Variants of global Carleman weights in one-measurement inverse problems and fluid-structure controllability problems

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–8) Variants of global Carleman weights in one-measurement inverse problems and fluid-structure controllability problems 1L. Baudouin, 2M. Boulakia, 3A. Doubova, 4A. Mercado, 4A. Osses, 5J.-P. Puel 1Laboratoire d’Analyse et d’Architecture des Syst`emes, LAAS - CNRS, 7 avenue du Colonel Roche, 31 077 Toulouse Cedex 04, France. E-mail: [email protected]. 2Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France. E-mail: [email protected]. 3Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, E-41012 Sevilla, Spain. E-mail: [email protected]. 4Departamento de Ingenier´ıa Matem´atica, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile and Centro de Modelamiento Matem´atico, UMI 2807 CNRS-Uchile, Chile. E-mails: http://www.dim.uchile.cl/∼axosses, [email protected], [email protected]. 5Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Versailles St-Quentin, 45 avenue des Etats Unis, 78035 Versailles cedex, France. E-mail: [email protected]. Keywords: Carleman estimates, Inverse Problems, Controllability Abstract We review some recent results on variants of global Carleman weights and Carleman inequalities applied to singular controllability and inverse problems partially developed in collaboration with the authors in a series of papers. First of all, we explain how we can modify weights to study one measurement inverse problems for the heat and wave equations with discontinuous coefficients in the principal part, in a case of locally supported boundary observations for recovering coefficients in the wave equation and we mention also some recent results for the Sch¨odinger equation. As another important application, we show how time-variable global Carleman weights are applied to study the nullcontrollability for a Navier-Stokes-rigid solid problem in moving domains. 1 L. Baudouin, A. Doubova, M. Boulakia, A. Mercado, A. Osses, J.-P. Puel 1 Carleman weights Let Ω ⊂Rnbe a bounded domain and P∗a second order adjoint operator in Q= Ω ×I, where Iis a time interval. We suppose that P∗depends on some stationary parameter q∈L∞(Ω). Given some regular weight function Φ defined in Q, we perform the following change of variables in the partial differential equation P∗z=fcalled conjugation: w=ρ z, ρ = exp (−sΦ), s > 0,(1) P∗z=f⇔ρP∗(ρ−1w) = ρ f. (2) We also introduce a function ϕ(x, t) such that ∇Φ = −λ∇ψ ϕ. Typical examples of weights Φ are: Heat equation:P∗=−δt−∆ + q,Q= Ω ×(0, T) Φ(x, t) = exp(λα)−exp(λψ(x)) T−t(3) for some λ > 0 and αlarge enough, where ψis some suitable regular and bounded function defined in Ω (see for instance Table 1 for some conditions on ψand Figure 1 for typical shapes of ψ). Wave equation:P∗=δtt −∆ + q,Q= Ω ×(−T, T) Φ(x, t) = −exp(λ(ψ(x)−β t2)), ψ(x) = |x−x0|2,(4) where x0is some given point outside Ω and β∈(0,1) is some suitably chosen parameter. Schr¨ odinger equation:P∗=i∂t+ ∆ + q,Q= Ω ×(−T, T) Φ(x, t) = exp(λα)−exp(λψ(x)) (T−t)(T+t), ψ(x) = |x−x0|2,(5) for some λ > 0 and αlarge enough, where x0is some given point outside Ω. Let ω⊂⊂ Ω be an internal observational or control region and let Γ0⊂∂Ω be a boundary observational or control region. We will work with global Carleman inequalities of the form p1(s, λ)kϕ1/2ρ∇zk2 L2(Q)+p0(s, λ)kϕ3/2ρ zk2 L2(Q)≤ C³kρ fk2 L2(Q)+p1(s, λ)kϕ3/2ρ∇z·nk2 L2(Γ0×I)+p0(s, λ)kϕ1/2ρ zk2 L2(ω×I)´,(6) where nis the unit exterior normal to Ω, piare the polynomial weights given in Table 1. Notice that ρ→0 