/2y, 0). It follows that

the complete class of such pairs is given by:

B∗εy = (ε−1/2(y − yu(T )), u) : u ∈ Fε.

On the other hand, let us recall that by definition of the norm in a quotient space we

48


have:

‖B∗εy‖H−1×Fε/Hε

= ınf|h|H−1(Ω0) + |u|L2(ω×(0,T )) : y = yu(T )|Ω0 + ε1/2h. (I.70)

Then, by (I.66), (I.70) and Lemma 8 we have:

Proposition 14. For all y ∈ L2(Ω0) and for all ε > 0 there exist uε ∈ L2(ω × (0, T )) and

hε ∈ L2(Ω0) such that:

yε(T )|Ω0 + ε1/2hε = y,

|uε|L2(ω×(0,T )) + |hε|H−1(Ω0) ≤ C|y|L2(Ω0).

Moreover, we can prove in the same way the analogue statements of Lemmas 7 and 8 and

Proposition 14 in the framework of spaces H−δ and Hδ.

In order to state explicitly those facts in terms of the cost of the control, let us define:

Definition 4. For each λ, ε, γ > 0 and f ∈ L2(Ω0), denoting yε as the solution of (I.63) with

control h ∈ L2(ω × (0, T )), we define

Cλε (f, γ) = ınf|h|L2(ω×(0,T )) : |yε(T ) − f |H−λ(Ω0) ≤ γ.

Then, from Proposition 14 we have:

Theorem 2. There exists C > 0 such that for each f ∈ L2(Ω0), γ > 0 and λ ∈ (0, 1], there

exists ε0 > 0 such that

Cλε (f, γ) ≤ C|f |L2(Ω0) ∀ε ≤ ε0.

Remark 7. In the proof of Proposition 14 we can see that the explicit dependence of ε0 with

respect to f, λ and γ is

ε0 =

(γ

|f |L2(Ω0)

)2/λ

.

49


5 Fokker-Planck equation

In this section, we study another equation where are present transport and diffusion:

Fokker-Planck equation.

Let X ⊂ Rn and V ⊂ R

n be open regular sets.

We define:

Ω = X × V

Γv = X × ∂V

Γ− = (x, v) ∈ ∂X × V | v · n(x) < 0.Γ+ = (x, v) ∈ ∂X × V | v · n(x) > 0.We consider the evolution equation given by:

ut + v · ∇xu− µ∆vu = f Ω × (0, T )

u = 0 Γv ∪ Γ− × (0, T )

u(0) = u0 Ω

(I.71)

We know that, for each f ∈ L2((0, T );L2(Ω)), and u0 ∈ L2(Ω), equation (I.71) has a

unique weak solution u ∈ C([0, T ];L2(Ω)), (see [22], for example).

5.1 Transport equation (µ = 0)

We consider the equation given by:

ut + v · ∇xu = h11ω1 Ω × (0, T )

u = h21ω2 Γ− × (0, T )

u(0) = u0 Ω

(I.72)

The adjoint equation of problem (I.72) is:

−ϕt − v · ∇xϕ = 0 Ω × (0, T )

ϕ = 0 Γ+ × (0, T )

ϕ(T ) = ϕT Ω

(I.73)

50


In a similar way than in the case of the transport equation studied in section (3), the

solution of problems (I.72) and (I.73) can be explicitly stated:

u(x, v, t) = u0(x− tv, v)1(0,τ−(x,v))(t)

+h2(x, v, t)1(τ−(x,v),T ) +

∫ t

maxt−τ−(x),0f(x− (t− s)v, s)ds

and

ϕ(x, t) = ϕT (x+ (T − t)v, v)1(T−τ+(x),T ).

where we have defined:

τ−(x, v) = ınft ∈ (0, T ) | x− vs ∈ Ω for all 0 ≤ s ≤ t,

τ+(x, v) = supt ∈ (0, T ) | x+ vs ∈ Ω for all0 ≤ s ≤ t,

The results on controllability for equation (I.72) are completely analogous than those

results of transport which we have developed in subsection (3). We only state the main

results.

For the approximate controllability, we have:

Proposition 15. Problem (I.71) is approximately controllable if and only if

For all (x, v) ∈ X × V there exists t ∈ [0, τ−(x, v)] such that x− tv ∈ ω1 ∪ ω2.

Proposition 16. If the initial condition u0 is null-controllable with the control zone ω1 ∪ω2

then it is null-controllable with ω1.

Proof:

Take A = (x, v)‖τ−(x, v) ≥ T and ϕT = ϕT 1A. Let ϕ be the solution of equation (I.73).

We have

ϕ(0) = ϕ(0) and ϕ|Γ−= 0

51


Hence

∫Ω |ϕ(0)|2 =

∫Ω |ϕ(0)|2

≤ C(∫ T0

∫ω1

|ϕ|2 +∫ T0

∫ω2

|ϕ|2)≤ C

∫ T0

∫ω1

|ϕ|2

Following that he have done in the third section of this chapter, we will find a weight

function for an observability inequality. Consider:

yt + v · ∇y = 1ω Ω × (0, T )

y = 0 Γ− × (0, T )

y(0) = 0 Ω

(I.74)

Completely analogous than in section 3, as a weight function for observability inequality

which corresponds to the null-control, we can take ρ = φ(0), where:

−φt − v · ∇φ = 0 Ω × (0, T )

φ = 1 Γ+ × (0, T )

φ(T ) = y(T ) Ω

(I.75)

Proposition 17. Let ω ⊂ Ω and T be such that

Ω ⊂ Ψv,T (ω ∪ Γ−).

Then ρ(x, v) > 0 in X × V

Proof:

Take Ω1 = (x, v) ∈ X × V ‖τ+(x, v) > T and Ω2 = Ω \ Ω1.

Thanks to the boundary condition on the problem solved by φ, we have ρ(x, v) =

1∀(x, v) ∈ Ω2.

Now, take (x, v) ∈ Ω1.

Then ρ(x, v) = y(T )(x+ Tv, v) = λ(t ∈ [0, T ]‖(x + tv, v) ∈ ω).

52


Then there exists a t such that (x+ tv, v) ∈ ω ∪ Γ−. Since x ∈ Ω we have (x+ tv, v) ∈ ω.

Finally, the set ω is open, thus the measure of the set of those t is positive. Therefore

ρ(x, v) > 0.

From Proposition (17) we have directly:

Corollary 4. Assume Ω ⊂ Ψv,T (ω)and let ρ = φ(0). If u0 ∈ L2ρ−1(Ω), then there exists

h ∈ L2(ω × (0, T )) such that the solution u of (I.72) satisfies u(T ) = 0.

5.2 Fokker-Planck equation (µ > 0)

In order to see if equation (I.71) is approximately controllable (with interior control,

supported in ω ⊂ Ω, say), by the dual characterizations (see section 2), we have to ask if an

unique continuation principle is true for the adjoint equation

−ϕt − v · ∇xϕ− µ∆vϕ = 0 Ω × (0, T )

ϕ = 0 Γv ∪ Γ− × (0, T )

ϕ(T ) = ϕT Ω

(I.76)

Now, we deal with the differential operator P = −∂t − v∂x − ∆v in X × V × (0, T ). It is

clear that P has analytic coefficients. Also, its principal part (−∆2v) has constant coefficient.

Then it is not so difficult to prove that the Holmgreen uniqueness theorem (see [35]) holds

for P .

Now, since the principal symbol of P is p(ξ1, ξ2, s) = |ξ2|2, we get that the characteristic

hyper-planes of P are given by Π = (x, v, t) : a · x+ bt = c, with a ∈ Rn and b, c ∈ R.

That is, Π is parallel to V . Then, given Π with non-empty intersection with Ω, all the sets

given by X × v0 × (0, T ) will intersect Π. This is exactly the hypothesis of the Holmgreen

uniqueness theorem. Then, we have proved the following

Proposition 18. Let ω ⊂ X×V be an open set. If there exists v0 ∈ V such that X×v0 ⊂ ω,

then there are unique continuation of P in ω × (0, T ).

53


By duality we have directly:

Corollary 5. Given an open set ω ⊂ Ω like in the previous theorem, then the equation (I.71)

is approximately controllable with controls h ∈ L2(ω × (0, T )).

54

CAPITULO II

Transmission wave equation.

55

CAPITULO II. TRANSMISSION WAVE EQUATION.

This chapter is a generalization to the n-dimensional case (with n ≥ 2) of the results presented in the paper A

global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem to

appear in Inverse Problems, realized in collaboration with Lucie Baudouin and Axel Osses.

In this chapter we study the wave equation with discontinuous principal coefficient. We

consider the case where the speed coefficient is piecewise constant, which set of discontinuities

is some closed hyper-surface. We regard this equation as a system of two wave equations

coupled by a transmission conditions. We prove a global Carleman inequality for this system

under some convexity hypothesis for the inner domain and some monotonicity conditions

on the velocity coefficients. We obtain, as a consequence of this inequality, the uniqueness

and Lipschitz stability for the inverse problem of retrieving a stationary potential for the

wave equation with Dirichlet data and discontinuous principal coefficient from a single time-

dependent Neumann boundary measurement.

