Cours-Udine-Puel-Mod1-06-04

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Global Carleman inequalities for the waveequations and applications to controllabilityand inverse problems1Jean-Pierre Puel1J.-P. Puel: ([e-mail]) Laboratoire de Math´ematiques Ap-pliqu´ees, Universit´e de Versailles Saint-Quentin, 45 avenue des Etats Unis, 78035 VersaillesChapter 1Presentation of the problemsExact controllability problems for wave equations have been extensively studied inthe last fifteen years (see [13] for example) and this problem corresponds to manyapplications like how tostop vibrationsbyaboundaryaction forexample andmanyothers. We will consider here the case of wave equations with bounded potentialsand present a direct approach to this problem. Another problem which will be con-sidered here is to retrieve an unknown potential in a wave equation from boundarymeasurements. This is now an inverse problem and again it corresponds to impor-tant applications, even if a moreinteresting problem would beto retrieve a diffusioncoefficient or elasticity coefficients. The situation considered here constitutes onestep in this direction.The results we present here are not new but they are given in a unified way andthe attempt is to make them rather simple and comprehensible despite of some longcomputations.The link between the two questions appears clearly from Lions’ Hilbert Unique-ness Method for example. Here we obtain results on the two problems using thesame mathematical machinery.We will present this machinery which ...

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• Publié par: Nisig
• Nombre de lectures: 60
• Langue: English

Extrait

Global Carleman inequalities for the wave equations and applications to controllability and inverse problems

Jean-Pierre Puel 1

1 J.-P. Puel: ([e-mail])Labora´htatameriotMedeueiqp-sA plique´es,Universite´deVersaillesSaint-Quentin,45avenuedesEtatsUnis,78035Versailles

Chapter 1

Presentation of the problems

Exact controllability problems for wave equations have been extensively studied in the last fifteen years (see [13] for example) and this problem corresponds to many applications like how to stop vibrations by a boundary action for example and many others. We will consider here the case of wave equations with bounded potentials and present a direct approach to this problem. Another problem which will be con- sidered here is to retrieve an unknown potential in a wave equation from boundary measurements. This is now an inverse problem and again it corresponds to impor- tant applications, even if a more interesting problem would be to retrieve a diffusion coefficient or elasticity coefficients. The situation considered here constitutes one step in this direction. The results we present here are not new but they are given in a unified way and the attempt is to make them rather simple and comprehensible despite of some long computations. The link between the two questions appears clearly from Lions’ Hilbert Unique- ness Method for example. Here we obtain results on the two problems using the same mathematical machinery. We will present this machinery which is the very technical but powerful math- ematical tool given by global Carleman estimates in the present context of wave equations with potentials. We will give a complete proof of these estimates for strong solutions of wave equations, following the method developped in [8]. Then we give applications to the two above mentionned problems. We start with well known preliminaries on wave equations which we will need all along this study. This text corresponds to lectures which have been given during the advanced school “Control of Solids and Structures: Mathematical Modelling and Engineering Applications” at the C.I.S.M. in Udine in June 2004 and to a part of D.E.A. course at the University Pierre et Marie Curie in 2003-2004.