exponentially as sΦ→+∞. The internal observational or control region ωappearing at the right hand side of the global Carleman inequality is such that the pseudoconvexity of Φ with respect to P∗holds outside ω(for pseudoconvexity notion see [15], [26]), in particular, |∇ψ(x)|>0 outside ω. On the other hand, the boundary observational or control region Γ0is such that a strong Lopatinskii condition holds outside Γ0, that is ∇ψ(x)·n < 0 outside Γ0(see [26] for a more general statement of global Carleman inequalities in this cases). In this communication we present a collection of Carleman weights whose applications illustrate the extent of Carleman inequalities when they are applied to the study of some inverse and controllability problems. 2 Variants of global Carleman weights p1p0 Heat sλ2s3λ4 Wave sλ s3λ3 Schr¨odinger sλ s3λ4 Table 1: Polynomial weights in global Carleman inequalities 2 Inverse source problem for heat transmission We consider a heat operator with discontinuous coefficients in the principal part. In this case, the function ψhas to be well adapted to this new situation and then specific global Carleman estimates can be derived. As an application of the Carleman inequality is the study of one measurement inverse problems using the general Bukhgeim-Klibanov approach [9]. The results of this section have been collected from [11], [6], [7] and [2]. Given Ω ⊂Rnbe a bounded and regular subset. Take Ω1⊂Ω and let set Ω0= Ω\Ω1. Define Sas the interface between Ω0and Ω1with unit normal nexterior to Ω1and let S+and S−be its outer and inner sides with respect to nand Σ+=S+×(0, T), Σ−=S−×(0, T). Let us consider the transmission problem    yt−div (a0(x)∇y) = f(x)g(x, t) in Ω0×(0, T) yt−div (a1(x)∇y) = f(x)g(x, t) in Ω1×(0, T) y|Σ+=y|Σ−, a0∂y ∂n |Σ+=a1∂y ∂n |Σ−, y = 0 on ∂Ω×(0, T) (∗) (7) with ai≥c0>0 a.e. in Ω and let us introduce the space V={y∈C2(Ωi×[0, T]), i = 0,1, y satisfies (∗)}. The inverse source problem consists in retrieving the source f(x) from the knowledge of g(x, t), the local trace of the solution yin ω0×(0, T), where ω0⊂Ω0and from a time slice y(·, T0) for some T0∈(0, T), but without any knowledge of the initial condition y(·,0). We have to assume also that some technical isotopy type condition is satisfied, see details in [11]. The inverse stability result that is obtained using a Carleman estimate for the heat operator with discontinuous coefficients is Theorem 2.1 ([6], [7]) Let T0∈(0, T)and ω0⊂Ω0and let us assume that Ω1and Ω0satisfy the isotopy type conditions of [11]. Assume that ysolution of (7) is such that y, yt∈V. Assume that a1|S−−a0|S+≥0and that g∈C2(Ω ×[0, T]),|g(·, T0)| ≥ r0>0 a.e. in Ω. Then there exists a constant C=C(g, ω0, T0)such that for all f∈L2(Ω) kfkL2(Ω) ≤C¡ky(·, T0)kH2(Ω0)+ky(·, T0)kL2(Ω1)+kykH1(0,T;L2(ω0))¢.(8) The global Carleman estimate for (7) stated in [11] was firstly used in order to prove the exact controllability to trajectories for a semilinear system similar to (7) that is controlled in ω0×(0, T). In the general case when Ω1is not simply connected, and in order to construct the weight functions, an isotopy type condition between Sand the boundary of two disjoint open subsets Oi,i= 1,2 of Ω1is used. Two weights similar to (3) are then constructed of the form Φi(x, t) = exp(λα)−exp(λψi(x)) T−t, i = 1,2,(9) 3 L. Baudouin, A. Doubova, M. Boulakia, A. Mercado, A. Osses, J.