1 Introduction.

Consider two embedded domains, where the speed coefficients are a1 > 0 in the inner

domain and a2 > 0 in the outer domain. Stability of the inverse problem we study here is

obtained by deriving a global Carleman estimate for the wave equation with discontinuous

coefficients. We prove this Carleman inequality under some convexity hypothesis for the

inner domain and the speed is monotonically increasing from the outer to the inner layers,

i.e. a1 > a2. This last situation is, incidentally, the general case in seismic prospection, where

data of a single wave that propagates through the Earth is considered.

Figure 1 illustrates the role of these hypothesis and gives some intuition with the help

of Snell’s law. In the case a1 > a2 (see Figure 1, left) the incident rays coming from the

inner domain toward the outer domain become closer to the normal at the interface since

sin(θ1) > sin(θ2), where θi, i = 1, 2 are the corresponding incident angles. Therefore, all

the rays coming from the inner ball with any incident angle θ1 in (−π/2, π/2) succeed in

crossing the interface. In the opposite case, when a1 < a2 (see Figure 1, center) we have

sin(θ1) < sin(θ2) and there is a critical incident angle θ∗ < π/2 such that the rays with

56


incident angles θ1 out of the range (−θ∗, θ∗) remain supported near the interface and do not

reach the outer domain, so this information does not arrive at the exterior boundary. Finally,

convexity of the inner domain avoids trapped rays (see Figure 1, right).

Figura II.1: Two domains with speed coefficients a1 (inner) and a2 (outer). In the first figure (left),if a1 > a2 by Snell’s law all the inner rays reach the exterior boundary independently of their incidentangles. Conversely, in the second figure (center) if a1 < a2 some rays with large incident angles remaintrapped near the inner interface. The last figure (right) shows a trapped ray into a captive domain.

Global Carleman estimates and the method of Bukhgeim-Klibanov [14], [13] are especially

useful for solving the one measurement inverse problems. It is possible to obtain local Lipschitz

stability around the single known solution, provided that this solution is regular enough

and contains enough information [42] (see also [41] and [66]). Many other related inverse

results for hyperbolic equations use the same strategy. To cite some of them see [56] and

[66] where Dirichlet boundary data and Neumann measurements are considered and [38], [39]

where Neumann boundary data and Dirichlet measurements are studied. These references

are all based upon the use of local or global Carleman estimates. Related to this, there are

also general pointwise Carleman estimates that are also useful in similar inverse problems

[27, 28, 43].

Recently, global Carleman estimates and applications to one-measurement inverse pro-

blems were obtained in the case of variable but still regular coefficients, see [37] for the

isotropic case, and [46] and [5] for the anisotropic case. It is interesting to note that these

authors require a bound on the gradient of the coefficients, so that the idea of approximating

discontinuous coefficients by smooth ones is not useful.

There are a number of important works [57, 32, 58, 63, 11] concerning the same inverse

problem in the case that several boundary measurements are available. In these cases, it is

57


possible to retrieve speed coefficients and even discontinuity interfaces without any restrictive

hypothesis of convexity or speed monotonicity. For instance, one can retrieve the interface by

observing the traveltime reflection of several waves. Indeed, it is well known that the interface

can be recovered as the envelope of certain curves as shown in Figure 1 (see also [34] and the

references therein). This method works independently of the sign of a1 −a2 and this explains

in part why there are no geometrical or speed monotonic hypotheses for these kind of inverse

results.

x

Figura II.2: Recovering the interface as the envelope of circumferences. Each circumference, centeredat some point x on the exterior boundary, represents the possible locations of the nearest point of theinner interface where the reflection took place for a given traveltime measured at x.

Let us now give some insight into the relationship between this work and exact controlla-

bility or energy decay for the wave equation with discontinuous coefficients.

First of all, the global Carleman estimate we obtain immediately implies a particular

case of a well known result of exact controllability for the transmission wave equation [48].

Roughly speaking, the result of [48] states that we can control internal waves from the exterior

boundary in a layered speed media if the speed is monotonically increasing from the outer to

the inner layers and the inner domain is star shaped, a weaker assumption than convexity.

Moreover, if the speed monotonicity is inverted, there are non controllable solutions with

concentrated energy near the interface [51], [19].

Secondly, there exist several results about the growth of the resolvent for the spectral

stationary transmission problem, from where it is possible to derive the speed of local energy

decay for the evolution wave equation with transmission conditions [17]. In the case a1 > a2

and if the inner domain is strictly convex, it has been shown using micro-local analysis

58


[54, 55] that the speed of the energy decay is exponential if the dimension of space is odd

and polynomial otherwise. In the general case, including the cases when a1 < a2 or the

inner domain is not strictly convex, it has been proved using micro-local analysis and global

Carleman estimates for the spectral problem [4] that the energy decays as the inverse of the

logarithm of the time.

Notice that we shall only consider here the case of a discontinuous coefficient which

is constant on each subdomain (i.e. a1 and a2 constants). We will indeed concentrate our

discussion on the main difficulty, namely the discontinuity at the interface. However, we

could also consider variable coefficients a1(x) and a2(x) such that their traces at the interface

are constant, under additional assumptions of boundedness of ∇aj similar to those appearing

in [37].

Finnaly, we note that a global Carleman estimate [24] has also been obtained for the heat

equation with discontinuous coefficients. That work was initially motivated by the study of

the exact null controllability of the semilinear wave equation, but the estimate has been

recently used to prove local Lipschitz stability for a one measurement inverse problem for the

heat equation with discontinuous coefficients [7], [10].

2 The transmission wave equation.

Let n ≥ 2, and let Ω and Ω1 be two open bounded subsets of Rn with smooth boundaries

Γ and Γ1. Suppose that Ω1 is simply connected, Ω1 ⊂ Ω and set Ω2 = Ω \Ω1. Thus, we have

∂Ω2 = Γ ∪ Γ1. We also set:

a(x) =

a1 x ∈ Ω1

a2 x ∈ Ω2

with aj > 0 for j = 1, 2. We consider the following wave equation:

utt − div(a(x)∇u) + p(x)u = 0 (x, t) ∈ Ω × (0, T )

u = 0 (x, t) ∈ Γ × (0, T )

u(0) = u0 x ∈ Ω

ut(0) = u1 x ∈ Ω.

(II.1)

59


We know that [50, 22] for each p ∈ L∞(Ω), u0 ∈ H10 (Ω) and u1 ∈ L2(Ω), there exists a

unique weak solution u(p) of equation (II.1) such that

u(p) ∈ C([0, T ];H10 (Ω)), ut ∈ C([0, T ];L2(Ω)).

We introduce here our main result concerning a global Carleman estimate for the solutions

of problem (II.1) extended to the time interval (−T, T ). We set Q = Ω × (−T, T ), Σ =

Γ × (−T, T ), Σ1 = Γ1 × (−T, T ), Qj = Ωj × (−T, T ), uj = u1Qjand νj the outward unit

normal to Ωj , for j = 1, 2.

We will work with an equivalent formulation of (II.1). Notice that for each f ∈ L2(Q), u

solves the equation

utt − div(a∇u) + pu = f in Q (II.2)

if and only if, for each j ∈ 1, 2, uj solves (see [48])

uj,tt − aj∆uj + puj = f1Qjin Qj (II.3)

together with the transmission conditions

u1 = u2 on Σ1

a1∂u1

∂ν1+ a2

∂u2

∂ν2= 0 on Σ1.

(II.4)

What we will do is to prove a Carleman inequality for functions satisfying the wave

equations (II.3), but with non-zero boundary values. We deal with the traces on the boundary

with the aid of the transmission condition (II.4) and the properties of the weight function we

will define in the following section.

3 The Carleman estimate

In this part of the work we construct the function which we will use as a weight function

in the Carleman estimate for the Transmission wave equation.

60


Essentially, we need a function φ satisfying the following two conditions:

1. The functions φ|Ω1 and φ|Ω2 can be used as a weight function for the wave equation in

Ω1 and Ω2, respectively.

2. The traces on the interface coming from the integration by parts in the Carleman

estimate can be controlled.

Motivated by this conditions, the function we propose is constructed modifying the func-

tion |x−x0|2 −βt2 which is often used in the literature as a weight function in the Carleman

estimates for the wave equation, in such a way that it is constant on the interface, and it

fulfills the Transmission condition (II.4). Thanks to this, it will not appear tangential deri-

vatives in the interface from the computations in the Carleman estimate. This fact together

with transmission conditions (II.4) and hypothesis a2 < a1 will imply that the sum of all the

traces in the interface is positive, and the second condition stated above is satisfied.

On the other hand, under some convexity assumption on the inner domain, we shall prove

that the function we have defined can be used in a Carleman estimate for the wave equation

in each domain Ω1 and Ω2. Basically, this comes from the fact that the Hessian matrix of the

weight function is positive definite.

Precisely, we set the following definition:

Definition 5. Let U ⊂ Rn be a open, bounded, convex set which boundary ∂U is a C2

manifold. We say U is strongly convex if for each plane Π ⊂ Rn intersecting U , the

boundary of the two-dimensional convex set U ∩ Π has strictly positive curvature at each

point.

In particular, strong convexity implies strictly convexity, but the converse is not true (see

for example the graph of x4 at zero).

Our main hypothesis concerning the domains Ω1 is given by

Assumption 1

The open set Ω1 is strongly convex.

61


3.1 The weight function.

We shall construct now the weight function for the Carleman estimate for the transmission

wave equation.