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1.1 Basic properties for the wave equation In the whole text, Ω is a bounded (regular enough) open subset of IR N , Γ denotes its boundary andT We will sometimes use the notationis a strictly positive number. Q= Ω×(0 T will call). Weνthe outward unit normal vector on Γ. We will consider various kinds of wave equations and they will all be of the following type :  ∂ 2 2 u −Δ =fin Ω×(0 T)  (1.1.1)    ∂ u u( t 0 = ) g =u o 0 n u Γ ; + ∂ ∂ × p t u ( u 0 0 (  ) T = ) u  1 in Ω A very important special situation is the case of homogeneous Dirichlet boundary condition, correponding tog= 0 which is   ∂ 2 2 u −Δu+pu=fin Ω×(0 T) u= 0on Γ×(0 T) (1.1.2)    u ∂ ( t 0) =u 0 ; t∂ u∂ (0) =u 1 in Ω 1.1.1 Existence of finite energy solutions We recall here the basic results concerning existence of finite energy solutions for equation (1.1.2) in the case of homogeneous Dirichlet boundary conditions (g= 0). We can refer to [5] for a complete study of the wave equations. Theorem 1.1.1Suppose thatp∈L ∞ (Ω×(0 T)) for every. Thenf∈L 1 (0 T;L 2 (Ω)), u 0 ∈H 01 (Ω)andu 1 ∈L 2 (Ω), there exists a unique solutionuto (1.1.2) with u∈C([0 T];H 01 (Ω)) t∂ u∂ ∈C([0 T];L 2 (Ω)) Proof. (idea) The proof is based on an energy estimate which can be justified on an approx- imate equation (finite dimensional or regularized...) and which works as follows. Multiply equation (1.1.2) by u∂ t∂ and integrate in Ω, we obtain 1 2td d ( Z Ω | ∂ t∂ u | 2 dx+ Z Ω |ru| 2 dx) + Z Ω xd u∂ t∂ up= Z Ω  t∂ xdf u∂ If we define the energy by E(t( 2 1 =) Z Ω | ∂t ∂u (t)| 2 dx+ Z Ω |ru(t)| 2 dx) 2

we have d d t E(t)≤ ||p|| L ∞ (Q) |u(t)| L 2 (Ω) | ∂u (t)| L 2 (Ω) +|f(t)| L 2 (Ω) | ∂u ∂t (t)| L 2 (Ω) ∂t AsΩisboundedthePoincar´einequalitysaysthatthereexistsaconstantC 0 such that ∀v∈H 01 (Ω)|v| L 2 (Ω) ≤C 0 |rv| L 2 (Ω)  Therefore we have d dt E(t)≤C 0 ||p|| L ∞ (Q) E(t 2) +|f(t)| L 2 (Ω) E(t) so that d d t E(t)≤ C 2 0 ||p|| L ∞ (Q) E(t 1 +) 2 |f(t)| L 2 (Ω)  Using Gronwall’s lemma we obtain ∀t∈(0 T) E(t)≤( √ 1 2 Z 0 T |f(s)| L 2 (Ω) ds+E(0))exp( C 2 0 ||p|| L ∞ (Q) T) Remark 1.1.2In fact we only needp∈L 1 (0 T;L ∞ (Ω)). This gives an explicit bound on the energyE(t) in terms ofE(0) (which only depends on the initial datas), the potentialpand the right hand sidefand this is the essential part of the proof of Theorem 1.1.1. We also have an existence and uniqueness result for finite energy solutions in another context. Theorem 1.1.3We still suppose thatp∈L ∞ (Ω×(0 T)). Then for everyf∈ W 11 (0 T;H −1 (Ω)),u 0 ∈H 01 (Ω)andu 1 ∈L 2 (Ω), there exists a unique solutionu to (1.1.2) with u∈C([0 T];H 01 (Ω))  ∂u ∈C([0 T];L 2 (Ω)) ∂t The difference in the proof comes from the term < f(t) ∂u ∂t (t)> which is treated as ∂ ∂ t <f(t) u(t)>− f∂ < ∂ (t t )  u(t)>  3