-P. Puel where ψi∈Vand ∇ψ= 0 only in Oi(see Figure 1 left). Notice that you can also consider the opposite case when Ω0⊂Ω and Ω1= Ω \Ω0, and always ω0⊂Ω0. In this case, an isotopy type condition between ∂Ω and Sis a sufficient condition. See Figure 1 right). O2 a0 a0a1 ψ2 O 1 Ω1 Ω0 O1O2 ψ 1 ω n ω ω n 1 Ω0 a1 a0 a1 ω ψ Ω n ω n Figure 1: Construction of the global Carleman weight (bottom curves) for the heat equation with discontinuous coefficients such that a1(S−)−a0(S+)>0 (middle curves). In the case Ω0⊂Ω (left) two combined weights are used and in the case Ω1⊂Ω (right) one weight suffices. In both cases the observation zone ωis represented by a black dot. 3 Inverse problem for waves from partial boundary data Here we focus on one measurement inverse problems for the wave equation from local boundary observations. In this case, the function ψis modified in order to obtain some strong Lopatinskii condition of the form (x−x0)·Tn < 0, where Tis some linear transformation of the normal field. For further details we refer to [12]. The main idea is to modify the weight function Φ given in (4) in such a way that its gradient ∇Φ is a rotation of the original field (x−x0) with a radially dependent magnitude. This concept come up from multipliers techniques commonly used in controllability [18]. Let Ω be a domain in Rn,n= 2,3. In order to solve the Dirichlet to Neumann one measurement inverse problem, it suffices to measure on a rotated exit part of the boundary Γrwhich corresponds in fact to a particular case of the geometrical optics condition BLR [1], [19]. If n= 2, this region depends on a point x0∈Rnand on a rotation Tθin an angle θ∈(−π/2, π/2). If n= 3, it depends also on a unit direction α∈R3and the rotation Tθ is considered on the orthogonal plane to αdenoted here by α⊥. More precisely, f we use the notation v⊥=v−(v·α)αfor the projection of the field von α⊥ Γr={x∈∂Ω|(x−x0)·Tθn > 0}if n= 2 (10) Γr={x∈∂Ω|(x−x0)·(cos θ(n−n⊥) + Tθn⊥)>0}if n= 3.(11) The main stability result is the following in the simplest case x06∈ Ω (see [12]) 4 Variants of global Carleman weights Theorem 3.1 ([12]) Let P∗=∂tt −∆ + qand let u(q)and u(q)be the respective solutions of P∗u= 0 with Dirichlet boundary conditions associated to q, q ∈L∞(Ω) and with Neumann measurements ξand ξon Γr×(0, T )respectively. There exists a time T > 0 such that if T > T, if u(q)∈H1(0, T ;L∞(Ω)) and if |u(0)| ≥ α0>0a.e. in Ω, then there exists a positive constant CMdepending on M=kqkL∞(Ω) such that ||q−q||L2(Ω) ≤CM° °ξ−ξ° °H1(0,T;L2(Γr)) ∀qwith kqkL∞(Ω) ≤M . (12) The proof of this Theorem is based on a global Carleman estimate using (compare with (4)) Φ(x, t) = −λexp ³cos θ|x−x0|2exp ³2 tan θarg(x−x0)⊥´−β t2´(13) for some suitable constant β∈(0,1). The main steps in the deduction of such inequality are taken from [22] following a well known technique due to Bukhgeim and Klibanov [9]. There are also geometrical exit type conditions for the analogous inverse problem in the case of the Scrh¨odinger equation (see [3]). The present method should also work in this case because the spatial part used in Carleman weights for wave and Sch¨odinger equations are the same (compare (4) with (5)) but this is an open problem. Nevertheless, in the case of the Scrh¨odinger operator, it is certain that the geometrical optics condition is not necessary to solve the one measurement inverse problem (see [20]). Other completely different problem is the case when you have Dirichlet to Newmann map measurements. In this case, a suitable arbitrarily small boundary of measurements is enough to solve the inverse problem both for Sch¨odinger and wave equations (see [17]). Finally, it has been shown [5] that you can solve the one measurement problem for the wave equation with an arbitrarily boundary measurement region (in the case of Neumann boundary conditions and Dirichlet measurements), but the corresponding inequality analogous to (12) is logarithmic. 