Fix x0 ∈ Ω1, and for each x ∈ Ω \ x0 define ℓ(x0, x) = x0 + λ(x− x0) : λ ≥ 0. Since

Ω1 is convex and bounded there is exactly one point y(x) such that

y(x) ∈ Γ1 ∩ ℓ(x0, x). (II.5)

We define the function ρ : Ω \ x0 −→ R+ by:

ρ(x) = |x0 − y(x)|. (II.6)

Let ε > 0 be such that Bε ⊂ Ω1 (and small enough in a sense that we will precise later)

and let 0 < ε1 < ε2 < ε. Then we consider a cut-off function η ∈ C∞(R) such that

0 ≤ η ≤ 1, η = 0 in Bε1(x0), η = 1 in Ω \Bε2(x0). (II.7)

For each j ∈ 1, 2 we take k such that j, k = 1, 2 and we define the following functions

in the whole domain Ω × R

φj(x, t) = η(x)ak

ρ(x)2|x− x0|2 − βt2 +Mj (x, t) ∈ Ω × R, (II.8)

where β, M1 and M2 are positive numbers that will be chosen later. Then, the weight function

we will use in this work is

φ(x, t) =

φ1(x, t) (x, t) ∈ Ω1 × R

φ2(x, t) (x, t) ∈ Ω2 × R.

(II.9)

In order to prove that the function φ satisfies enough properties for being a weight function

in a Carleman estimate, firstly we will examine their spatial part. Without loss of generality

62


take x0 = 0, and let ρ be the corresponding function defined in (II.6).

We define the function φ0 : Rn \ 0 → R by

φ0(x) =|x|2ρ(x)2

.

We have the following

Proposition 19. If Ω1 is a strongly convex domain in Rn, with C2 boundary, then D2(φ0)

is positive definite at every x ∈ Rn \ 0.

Proof: We begin recalling the definition and some properties of the Minkowski functional (see

Chapter 1 of [59]): If X is a vector space, and C ⊂ X is an absorbing set with 0 ∈ C, the

Minkowski functional of C is given by

µC(x) = ınfr > 0 :1

rx ∈ C. (II.10)

If C is convex then µC is sub-additive and positive-homogenous, thus in particular convex.

Let us show that φ0 = µ2Ω1

. Givenx

r∈ Ω1, we have

|x| = r∣∣∣x

r

∣∣∣ ≤ r|ρ(x/r)| = r|ρ(x)|

and then|x|ρ(x)

≤ r. Hence|x|ρ(x)

≤ µΩ1(x).

On the other hand, if r >|x|ρ(x)

, then |x| < rρ(x) = r|y(x)| and thus |xr | < |y(x)| with

y(x) ∈ Γ1. By the -strictly- convexity of Ω1, we have 1rx ∈ Ω1, and thus µΩ1(x) ≤ r.

We conclude that µΩ1(x) = |x|ρ(x) , and hence φ0 = µ2

Ω1.

Since Ω1 is convex, we have φ0 is the square of a convex function, and hence it is convex.

Therefore D2(φ0) is positive semi-definite at every point x ∈ Rn \ 0, and thus, we have to

prove that

det(D2(φ0)) 6= 0 in Rn \ 0. (II.11)

Now, take x, v ∈ Rn \ 0, and define g(t) = φ0(x + tv). Then g depends only on the

restriction of φ0 at the space Π = 〈x, v〉. Moreover, by definition of ρ, it is not difficult to

63


see that ρ|Π = ρ′, where ρ′ is the function defined in Π as in (II.6) relative to the closed curve

∂(Π ∩ Ω1), wich by hypothesis is strongly convex. Then g(t) = φ′0(x+ tv), with φ′0 = |x|2

ρ′ .

Taking in account thatdg

dt(0) = D2(φ0)(x)(v, v), we conclude that it suffices to prove

(II.11) in the two-dimensional case.

Until the end of this proof we set n = 2, and we work in polar coordinates. The expression

for the Hessian matrix of second derivatives in polar coordinates is

D2(φ) = QθH(φ)QTθ

where Qθ is the rotation matrix by angle θ, and

H(φ) =

∂2φ

∂r21

r

(∂2φ

∂r∂θ− 1

r

∂φ

∂θ

)

1

r

(∂2φ

∂r∂θ− 1

r

∂φ

∂θ

)1

r2∂2φ

∂θ2+

1

r

∂φ

∂r

.

Now, we have that

φ0(θ, r) =r2

ρ(θ)2

and one can notice that φ0 is well defined and smooth for all r > 0. It is also clear that ρ is

constant with respect to r and only depends on θ such that ∂ρ∂r = 0. Hence, we have that

H(φ0) =2

ρ2

1 −ρθ

ρ

−ρθ

ρ

1

ρ2(3ρ2

θ − ρρθθ + ρ2)

, (II.12)

where we have denoted ρθ = ∂ρ∂θ and so on.

We will use the following formula (see for example [30]) concerning curves in the plane:

Lemma 9. Let γ be a C2 curve in the plane parameterized in polar coordinates by its angle,

that is

γ(θ) = (r(θ) cos θ, r(θ) sin θ),

64


then the curvature of γ, at the point γ(θ) is given by the formula

κγ(θ) =r2 + 2r2θ − rrθθ

(r2 + r2θ)3/2

.

Since the polar parametrization of Γ1 is precisely the above function with r(θ) = ρ(θ) and

Ω1 is strongly convex, we obtain

κΓ1

(θ) =ρ2 + 2ρ2

θ − ρρθθ

(ρ2 + ρ2θ)

3/2> 0 ∀θ ∈ [0, 2π[. (II.13)

From (II.12) and (II.13) we have

det(H(φ)) =2

ρ4

(3ρ2

θ − ρρθθ + ρ2 − ρ2θ

)

=2

ρ4(ρ2 + ρ2

θ)3/2κ

Γ1> 0.

and then we get the desired result.

From the previous proof we can also deduce the following estimate on the laplacian of φ0,

which will be used in the Carleman estimate.

Corollary 6. If Ω1 is a strongly convex domain in Rn, then

∆φ0(x) >2

ρ2(x)

for all x ∈ Rn \ 0.

Proof: Let x ∈ Rn \ 0. Without loss of generality, we can suppose x ∈ 〈e1〉, where e1 =

(1, 0, . . . , 0) ∈ Rn (more precisely, we regard the Hessian matrix of φ0 in x after be factorized

by a sequence of n− 1 rigid rotations, given by the angles of x. The laplacian is the trace of

this matrix).

Then we get∂2φ0

∂x21

(x) =∂2φ0

∂r2(x) =

2

ρ2(x)(II.14)

65


Now, for each j = 2, . . . , n we can parameterize the plane 〈x, ej〉 in polar coordinates

as we have done en the previous proof. Then we have

∂2φ0

∂x2j

(x) =1

r2∂2φ0

∂θ2(x) +

1

r

∂φ0

∂r

=6

ρ4ρ2

θ −2

ρ3ρθθ +

2

ρ2

=2

ρ4

(3ρ2

θ − ρρθθ + ρ2)

=2

ρ4

(κj,Γ1(y(x))(ρ

2 + ρ2θ)

3/2 + ρ2θ

)(II.15)

where κj,Γ1 denotes the curvature of the curve given by the intersection of Γ1 with the plane

〈e1, ej〉. By hypothesis on Ω1 we have κj,Γ1 > 0 at every point.

Hence, from (II.14) and (II.15) we have that

∆φ0(x) =2

ρ2

1 +

n∑

j=2

1

ρ2(κj,Γ1(y(x))(ρ

2 + ρ2θ)

3/2 + ρ2θ)

≥ 2

ρ2

Throughout the rest of this part of the thesis, we shall use the following notation:

M = M11Q1 +M21Q2, a = a21Q1 + a11Q2, c(x) =a

ρ(x)2,

Ω0 = Ω1 ∪ Ω2, Ωx0 = Ω0 \Bε(x0),

Q0 = Ω0 × (−T, T ), Qx0 = Ωx0 × (−T, T ).

In order to the function φ satisfies the transmission condition (II.4), we need the following

Assumption 1

M1 −M2 = a1 − a2. (II.16)

Now, we prove the

66


Proposition 20. If Ω1 is a strongly convex domain of class C3, we can take ε, δ > 0 such

that:

(a) |∇φ| ≥ δ > 0 in Qx0 = (Ω1 ∪ Ω2) \Bε(x0) × (−T, T )

(b) ∇φ1(x, t) · ν1(x) ≥ δ > 0, ∀ (x, t) ∈ Σ1

where ν1 is the unit outward normal vector to Ω1.

If additionally (II.16) is satisfied, we also have:

(c) φ1(x, t) = φ2(x, t) = a2 − βt2 +M1 ∀ (x, t) ∈ Σ1

(d) a1∂α1+α2

∂xα11 ∂xα2

2

φ1(x, t) = a2∂α1+α2

∂xα11 ∂xα2

2

φ2(x, t) for all (x, t) ∈ Σ1 and α1, α2 ∈ N ∪ 0with α1 + α2 ≤ 3.

(e) ∆φ(x, t) ≥ 2c(x) ∀(x, t) ∈ Qx0 .

(f) D2(φ)(X,X) ≥ δ1|X|2 in Qx0 ∀X ∈ Rn for some δ1 > 0.

Proof: We have

∇φ = 2c(x)(x − x0) + |x− x0|2∇c(x).