1.1.2 Regularity of the normal derivative We now recall a regularity result for solutions given by Theorem 1.1.1. This result is known as a result of “hidden regularity” (see for example [13]). Theorem 1.1.4We assume the same hypotheses as in Theorem 1.1.1. the Then solutionuof (1.1.2) satisfies ∂ ∂ u ν ∈L 2 (0 T;L 2 (Γ)) Moreover, the mapping ∂u (f u 0  u 1 )→ ∂ν which is well defined for regular (and dense) datas, is linear continous from L 1 (0 T;L 2 (Ω))×H 01 (Ω)×L 2 (Ω)intoL 2 (0 T;L 2 (Γ)). Proof. (idea). It relies on the multiplier method. The computations are done in the case of regular solutions (for a dense set of regular datas). Letm∈W 1∞ (Ω; IR N ) (called multiplier) and let us multiply (1.1.2) bymru. We successively obtain (using the summation convention for repeated indices) Z 0 T Z Ω ∂ ∂ 2 t 2 m u rudxdt = Z Ω t∂ u∂ (T)m Z Ω u 0 dx− Z 0 T Z Ω mr t ∂u ∂t dxdt ∂ ∂ u ru(T)dx−u 1 mr = Z Ω ∂u ∂t (T)mru(T)dx− Z Ω u 1 mru 0 dx− 2 1 Z 0 T Z Ω mr(| t ∂u ∂ | 2 ) dxdt = Z Ω ∂ u t ∂ (T)mru(T)dx− Z Ω u 1 mru 0 dx+ 2 1 Z 0 T Z Ω (divm)| u∂ ∂t | 2 dxdt Z 0 T Z Ω (−Δu)mrudxdt = Z 0 T Z Ω u ∂ ∂ x i x∂ ∂m i j ∂x ∂u j dxdt+ Z 0 T Z Ω ∂ x∂ u i m j x∂ ∂ ( ∂ x∂ u i )dxdt− Z 0 T Z Γ (ruν)(mru)dσdt j T ∂u Z 0 Z Ω ∂x i ∂x ∂m j i ∂u ∂x j dxdt 2 1 + Z 0 T Z Ω m j ∂ x∂ j (|ru| 2 )dxdt− Z 0 T Z Γ |ruν| 2 (mν)dσdt = = Z 0 T Z Ω ∂u ∂x i ∂m ∂x j i x u ∂ ∂ j dxdt− 1 Z T Z Ω (divm)|ru| 2 dxdt− 1 2 Z 0 T Z Γ |ruν| 2 (mν)dσdt 2 0 4

We now take the multipliermas a lifting of the outward unit normalνso that mν obtain, gathering all is possible if Ω is regular enough. We= 1 on Γ. This terms, the following estimate 2 1 Z 0 T Z Γ |ruν| 2 dσdt≤C( Z 0 T E(t)dt+ Z 0 T Z Ω |p||u||ru|dxdt+ Z 0 T Z |f||ru|dxdt) Ω This gives the desired result. Remark 1.1.5Iff∈W 11 (0 T;H −1 (Ω)), we still have existence and uniqueness of a finite energy solution from Theorem 1.1.3. But in that case, we cannot obtain any regularity result on the normal derivative ∂u ∂ν . 1.1.3 Solution obtained by transposition We will also need a weaker notion of solution which is obtained by transposition of the previous results. These solutions are usually called “solution by transposition”. Let us consider a wave equation with non homogeneous Dirichlet boundary con- dition   ∂ ∂ 2 t 2 z −Δz+pz=fin Ω×(0 T) (1.1.3) z=gon Γ×(0 T)    z(0) =z 0 ; ∂ t z ∂ (0) =z 1 in Ω Theorem 1.1.6For everyf∈L 1 (0 T;H −1 (Ω)),g∈L 2 (0 T;L 2 (Γ)),z 0 ∈L 2 (Ω) andz 1 ∈H −1 (Ω), there exists a unique solutionzof (1.1.3) withz∈C([0 T];L 2 (Ω)) and t∂ ∂z ∈C([0 T];H −1 (Ω)). Proof. . Leth∈L 1 (0 T;L 2 (Ω)) and let u be solution of  ∂ ∂ 2 t 2 u −Δu+pu=hin Ω×(0 T) (1.1.4)   u= 0on Γ×(0 T)   u(T) = 0 ; ∂ u∂ t (T) = 0in Ω We know from Theorem 1.1.1 thatu∈C([0 T];H 01 (Ω)) and from Theorem 1.1.3 that ∂ν ∂u ∈L 2 (0 T;L 2 (Γ)). Now let us define L(h) = Z 0 T < f(t) u(t)> dt− Z Ω z 0 t∂ u∂ (0)dx+< z 1  u(0)>− Z 0 T Z Γ g ∂ν ∂u dtdσ 5