By definition, ρ(x) (thus also c(x)) is constant in the direction of x− x0. Therefore

(x− x0) · ∇c(x) = 0

and

|∇φ|2 = 4c2(x)|x− x0|2 + |x− x0|4|∇c(x)|2

≥ 4c2(x)|x− x0|2

≥ 4

(a

diam(Ω)2

)2

ε2 in Ωx0

and (a) is proved.

Now, it is clear that φ1(x, t) = a2 − βt2 +M1 for each (x, t) ∈ Σ1, so Γ1 × t is a level

curve of φ1(·, t) for each t ∈ [−T, T ]. Since φ1(x, t) < a2 −βt2 +M1 < φ1(y, t) for any x ∈ Ω1

and y ∈ Ω2, we have ∇φ1 = |∇φ1|ν1 on Σ1, and thus (a) implies (b).

67


By definition ρ(x) = |x− x0| for all x ∈ Γ1, hence (c) is simply deduced from (II.16).

Without lost of generality, we can take x0 = 0. Writing ρ in polar coordinates, Γ1 can be

parameterized by

γ(θ) = (ρ(θ) cos θ, ρ(θ) sin θ). (II.17)

and then ρ is a C3 function. If D =∂α1+α2

∂xα11 ∂xα2

2

with α1 + α2 ≤ 3, we get

a1Dφ1(x, t) = a1a2D

( |x− x0|2ρ(x)2

)= a2Dφ2(x, t)

for all (x, t) ∈ Σ1 and (d) is proved.

Inequality (e) follows directly from Corollary 6, and from Proposition 19 we have that

D2(φ)) is positive definite. Since Ωx0 is compact, this implies (f) and the proof of Lemma 20

is complete.

We introduce now the last hypothesis we will need in order to get the Carleman inequality:

Assumption 2

β < mın

mına1, a2δ1

2,M

T 2

(II.18)

and we take γ ∈ (0, 1) such that

Assumption 3

γ >2β

β + a1a2diam(Ω)2

and (II.19)

Assumption 4

γ <2mına1, a2δ1

2β + maxa1, a2‖∆φ‖2L∞(Ω0)

. (II.20)

Remark 8. 1. We have to take M large enough in order to (II.18) and the hypothesis of

the inverse problem (see Theorem 5) become compatible.

2. Taking β small enough, (II.19) and (II.20) become compatible. But, as we will see in the

next section, the smaller is β, the bigger need to be the inversion time for the inverse

problem (see Theorem 5).

68


3. Actually, the optimal δ1 is the first eigenvalue of D2(φ). Having an explicit expression

for it could help to a better choice of β.

4. From the hypothesis of Theorem 3 we have a2 < a1 and the maximum and the minimum

in (II.18) and (II.20) are known.

Pseudoconvexity.

Let us denote by x = (t, x) the coordinates in [0, T ]×Ω and by ξ = (s, ξ) the corresponding

Fourier variables. Then the symbol of the hyperbolic operator P = ∂2t − a∆ is p = s2 − a|ξ|2.

The C2 function ψ = ψ(x, t) is called pseudoconvex with respect to P if it satisfies the

following conditions: (see [1], [36]).

p, p, ψ(x, ξ) > 0 on ξ 6= 0 : p(x, ξ) = p, ψ(x, ξ) = 0 (II.21)

and

p, ψ, p(x,∇ψ) > 0 on p(x,∇ψ) = 0. (II.22)

where p, q = ∇ξp · ∇xq −∇ξq · ∇xp is the usual Poisson bracket.

Let us see that Proposition 20 and assumption (H2) implies that φ is pseudoconvex with

respect to P in Ω1 ∪ Ω2\Bε(x0).

In the set ξ 6= 0 : p(x, ξ) = p, φ(x, ξ) = 0 we have s2 = a|ξ|2, and then:

p, p, φ(x, ξ) = −8βs2 + 4a2D2(φ)ξ · ξ

= −8βa|ξ|2 + 4a2D2(φ)ξ · ξ

= −8βa|ξ|2 + 4a2δ1|ξ|2

≥ 4|ξ|2a(−2β + aδ1) > 0

which proves (II.21).

69


On the other hand, in the set p(x,∇φ) = 0 we have 4β2t2 = a|∇φ|2, and subsequently:

p, φ, p(x,∇φ) = −32β3t2 + 4a2D2(φ)∇φ · ∇φ

= −8βa|∇φ|2 + 4a2D2(φ)∇φ · ∇φ

≥ −8βa|∇φ|2 + 4a2δ1|∇φ|2

= 4a|∇φ|2(aδ1 − 2β) > 0 in Ω1 ∪ Ω2\Bε(x0)

so (II.22) is also true and then φ is pseudoconvex.

Therefore, a Carleman estimate like (II.29) (without explicit dependence on the parameter

λ) can be derived directly (see [64], [1]). However, here we have the additional difficulty derived

from the fact that we deal with non-zero boundary values, that is, the traces of the solutions

on the interface Γ1.

Variable principal coefficient.

We can also prove a Carleman estimate in the more general case a(x) = a1(x)1Ω1 +

a2(x)1Ω2 with aj ∈ C1(Ωj), j = 1, 2, if each aj is constant in the interface Γ1, and under some

hypothesis on ∇a (similar to those of [37]). More precisely, if we check the pseudoconvexity

condition in this case, we will have that φ is pseudoconvex with respect to the operator

∂tt − div(a∇u) (in each domain Qj) if there exists θ ∈ (0, 1) such that:

|∇φ · ∇aj|2aj

< δ1(1 − θ), j = 1, 2 (II.23)

β <

√aj√

aj + T |∇aj |

(ajδ1θ

2

), j = 1, 2. (II.24)

It is easy to check that these hypothesis are compatible with the assumptions of Theorem 4.

Indeed, for T sufficiently large, there exists β satisfying both β > a1/T2 and (II.24). Nevert-

heless, in order to construct the weight function as we have done above (and deal with the

traces of the solutions on the interface Γ1), it is crucial for each function aj to be constant

on the interface.

70


3.2 Development of the L2-product .

We denote

L = ∂2t − a∆ and E(z) = |zt|2 − a|∇z|2.

As usual, we do the change of variables

ϕ = eλφ , λ > 0, w = esϕu , s > 0, P (w) = esϕL(e−sϕw) (II.25)

and after algebraical computations, we split P (w) into three terms as follows:

P (w) = P1(w) + P2(w) +R(w),

where for some fixed real number γ ∈ (0, 1)

P1(w) = wtt − a∆w + s2λ2ϕ2E(φ)w,

P2(w) = (γ − 1)sλϕL(φ)w − sλ2ϕE(φ)w − 2sλϕ(φtwt − a∇φ · ∇w),

R(w) = −γsλϕL(φ)w.

We will write P φ, P φ1 , P φ

2 , etc. if we want to make the dependence on φ explicit.

We will now develop the L2-product of P1(w) and P2(w). We will do formal computations,

by writing generically φ for the weight function and Q for the domain with boundary Σ.

We set 〈P1(w), P2(w)〉L2 =3∑

i,j=1Ii,j, where Ii,j is the integral of the product of the ith-term

in P1(w) and the jth-term in P2(w). Therefore,

I1,1 = −sλ(γ − 1)

∫∫

QϕL(φ)|wt|2 +

sλ2(γ − 1)

2

∫∫

Q|w|2ϕ(φtt + λ|φt|2)L(φ),

I1,2 = sλ2

∫∫

Q|wt|2ϕE(φ) − sλ2

∫∫

Q|w|2ϕ|φtt|2 −

5sλ3

2

∫∫

Q|w|2ϕφtt|φt|2

+sλ3

2

∫∫

Q|w|2aϕφtt|∇φ|2 −

sλ4

2

∫∫

Q|w|2ϕ|φt|2E(φ),

71


I1,3 = sλ

∫∫

Q|wt|2ϕ(φtt + λ|φt|2) − 2sλ2

∫∫

Qwtφtϕa∇w · ∇φ

+sλ

∫∫

Q|wt|2ϕa(∆φ+ λ|∇φ|2) − sλ

∫∫

Σ|wt|2aϕ∇φ · ν,

I2,1 = −sλ(γ − 1)

∫∫

ΣϕL(φ)wa

∂w

∂ν+ sλ(γ − 1)

∫∫

Qa|∇w|2ϕL(φ)

−sλ2γ − 1

2

∫∫

Q|w|2ϕaL(φ)(λ|∇φ|2 + ∆φ) − sλ

γ − 1

2

∫∫

Q|w|2aϕ∆(L(φ))

−sλ2(γ − 1)

∫∫

Q|w|2aϕ(∇φ · ∇L(φ)) + sλ

γ − 1

2

∫∫

Σ|w|2aϕ∇L(φ) · ν

+sλ2γ − 1

2

∫∫

Σ|w|2aϕL(φ)

∂φ

∂ν,

I2,2 = sλ2

∫∫

ΣaϕE(φ)w∇w · ν − sλ2

2

∫∫

Σa|w|2(λϕE(φ)∇φ + ϕ∇E(φ)) · ν

+sλ3

2

∫∫

Q|w|2aϕE(φ)(∆φ + λ|∇φ|2) − 2sλ3

∫∫

Q|w|2a2ϕD2(φ)(∇φ,∇φ)

+sλ2

2

∫∫

Q|w|2aϕ∆(E(φ)) − sλ2

∫∫

Q|∇w|2aϕE(φ),

I2,3 = sλ

∫∫

Q|∇w|2aϕL(φ) + sλ2

∫∫

Q|∇w|2aϕE(φ) + 2sλ2

∫∫

Qa2ϕ|∇φ · ∇w|2

−2sλ2

∫∫

Qaϕφtwt∇w · ∇φ+ 2sλ

∫∫

Qa2ϕD2(φ)(∇w,∇w)

+sλ

∫∫

Σ|∇w|2a2ϕ∇φ · ν + 2sλ

∫∫

Σaϕ(φtwt − a∇φ · ∇w)

∂w

∂ν,

I3,1 = s3λ3(γ − 1)

∫∫

Q|w|2ϕ3L(φ)E(φ),

I3,2 = −s3λ4

∫∫

Q|w|2ϕ3E(φ)2,

I3,3 = s3λ3

∫∫

Q|w|2ϕ3E(φ)L(φ) + 2s3λ3

∫∫

Q|w|2ϕ3(|φt|2φtt + a2D2(φ)(∇φ,∇φ))

+3s3λ4

∫∫

Q|w|2ϕ3E(φ)2 + s3λ3

∫∫

Σa|w|2ϕ3E(φ)

∂φ

∂ν.

72


Gathering all these terms,we get

〈P1(w), P2(w)〉L2(Q) = 2sλ

∫∫

Q|wt|2ϕφtt − γsλ

∫∫

Q|wt|2ϕL(φ)

+2sλ2

∫∫

Qϕ(|wt|2|φt|2 − 2wtφta∇w · ∇φ+ a2|∇φ · ∇w|2

)

+2sλ

∫∫

Qa2ϕD2(φ)(∇w,∇w)

+γsλ

∫∫

Qa|∇w|2ϕL(φ) + 2s3λ4

∫∫

Q|w|2ϕ3E(φ)2

+2s3λ3

∫∫

Q|w|2ϕ3(|φt|2φtt + a2D2(φ)(∇φ,∇φ))

+γs3λ3

∫∫

Q|w|2ϕ3L(φ)E(φ)

+X + J,

where J is the sum of all the boundary terms:

J = sλ

∫∫

Σ

(a2ϕ|∇w|2 ∂φ

∂ν− 2a2ϕ(∇φ · ∇w)

∂w

∂ν

)

+sλγ − 1

2

∫∫

Σ|w|2ϕa∇L(φ) · ν

+sλ2γ − 1

2

∫∫

Σ|w|2ϕaL(φ)

∂φ

∂ν

−sλ(γ − 1)

∫∫

Σwa

∂w

∂νϕL(φ) + sλ2

∫∫

ΣaϕE(φ)w

∂w

∂ν

−sλ3 1

2

∫∫

Σ|w|2ϕE(φ)a

∂φ

∂ν

−sλ2 1

2

∫∫

Σ|w|2aϕ∇E(φ) · ν

+2sλ

∫∫

Σaϕφtwt

∂w

∂ν− sλ

∫∫

Σ|wt|2ϕa

∂φ

∂ν

+s3λ3

∫∫

Σ|w|2ϕ3E(φ)a

∂φ

∂ν

and X is a the sum of the remaining terms, in such a way that:

|X| ≤ Csλ3

∫∫

Qϕ3|w|2

73


In the sequel, we denote by Aj , j = 1, ..., 8 the first eight integrals we have listed in the

product of P1(w) by P2(w). Thus, we have

〈P1(w), P2(w)〉L2(Q) =8∑

j=1

Aj +X + J. (II.26)

3.3 The inequality

Here we shall state the main result of this part of the thesis, the Carleman estimate for

the transmission wave equation. In order to do that, we will state first some definitions.

Given an open set U ⊂ Rn and T > 0, we define the following norm in H1(U × (−T, T ))

‖g‖2U,ϕ

= sλ

∫ T

−T

∫

U(|gt|2 + |∇g|2)ϕ+ s3λ3

∫ T

−T

∫

U|g|2ϕ3, (II.27)

and we define the space

X = u ∈ L2(−T, T ;L2(Ω)) | Luj ∈ L2(−T, T ;L2(Ωj)), j = 1, 2; u|Σ = 0,

u(±T ) = ut(±T ) = 0, and u satisfies (II.4).

Finally, we define

Σφ+ = (x, t) ∈ Σ : ∇φ(x, t) · ν(x) > 0. (II.28)

We have the following global Carleman estimate:

Theorem 3. Let Ω ⊂ Rn be an open bounded set. Assume Ω1 is a strongly convex domain

of class C3, and a1 > a2 > 0. Let xk ∈ Ω1, k = 1, 2 and let φk, ϕk, wk be the corresponding

functions defined for xk as we did before for x0 in (II.5), (II.6), (II.9) and (II.25). Let ν be

the unit outward normal to Ω. Then there exists C > 0, s0 > 0 and λ0 > 0 such that

2∑

k=1

(∥∥∥P φk

1 (wk)∥∥∥

2

L2(Q)+∥∥∥P φk

2 (wk)∥∥∥

2

L2(Q)+∥∥∥wk

∥∥∥2

Q,ϕk

)

≤ C

2∑

k=1

(∥∥∥P φk

(wk)∥∥∥

2

L2(Q)+ sλ

∫∫

Σφk

+

ϕk

∣∣∣∣a2∂wk

∂ν

∣∣∣∣2) (II.29)

74


for all u ∈ X, λ ≥ λ0 and s ≥ s0.

Notice that in the right-hand side of (II.29) we have the term

∥∥∥P φk

(wk)∥∥∥

2

L2(Q)=

∫∫

Qe2sϕk |Lu|2 .

Since we consider an equation given by the operator Lp = ∂tt − a∆ + p, it is important to

note that the same estimate with right-hand side equal to

∫∫

Qe2sϕk |Lpu|2 is also true for all

potentials p such that |p|L∞(Ω) ≤ m, with m already fixed. Indeed,

|Lu|2 ≤ 2|Lpu|2 + 2m2|u|2

and taking s large enough, the left hand side of the Carleman estimate of Theorem 3 can

absorb the term 2Cm2

∫∫

Qe2sϕ|u|2. That is, we have the following result.

Corollary 7. Under the hypothesis and notations of Theorem 3, given m ∈ R, there exists

C > 0 (depending on m), s0 > 0 and λ0 > 0 such that for all p ∈ L∞(Ω) with ‖p‖L∞(Ω) ≤ m

we have

2∑

k=1

(∥∥∥P φk

1 (wk)∥∥∥

2

L2(Q)+∥∥∥P φk

2 (wk)∥∥∥

2

L2(Q)+∥∥∥wk

∥∥∥2

Q,ϕk

)

≤ C

2∑

k=1

(∫∫

Qe2sϕk |Lpu|2 + sλ

∫∫

Σφk

+

ϕk

∣∣∣∣a2∂wk

∂ν

∣∣∣∣2) (II.30)

for all u ∈ X, λ ≥ λ0 and s ≥ s0.

4 Proof of the Carleman estimate.

In this section we shall prove Theorem 3. We take β, γ and M satisfying the Assumption

1 to 4, and φ as the corresponding weight function. We assume throughout all this section

the hypothesis of Theorem 3.

Recall the notation Q0 = (Ω1 ∪ Ω2) × (−T, T ) and Qx0 = (Ω0 \ Bε(x0)) × (−T, T ). We

apply the computations we have made in subsection 3.2 to w = esϕu with u ∈ X in each one

75


of the open sets Q1 and Q2. Adding the terms resulting in both cases, we have (recall that

u(±T ) = ut(±T ) = 0 for all u ∈ X ):

〈P1(w), P2(w)〉L2(Q0) =8∑

j=1

Aj,Q0(w) +XQ0(w) + JΣ1(w) + JΣ(w), (II.31)

where we have written Aj,Q0 instead of the integral Aj given in subsection 3.2 taken in the

set Q0, et cetera.

The proof of Theorem 3 is based on the next three facts:

The sum of the Aj-integrals in Qx0 can be minored.

The sum of terms in the interface given by JΣ1 is nonnegative, and:

We can introduce a second weight function centered at a point different to x0 in order

to deal with the integrals in Bε(x0).

The key points in each step of the proof are based on the properties of φ listed in Propo-

sition 20.

4.1 The interior

Proposition 21. There exist δ > 0, C > 0 and λ0 > 0 such that:

8∑

j=1

Aj,Q0(w) ≥ δ‖w‖Ω0 ,ϕ − C‖w‖Bε(x0),ϕ

for all λ ≥ λ0, for all u ∈ X.

Proof: We arrange the terms into four groups:

1. A1,Q0 +A2,Q0 = sλ

∫∫

Q0

|wt|2ϕ(−γL(φ) − 4β). For all (x, t) in Qx0 we have:

−γL(φ) − 4β = γ(2β + a∆φ) − 4β

≥ γ(2β + 2ac(x)) − 4β by Proposition 20

≥ γ(2β + 2aa

diam(Ω)2) − 4β by definition of c(x)

= δ > 0 by (II.19).