It is immediate to see thatLis a linear continuous functional onL 1 (0 T;L 2 (Ω)). Therefore there exists a unique functionz∈L ∞ (0 T;L 2 (Ω)) such that T ∀h∈L 1 (0 T;L 2 (Ω)) Z 0 Z Ω (h) zhdxdt=L = Z 0 T < f(t) u(t)> dt− Z Ω z 0 ∂ ∂ u t (0)dx+< z 1  u(0)>− Z 0 T Z Γ ∂ gd ν∂ u tdσ It is now standard to interpret the above equation and to give a sense to the value ofzon the boundary as well as the initial valuesz(0) and z∂ ∂t (0) to obtain (1.1.3). By taking first the datasf,z 0 andz 1 in a dense subset of regular functions, we can show easily that in factz∈C([0 T];L 2 (Ω)) and then that ∂ ∂ z t ∈C([0 T];H −1 (Ω)). This gives the result of Theorem 1.1.6.

1.2 Retrieving a potential from boundary measurements We will consider the inverse problem consisting in retrieving a potential, depending only on the space variable, in a wave equation from the knowledge of measurements of the flux on a part of the boundary during a period of time (0 T). More precisely we consider a wave equation  ∂ 2 u ×(0 T) (1.2.5)   u ∂t = 2 g − no Δu Γ + × p ( u 0 =f T)   in Ω   u(0) =u 0 ; u ∂ ∂ t (0) =u 1 in Ω where the potentialp=p(x) is unknown (but assumed to be bounded). For every p∈L ∞ (Ω) (assuming enough regularity onf,g,u 0 andu 1 ) we suppose that (1.2.5) a measure of ∂u on a set has a unique solutionu(p). On the other hand we know ∂ν Γ 0 ×(0 T) where Γ 0 part of the boundary Γ sayis a ∂u (1.2.6) =Hon Γ 0 ×(0 T) ∂ν The question is then : can we retrievepfrom the measureH? To be more precise we can ask two questions : •Uniqueness : do we have ∂ ∂ u(p ν ) = ∂ u∂( ν q  Γ ) on 0 ×(0 T)⇒p=q?

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• we have doStability : q) ||p−q|| X(Ω) ≤C|| ∂ u∂( ν p) − u∂ ∂ ( ν || Y(Γ 0 ) for suitable normsX(Ω) andY(Γ 0 )? Of course stability implies uniqueness. We will obtain here, following the work of [9] and [10] , stability results in a situation where we assume to know the solutionu(q) corresponding to a given potentialqwith enough regularity onu(q) (for example this solutionu(q) could be a result of computations) and under a geometrical hypothesis on Γ 0 together withTsufficiently large. results in the same direction have Previous been obtained for example in [15], [17] and [11] We can already notice that if we call z=u(p)−u(q) zis solution of the following wave equation :   ∂ ∂ 2 z t 2 −Δz+pz= (q−p)u(q)in Ω×(0 T) (1.2.7)  z= 0 on Γ×(0 T)   z(0) = 0 ; ∂z ∂t (0) = 0in Ω The problem is then to estimate (p−q) in terms of ν∂ z∂ . Notice that the potentialp appearing in the left hand side of the equation is unknown. In [3] the authors consider another inverse problem for retrieving a potential in a stationnary elliptic equation from measurements on a part of the boundary. More precisely they consider an elliptic equation −Δu+pu=fin Ω (1.2.8) ( on Γ u=ϕ wherepis unknown. They suppose to know the complete mapping Λ p :ϕ→ u ∂ν ∂ Γ 0  The question they address is then : If two such mappings (corresponding to two potentials) coincide, are the potentials equal? More precisely, do we have Λ p = Λ q ⇒p=q?