76


Therefore:

A1,Q0 +A2,Q0 ≥ δ1sλ

∫∫

Ωx0

|wt|2ϕ− Csλ

∫∫

Bε(x0)|wt|2ϕ

2. A3,Q0 = 2sλ2

∫∫

Q0

ϕ(φtwt − a∇φ · ∇w)2 ≥ 0

3. A4,Qx0+A5,Qx0

= sλ

∫∫

Qx0

ϕ(a2D2(φ)(∇w,∇w) + γaL(φ)|∇w|2

). Then, using Propo-

sition 20 and (II.20), we obtain

A4,Qx0+A5,Qx0

≥ sλ

∫∫

Qx0

ϕ(2a2δ1|∇w|2 + γaL(φ)|∇w|2

)

≥ sλ

∫∫

Qx0

ϕa (2aδ1 − γ(2β + a∆φ)) |∇w|2

≥ sλ

∫∫

Qx0

ϕa (2aδ1 − γ(2β + a‖∆φ‖L∞)) |∇w|2

≥ sλδ2

∫∫

Qx0

ϕ|∇w|2

Therefore

A4,Q0 +A5,Q0 ≥ δ2sλ

∫∫

Ωx0

ϕ|∇w|2 − Csλ

∫∫

Bε(x0)ϕ|∇w|2

4.8∑

j=6Aj,Q0 = s3λ3

∫∫

Q0

|w|2ϕ3Fλ(φ) where

Fλ(φ) = 2λE(φ)2 + 2|φt|2φtt + 2a2D2(φ)(∇φ,∇φ) + γL(φ)E(φ)

= 2λE(φ)2 + γL(φ)E(φ) − 16β3t2 + 2a2D2(φ)(∇φ,∇φ)

= 2λE(φ)2 + (γL(φ) − 4β)︸︷︷︸b(x)<0

E(φ) − 4βa|∇φ|2 + 2a2D2(φ)(∇φ,∇φ)

From Proposition 20 and (II.18), there exists d0 > 0 such that for all (x, t) ∈ Qx0 we

77


have

Fλ(φ) ≥ 2λE(φ)2 + b(x)E(φ) − 4βa|∇φ|2 + 2a2δ1|∇φ|2

≥ 2λE(φ)2 + b(x)E(φ) + a(2aδ1 − 4β)|∇φ|2

≥ 2λE(φ)2 − ‖b‖∞|E(φ)| + a(2aδ1 − 4β)|∇φ|2

≥ 2λE(φ)2 − ‖b‖∞|E(φ)| + d0

=

f1,λ(E(φ)) if E(φ) > 0

f2,λ(E(φ)) if E(φ) < 0

where

fj,λ : R −→ R

x 7−→ 2λx2 + (−1)j‖b‖∞x+ d0.

As d0 > 0, there exists λ0 > 0 such that for all λ ≥ λ0

mınR

(fj,λ) ≥ d0

2> 0 j = 1, 2.

Thus, for each λ ≥ λ0 we have

8∑

j=6

Aj,Q0 ≥ δ3s3λ3

∫∫

Qx0

|w|2ϕ3 − Cs3λ3

∫∫

Bε(x0)|w|2ϕ3.

By collecting all the terms Aj,Q0 together, we conclude the proof of Proposition 21.

4.2 The traces at the boundary

Since the interface Γ1 is a common boundary of Ω1 and Ω2, the term JΣ1 in (II.31) is

the sum of the integrals comming from each domain: JΣ1 = JΣ1(w1) + JΣ1(w2). We have the

following result:

78


Proposition 22. Suppose 0 < a2 < a1. Then there exists s0 > 0 such that

JΣ1 = JΣ1(w1) + JΣ1(w2) ≥ 0 ∀s ≥ s0

for all u ∈ X.

Proof: We enumerate the ten integrals arising in (II.26) associated with the common boundary

Σ1, and we denote by Ji the sum of the i-th integral in (II.26) which comes from Ω1 with the

respective one of Ω2: Ji = Ji(w1) + Ji(w2).

In order to prove the inequality, we arrange the terms into three groups. In each case, we

use Proposition 20 and the fact that w satisfies the transmission conditions.

1. Is not difficult to see that Jk = 0 for each k ∈ 3, 4, 7, 8, 9. Indeed, from (d) of

Proposition 20 we get L1(φ1) = L2(φ2) and a1∇E1(φ1) = a2∇E2(φ2) on Σ1, and the

desired result follows.

Now, let us denote by g the real function defined in Σ1 by

g(x, t) := E1(φ1) − E2(φ2) =

(1

a2− 1

a1

) ∣∣∣∣a1∂φ

∂ν1

∣∣∣∣2

Since a2 < a1, we have g > 0 in Σ1.

Thus we can prove:

2. J2 + J6 + 12J10 ≥ 0 ∀s ≥ s0

Indeed:

−J2 − J6 = sλ1 − γ

2

∫∫

Σ1

|w|2ϕa∇L(φ) · ν

+sλ3 1

2

∫∫

Σ1

|w|2ϕ(a1∂φ1

∂ν1

)g(x, t)

≤ 1

2J10

for all s ≥ s0, since ϕ ≥ 1.

3. J1 + J5 + 12J10 ≥ 0 ∀s ≥ s0

79


By construction, φ is constant on each level Γ1 × t of the interface (Proposition 20).

Thus

∇φj · ∇wj =∂φj

∂νj

∂wj

∂νjin Σ1 for j = 1, 2. (II.32)

Moreover, since w satisfies (II.4) we have

∣∣∣∣∂w1

∂τ1

∣∣∣∣ =

∣∣∣∣∂w2

∂τ2

∣∣∣∣ in Σ1 ∀u ∈ X.

Hence:2∑

j=1

∣∣∣∣∂wj

∂τj

∣∣∣∣2

a2jϕj

∂φj

∂νj=

∣∣∣∣∂w1

∂τ1

∣∣∣∣2

ϕ1

(a1∂φ1

∂ν1

)(a1 − a2) > 0 (II.33)

From (II.32) and (II.33) we get:

J1 ≥ −sλ∫∫

Σ1

(∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

ϕ1∂φ1

∂ν1+

∣∣∣∣a2∂w2

∂ν2

∣∣∣∣2

ϕ2∂φ2

∂ν2

)

= −sλ∫∫

Σ1

∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

ϕ1

(∂φ1

∂ν1+∂φ2

∂ν2

)

= sλ

∫∫

Σ1

∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

ϕ1

(1

a2− 1

a1

)(a1∂φ1

∂ν1

).

On the other hand,

−J5 = −sλ2

∫∫

Σ1

w1ϕ1

(a1∂w1

∂ν1

)g(x, t)

≤ 1

2s2λ3

∫∫

Σ1

|w1|2ϕ1g +1

2λ

∫∫

Σ1

ϕ1g

∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

=1

2s2λ3

∫∫

Σ1

|w1|2ϕ1g

+1

2λ

∫∫

Σ1

∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

ϕ1

(1

a2− 1

a1

) ∣∣∣∣a1∂φ

∂ν1

∣∣∣∣2

≤ 1

2δs2λ3

∫∫

Σ1

|w|2ϕ1g

(a1∂φ1

∂ν1

)

+C

2λ

∫∫

Σ1

∣∣∣∣a1∂w1

∂ν1

∣∣∣∣2

ϕ1

(1

a2− 1

a1

)(a1∂φ

∂ν1

)

≤ 1

2J10 + J1 ∀s ≥ s1.

80


Proposition (22) is proved.

Now, concerning the boundary Σ, since we deal with functions w such that w2 = 0 in Σ,

we have

JΣ = J1(w2) = −sλ∫∫

Σϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2(∂φ

∂ν

)

≥ −sλ∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2(∂φ

∂ν

)

≥ −sλ∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2 ∥∥∥∥∂φ

∂ν

∥∥∥∥L∞(Σ)

(II.34)

= −sλC∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2

,

where we have defined Σ+ = (x, t) ∈ Γ : ∇φ(x, t) · ν(x) > 0.

4.3 Proof of Theorem 3.

Here we use the analysis we have done in the previous sections, in order to prove Theorem

3.

From (II.31), (II.34) and Propositions 21 and 22, there exist s0, λ0, C ∈ R such that for

each s ≥ s0 and λ ≥ λ0 we have

‖w‖2Ω0,ϕ

− C ‖w‖2Bε(x0),ϕ

+ XQ0

−sλC∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2

≤ C 〈P1(w), P2(w)〉L2(Q0) (II.35)

Adding C2

(|P1(w)|2L2(Q0) + |P2(w)|2L2(Q0)

)at both sides of (II.35) we obtain

|P1(w)|2L2(Q0) + |P2(w)|2L2(Q0) + ‖w‖2Ω0,ϕ

−C ‖w‖2Bε(x0),ϕ

+XQ0 − sλC

∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2

≤ C |P1(w) + P2(w)|2L2(Q0)

= C |P (w) −R(w)|2L2(Q0) (II.36)

≤ C(|P (w)|2L2(Q0)

+ |R(w)|2L2(Q0)

)

81


Thanks to (II.18) we have λ ≤ Cϕ for λ large enough. Therefore

|XQ0| + C|R(w)|2L2(Q) ≤ Csλ3

∫∫

Qϕ2|w|2 ∀λ ≥ λ1

≤ 1

2‖w‖2

Ω0,ϕ∀s ≥ s1.

(II.37)

From (II.36) and (II.37) we get, for all s ≥ maxs0, s1, λ ≥ maxλ0, λ1:

|P1(w)|2L2(Q0) + |P2(w)|2L2(Q0)+ ‖w‖2

Ω0,ϕ≤ C|P (w)|2L2(Q0)

+C ‖w‖2Bε(x0),ϕ

+ sλC

∫∫

Σ+

ϕ

∣∣∣∣a2∂w

∂ν

∣∣∣∣2

.(II.38)

In the last step we will remove the integral in Bε(x0) from the right hand side of (II.38).

In order to do that, first remark that x0 can be arbitrarily chosen in Ω1 since Ω1 is convex.

Thus, we can take two different points in Ω1 and we have the two respective inequalities

given by (II.38). Now, we will show that the left hand side of each inequality can absorb the

term ‖ · ‖Bε(x0) from the other inequality provided that ε is small and λ is large enough:

Denote by x1, x2 two points in Ω1, and φ1, φ2 their respective weight functions. In order

to have ‖ · ‖Bε(x1),ϕ1 absorbed by the term ‖w2‖Ω0,ϕ2 it suffices that

Cϕ1 <1

2ϕ2 in Bε(x1)

i.e.

eλ(φ2−φ1) > 2C in Bε(x1)

Thus, if we show that it is possible to have φ2 − φ1 > δ > 0 in Bε(x1) by taking λ large

enough we are done.

In fact, let be d = 12 |x1 − x2| and assume that ε < d. Then, for all x ∈ Bε(x1) we have:

φ1(x, t) ≤ a

ρ21

ε2 − βt2 +M

≤ a

α21

ε2 − βt2 +M, (II.39)

82


where α1 = d(x1,Γ1) > 0.

In the same way, if we denote D2 = maxy∈Γ1

d(y, x2), we get

φ2(x, t) ≥ a

ρ22

d2 − βt2 +M

≥ a

D22

d2 − βt2 +M ∀x ∈ Bε(x1).

Consequently, we have

φ2 − φ1 ≥ a

(d2

D22

− ε2

α21

)∀x ∈ Bε(x1). (II.40)

It is clear that an analogous result is true by interchanging x1 and x2 (now with α2 and

D1). Thus, taking ε < mın(

dα1D2, dα2

D1

)we can absorb the desired terms in the inequality and

Theorem 3 is proved.

5 Application to the inverse problem

In this section we apply the Carleman inequality of Theorem 3 to solve an inverse problem

for the transmission wave equation.

Given the equation

utt − div(a(x)∇u) + p(x)u = 0 (x, t) ∈ Ω × (0, T )

u = 0 (x, t) ∈ Γ × (0, T )

u(0) = u0 x ∈ Ω

ut(0) = u1 x ∈ Ω,

(II.41)

we shall prove the well-posedness of the inverse problem consisting of retrieving the potential

p involved in equation (II.41), by knowing the flux (the normal derivative) of the solution u(p)

of (II.1) on the boundary. We will prove uniqueness and stability of the non linear inverse

83


problem characterized by the non linear application

p|Ω 7−→ a2∂u

∂ν

∣∣∣∣Γ×(0,T )

. (II.42)

More precisely, we will answer the following questions.

Uniqueness :

Does the equality ∂u(q)∂ν = ∂u(p)

∂ν on Γ × (0, T ) imply q = p on Ω ?

Stability :

Is it possible to estimate (q − p)|Ω by(

∂u(q)∂ν − ∂u(p)

∂ν

)∣∣∣Γ×(0,T )

in suitable norms ?

The idea is to reduce the nonlinear inverse problem to some perturbed inverse problem

which will be solved with the help of a global Carleman estimate. More precisely, we will

give a local answer about the determination of p, working first on the perturbed version of

the problem. Assuming that p ∈ L∞ is a given function, we are concerned with the stability

around p. That is to say, p and u(p) are known while q and u(q) are unknown.

In order to sate the precise result about the inverse problem, let us define, for x1, x2 ∈ Ω1,

the quantities given by

αj = d(xj ,Γ1), Rj = sup|x− yj(x)| : x ∈ Ω2 (II.43)

for each j = 1, 2, where yj is defined in (II.5) with x0 = xj.

Also, set

D0 = max

R1 + α1

α1,R2 + α2

α2

. (II.44)

With the aid of the Carleman estimate we have shown in the previous section, we will

prove here the following result, which states the stability of the inverse problem.

Theorem 4. Assume Ω is bounded, Ω1 ⊂⊂ Ω is a strongly convex domain with boundary Γ1

of class C3 and a1 > a2 > 0. If T > 0, p ∈ L∞(Ω), u0 ∈ H10 (Ω), u1 ∈ L2(Ω) and r > 0 satisfy

T > D0

√a1β

84


|u0(x)| ≥ r > 0 a. e. in Ω, and

u(p) ∈ H1(0, T ;L∞(Ω))

then, given a bounded set U ⊂ L∞(Ω), there exists a constant

C = C(a1, a2,Ω1,Ω2, T, ‖p‖L∞(Ω), ‖u(p)‖H1(L∞),U , r) > 0 such that:

‖p − q‖L2(Ω) ≤ C

∥∥∥∥a2∂u(p)

∂ν− a2

∂u(q)

∂ν

∥∥∥∥H1(0,T ;L2(Γ))

for all q ∈ U , where u(p) and u(q) are the solutions of (II.41) with potential p and q, respec-

tively.

Remark 9. In section 5 is given an estimate for T0 in function of a1, a2, Ω1 and Ω2. See

Theorem 5.

Let us remark that as a direct consequence of the local stability of Theorem 4 we have

the following global uniqueness for our inverse problem:

Corollary 8. If u(p) and u(q) are two solutions of (II.1) for potentials p and q in L∞(Ω)

with u(p), u0, u1, a1, a2, Ω1, Ω2 and T satisfying the hypothesis of Theorem 4 and such that

∂u(q)∂ν = ∂u(p)

∂ν on Γ × (0, T ) then p = q.

5.1 The perturbed inverse problem.

In order to prove the local stability stated in Theorem 4, we follow the ideas of [14] and

[42]. For a principal coefficient a piecewise constant and p ∈ L∞(Ω), we consider the wave

equation

utt − div(a(x)∇u) + p(x)u = g(x, t) Ω × (0, T )

u = h Γ × (0, T )

u(0) = u0 Ω

ut(0) = u1 Ω.

(II.45)

If g ∈ L1(0, T ;L2(Ω)), h ∈ L2(0, T ;L2(Γ)) and u0 ∈ H10 (Ω), u1 ∈ L2(Ω), then [50, 22]

equation (II.45) has a unique weak solution u ∈ C([0, T ];H10 (Ω)) ∩ C1([0, T ];L2(Ω)) with

continuous dependence in initial conditions and such that ∂u∂ν ∈ L2(0, T ;L2(Γ)).

85


We will first consider a linearized version of the inverse problem, what means working on

the wave equation

ytt − div(a(x)∇y) + p(x)y = f(x)R(x, t) Ω × (0, T )

y = 0 Γ × (0, T )

y(0) = 0 Ω

yt(0) = 0 Ω

(II.46)

given p and R, and proving the stability of the application f |Ω 7−→ ∂y∂ν

∣∣∣Σ.

We shall prove the following result:

Theorem 5. With the hypothesis of Theorem 4, and T , β satisfying (II.16), (II.18), (II.19)

and (II.20), suppose that

‖p‖L∞(Ω) ≤ m

T > D0

√a1β

R ∈ H1(0, T ;L∞(Ω))

0 < r < |R(x, 0)| almost everywhere in Ω.

Then there exists C > 0 such that for all f ∈ L2(Ω), the solution y of (II.46) satisfies

‖f‖2L2(Ω) ≤ C

∥∥∥∥a2∂y

∂ν

∥∥∥∥2

H1(0,T ;L2(Γ))

.

Proof: For each f ∈ L2(Ω) and R ∈ H1(0, T ;L∞(Ω)), let y be the solution of (II.46). We take

the even extension of R and y to the interval (−T, T ). We call this functions in the same way,

and in this proof we denote Q = Ω × (−T, T ) and Σ = Γ × (−T, T ) the extended domains.

Therefore, z = yt satisfies the following equation:

ztt − div(a∇z) + pz = f(x)Rt(x, t) Q

z = 0 Σ

z(0) = 0 Ω

zt(0) = f(x)R(x, 0) Ω

(II.47)

86


and we have the usual energy estimate

‖z‖H1(−T,T ;L2(Ω)) ≤ C‖fRt‖L2(−T,T ;L2(Ω)) + C‖fR(0)‖L2(Ω)

that gives, since R ∈ H1(0, T ;L∞(Ω)),

‖z‖H1(−T,T ;L2(Ω)) ≤ C‖f‖L2(Ω)‖R‖H1(0,T ;L∞(Ω)) ≤ C‖f‖L2(Ω). (II.48)

In order to apply Theorem 3 and use the appropriate Carleman estimate, we need a

solution of the wave equation that vanishes at time t = ±T . Thus, for 0 < δ < T we take the

cut-off function θ ∈ C∞0 (−T, T ) such that

0 ≤ θ ≤ 1

θ(t) = 1, for all t ∈ (−T + δ, T − δ)

and we define v = θz. Then v satisfies:

vtt − div(a∇v) + pv = θ(t)f(x)Rt(x, t) + 2θtytt + θttyt Q

v = 0 Σ

v(0) = 0 Ω

vt(0) = f(x)R(x, 0) Ω

v(±T ) = vt(±T ) = 0 Ω.