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Notice that the knowledge of Λ p = Λ q implies an infinite number of measurements, but in our case, the knowledge of u∂p ∂ν = ν∂ u∂q on Γ 0 ×(0 T) means the knowledge of one measurement but during a time period (0 T) so that it also means an infinite number of measurements. The positive result they obtain requires hypotheses which are analogous to the ones we will assume here and the techniques involved in their proof is similar to the one we will describe here for the wave equation. However, up to our knowledge, the precise relation between the two problems is unknown at the moment and it would be very interesting know if such a relation exists. AsimilarquestioncanbetreatedforSchro¨dingerequationsandtheanswersare similar (see [2]). The basic tool we will use is a global Carleman inequality for the wave equation. The obtention of such an inequality will be the subject of Chapter 2. 1.3 Exact controllability for wave equations with bounded potentials In this part we are given a wave equation  ∂ ∂ 2 t 2 u −Δu+pu= 0in Ω×(0 T) (1.3.9)   u u 0 = = v  Γ Γo( on n 0 − × Γ (0 0  ) T × ) (0 T)   u(0) =u 0 ; t∂ u∂ (0) =u 1 in Ω wherep∈L ∞ (Ω×(0 T)). We know from Theorem 1.1.6 that for everyu 0 ∈ L 2 (Ω),u 1 ∈H −1 (Ω) and for everyv∈L 2 (0 T;L 2 (Γ 0 )), there exists a unique solutionuto (1.3.9) (defined by transposition) withu∈C([0 T];L 2 (Ω)) and ∂ u t ∂ ∈ C([0 T];H −1 (Ω)). The problem of exact controllability is then the following : Given any target (z 0  z 1 )∈L 2 (Ω)×H −1 (Ω), can we find a controlv∈L 2 (0 T;L 2 (Γ 0 )) such that the state reaches the target at timeTwhich means u(T) =z 0 and t ∂ ∂ u (T) =z 1  Because of the linearity and the reversibility of (1.3.9) it is easy to see that it suffices to treat the case whenz 0 = 0 andz 1 = 0. Moreover, since the work of Lions ([13]) and his development of the Hilbert Uniqueness Method (HUM), it is well known that solving the exact controllability 8

problem is equivalent to proving an observability inequality on the adjoint equation. Let us consider the adjoint equation  ∂ 2 ϕ −Δϕ+pϕ= 0in Ω×(0 T)   ∂t ϕ= 2 0on Γ×(0 T) (1.3.10)   ϕ(0) =ϕ 0 ; ϕ∂ t∂ (0) =ϕ 1 in Ω whereϕ 0 ∈H 01 (Ω) andϕ 1 ∈L 2 (Ω). If the initial energy is defined by E 0 = 1 2 ( Z Ω |ϕ 1 | 2 dx+ Z Ω |rϕ 0 | 2 dx) the observability inequality to be proved (which is equivalent to solving the exact controllability problem) is the following (1.3.11)∃C 0 >0such thatE 0 ≤C 0 Z 0 T Z Γ 0 | ν∂ ∂ϕ | 2 dσdt Using the same basic tool as for the previous inverse problem, namely a global Carleman inequality, we will show inequality (1.3.11) under a geometrical hypothesis on Γ 0 and forTsufficiently large.

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Chapter 2

Global Carleman inequality for the wave equation

Carleman estimates for regular functions with compact support (which we shall call local) go back to [4] and have been developped in [7]. Here we are interested in proving global Carleman estimates (up to the boundary) corresponding to the wave operator. We consider a functionv∈L 2 (−T  T;L 2 (Ω)) such that L 0 v= ∂ ∂ 2 t 2 v −Δv∈L 2 (−T  T;L 2 (Ω)) We then call Lv=L 0 v+pv= ∂ ∂ 2 t 2 v −Δv+pv and we also have Lv∈L 2 (−T  T;L 2 (Ω)) We can then define in a very weak sense the traces on Γ×(−T  T) ofvand ∂ v∂ ν because Δv∈H −2 (−T  T;L 2  can also define the values of(Ω)). Wevand ∂ t v ∂ at times−TandTbecause ∂ ∂ 2 t 2 v ∈L 2 (−T  T;H −2 (Ω)). We know consider the set of functionsvsuch that v∈L 2 (−T  T;L 2 (Ω)) L 0 v∈L 2 (−T  T;L 2 (Ω)) v= 0 on Γ×(−T  T) v(−T) =v(T) = 0 ; v ∂ ∂ t (−T) = t∂ v∂ (T) = 0 It is classical to show that this set contains a dense subset of “regular” functions.

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