(II.49)

Take j ∈ 1, 2, and let y be the function defined in (II.5) and φ the weight function,

corresponding to the point xj ∈ Ω1. Notice that

φ(x, t) ≤ φ(x, 0) ∀(x, t) ∈ (0, T ) × Ω. (II.50)

Moreover, by definition of ρ and a (see (II.6) and the definitions below) we also have

|x− xj |ρ(x)

≤ 1 +|x− y(x)|ρ(x)

≤ 1 +Rj

αj≤ D0

87


and then

φ(x, t) = a|x−xj|

2

ρ2(x)− βt2 +M

≤ aD20 − βt2 +M. (II.51)

Then, by the choice of T > D0

√a1β we get

φ(x,±T ) < M ≤ φ(x, 0). ∀x ∈ Ω (II.52)

Thus, taking δ small enough, it is also true that

φ(x, t) < M ≤ φ(x, 0). (II.53)

for all x ∈ Ω and t ∈ [−T,−T + δ] ∪ [T − δ, T ].

From now on, C > 0 will denote a generic constant depending on Ω, T , β, θ, x1, x2, δ, s0

and λ0 but independent of s > s0 and λ > λ0. We will occasionally use the notation ∂t for

the time derivative.

As in the proof of Theorem 3, we set ϕ = eλφ, wj = esϕvj and

P1(w) = wtt − a∆w + s2λ2ϕ2E(φ)w.

It is easy to check that

2∑

j=1

〈P1wj , ∂twj〉L2(Ωj×(0,T )) =1

2

∫

Ω|∂tw(0)|2 +X (II.54)

where X is a sum of negligible terms such that

2∑

j=1

〈P1wj, ∂twj〉L2(Ωj×(0,T )) ≥1

2

∫

Ω|∂tw(0)|2 −Cs2λ3

∫ T

0

∫

Ωϕ3|w|2.

Since we have wt(0) = esϕ(0)vt(0) = esϕ(0)f(x)R(x, 0) and |R(x, 0)| ≥ r, we get (recall that

88


Q = Ω × (−T, T ), Qj = Ωj × (−T, T ) and so on).

r2∫

Ωe2sϕ(0)|f |2 ≤ C

2∑

j=1

〈P1wj, ∂twj〉L2(Ωj×(0,T )) + s2λ3

∫∫

Qϕ3|w|2

.

In order to apply the Carleman estimate (Corollary 7) we consider both weight functions ϕ1

and ϕ2, corresponding to x1 and x2 ∈ Ω1 and we apply the previous estimates to wkj = esϕ

kvj

for j, k = 1, 2 and sum up the inequalities. We obtain, for s > s0 and λ > λ0, using Cauchy-

Schwarz inequality, the following:

r2∫

Ω(e2sϕ1(0) + e2sϕ2(0))|f |2

≤ C

2∑

k=1

2∑

j=1

⟨P φk

1 wkj , ∂tw

kj

⟩L2(Ωj×(0,T ))

+ s2λ3

∫∫

Q(ϕk)3|wk|2

≤ C

2∑

j,k=1

(1√s

∣∣∣P φk

1 wkj

∣∣∣2

L2(Qj)+

√s∣∣∣∂tw

kj

∣∣∣2

L2(Qj)

)

+ C s2λ32∑

k=1

∫∫

Q(ϕk)3

∣∣∣wk∣∣∣2

≤ C√s

2∑

j,k=1

(∣∣∣P φk

1 wkj

∣∣∣2

L2(Qj)+∥∥∥wk

j

∥∥∥2

Qj ,ϕk

)

Now, applying Corollary 7, we get

r2∫

Ω(e2sϕ1(0) + e2sϕ2(0))|f |2

≤ C√s

2∑

k=1

(∣∣∣P φk

1 wk∣∣∣2

L2(Q0)+∥∥∥wk

∥∥∥2

Q0,ϕk

)

≤ C√s

2∑

k=1

(∣∣∣esϕk

Lpv∣∣∣2

L2(Q0)+sλ

∫∫

Σφk

+

ϕk

∣∣∣∣a2∂wk

∂ν

∣∣∣∣2)

On the one hand, since we have θt = 0 in [−T + δ, T − δ], then from estimate (II.48), (II.50)

89


and (II.53), we obtain ∀k = 1, 2

∫∫

Q0

e2sϕk |Lpv|2 =

∫∫

Q0

e2sϕk |θfRt + 2θtytt + θttyt|2

≤ C

∫∫

Q0

e2sϕk |f |2|Rt|2 + C

∫∫

Q0

e2sϕk (|θtzt|2 + |θttz|2)

≤ C

∫∫

Q0

e2sϕk(0)|f |2|Rt|2 + C

(∫ −T+δ

−T+

∫ T

T−δ

)∫

Ω0

e2seλM (|zt|2 + |z|2)

≤ C‖R‖H1(0,T ;L∞)

∫

Ωe2sϕk(0)|f |2 +Ce2seλM‖z‖2

H1(−T,T ;L2(Ω))

≤ C

∫

Ωe2sϕk(0)|f |2 + Ce2seλM‖f‖L2(Ω)

≤ C

∫

Ωe2sϕk(0)|f |2.

Now, recalling the notation stated in (II.28), we have Σφ1

+ ∪ Σφ2

+ ⊂ Σ and

∣∣∣∣∂wk

∂ν

∣∣∣∣ = esϕk

∣∣∣∣∂v

∂ν

∣∣∣∣on Σ for each k, so we finally obtain

r2∫

Ω(e2sϕ1(0) + e2sϕ2(0))|f |2

≤ C√s

∫

Ω

(e2sϕ1(0) + e2sϕ2(0)

)|f |2 (II.55)

+ C√sλ

∫∫

Σ

(ϕ1e2sϕ1

+ ϕ2e2sϕ2) ∣∣∣∣a2

∂v

∂ν

∣∣∣∣2

.

For s large enough, the left hand side in (II.55) can absorb the first term of the right hand

side. Therefore, since ϕk and θ are bounded on Σ and∂z

∂νis an even function with respect

to t ∈ [−T, T ], we obtain

∫

Ω|f |2 ≤ C

∫∫

Σ

(ϕ1e2sϕ1

+ ϕ2e2sϕ2) ∣∣∣∣a2

∂v

∂ν

∣∣∣∣2

≤ C

∫∫

Σ

∣∣∣∣a2∂z

∂ν

∣∣∣∣2

= 2C

∫ T

0

∫

Γ

∣∣∣∣a2∂z

∂ν

∣∣∣∣2

.

and this ends the proof of Theorem 5.

90


5.2 Proof of Theorem 4.

We conclude this part of the thesis with the proof of Theorem 4, which is a direct conse-

quence of Theorem 5. Indeed, if we set y = u(q) − u(p), f = p − q and R = u(p), then y is

the solution of

ytt − div(a∇y) + (p− f)y = f(x)R(x, t) (0, T ) × Ω

y = 0 (0, T ) × Σ

y(0) = 0 Ω

yt(0) = 0 Ω

(II.56)

where q = p− f ∈ U , with U bounded in L∞(Ω) from the hypothesis of Theorem 4. The key

point is that in the proof of Theorem 5, all the constants C > 0 depend on the L∞-norm

of the potential as stated in Corollary 7. Thus, with q ∈ U , we are actually, with equation

(II.56), in a situation similar to the linear inverse problem related to equation (II.46) and we

then obtain the desired result.

91

Conclusiones

92

Conclusiones

There are several interesting problems which can be considered as a logical continuation

of this thesis. It was only a problem of time the reason we have not -yet- attacked some of

this problems. Here we present some of them:

Transport-difussion equation.

1. With respect to the estimation about the cost of the control for the transport diffusion

equation, with our approach we take as a target any L2 function, and we measure the

error with H−1-norm . It is natural to ask if we can measure the error with the L2-norm

(with targets in H1, for example).

2. With our approach, it is not clear if we could say anything about the null-control, and

its behavior when ε→ 0.

3. Clearly it would be interesting to look for some Carleman estimate for the Fokker-Planck

(FP) equation, and look if it implies the null controllability, for example. A difficulty

is the estimation of the terms with x−derivatives, so maybe a precise hypo-elliptic

inequality would help.

4. In order to apply the same methods to the (FP) equation with coefficient ε, we would

need to prove that the convergence to zero of the right hand side of (FP) in H−1(X×V )

implies some convergence to zero for the solution of (FP). Again, maybe some hypo-

elliptic inequality could help.

Carleman for transmission wave equation.

1. First of all, it should be possible to state the hypothesis of strong convexity on the inner

domain in more direct terms concerning its geometry (like some property with tangents

hyper-planes, etc). We should look carefully into the Differential Geometry concepts.

2. There are several works where it is solved the inverse problem of recovering the main

coefficient of a wave equation [37], [46], (assuming it is smooth). It would be very

interesting to apply the Carleman estimate we have developed in this work for that

inverse problem.

93

Conclusiones

3. In the articles [6] and [8] it is solved the inverse problem consisting in the estimation

of the potential and the main coefficient of the wave equation, respectively, by a single

measurement in a subset of the boundary, without any geometrical restriction, obtaining

a logarithmic Lipschitz estimate. It would be very interesting the same problem for the

transmission system.

4. In [9] it is solved the inverse problem of recovering the potential for the Schrodinger

equation from one boundary measuremet, by some suitable Carleman estimates. It is

natural to wonder if we can adapt our construction for the Schrodinger equation with

discontinuous coefficients.

94

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