4 Inverse coefficient problem for wave transmission Notice that recently, Global Carleman estimates and applications to one measurement inverse problems for the wave equation were obtained in the case of variable but still regular coefficients [4], [16]. The inverse problem of retrieving coefficients from a wave equation with discontinuous coefficients from single boundary measurements arise naturally in geophysics and in seismic prospection [27]. Let Ω and Ω1⊂Ω be two open subsets of R2with smooth boundaries Γ and Γ1 respectively and let Ω2= Ω \Ω1. To fix ideas we assume that Ω1is simply connected. We set: a(x) = a1in Ω1and a2in Ω2with aj>0 for j= 1,2, for each q∈L∞(Ω), we consider u(q) as the solution of the following wave transmission equation        utt −div(¯a(x)∇u) + q(x)u= 0 in Q= Ω ×(0, T) u= 0 on Σ = Γ ×(0, T) u(0) = u0in Ω ut(0) = u1in Ω. (14) The following inverse stability result holds (see the preprint [2]): 5 L. Baudouin, A. Doubova, M. Boulakia, A. Mercado, A. Osses, J.-P. Puel Theorem 4.1 ([2]) Assume Ω1is strictly convex and a1> a2>0. Let Ube a bounded subset of L∞(Ω),q∈L∞(Ω) and r > 0. If |u0(x)| ≥ r > 0a. e. in Ωand u(q)∈ H1(0, T;L∞(Ω)), then there exists C=C(Ω, T, kqkL∞(Ω),U)>0such that: kq−qkL2(Ω) ≤Ck∂nu(q)−∂nu(q)kH1(0,T;L2(Γ)) for all u0∈H1 0(Ω) and q∈ U. This Theorem is proved by combining the Carleman inequality for the wave equation with discontinuous coefficients proved in [2] and the method of Bukhgeim-Klibanov explained in section 3. To this end, system (14) is viewed as two wave equations with constant coefficients coupled with transmission conditions (see [18]). Then, a global Carleman inequality is found out for this transmission problem by working with variants of Carleman weights of the form Φ = −exp(λg) where (compare with (4)) g(x, t) =    η(x)a2 r(x)2|x−x0|2−βt2+M1in Ω1×(−T, T) a1 r(x)2|x−x0|2−βt2+M2in Ω2×(−T, T) (15) where M1and M2are constants such that M1−M2=a1−a2,r(x) = |x0−y(x)|, y(x) = Γ1∩[x0, x] and ηis some cut-off function with support in Ω1centered at x0. We also combine the Carleman inequalities obtained from two different interior points as we did in section 2, see also Figure 1, left. The convexity hypothesis on Ω1comes from the fact that the positiveness of the Hessian of the weight Φ is related with the curvature of Γ1with respect to x0. There are a lot of important works concerning this inverse problem in the case that a wide class of measurements are available. In these cases, microlocal analysis has been used and it gives positive answer to the problem of retrieving coefficients and discontinuity interfaces without restrictive hypothesis of convexity of the interfaces or monotonicity of the speed of waves. These kinds of results are fundamental for seismic prospection. For an overview on this subject see [27] and the references therein. 5 Controllability problems in fluid-structure interaction Here we consider the case of mobile domains in fluid-structure problems, when studying the boundary null controllability of an immersed solid into a viscous Navier-Stokes fluid. In this case, the weight function ψdepends also on time, and the global Carleman inequality is more complicated than (6) due to the incompressibility in Navier-Stokes and the presence of the structure. The results we present in this communication were adapted from the article [8]. The first result of this kind using global Carleman estimates were obtained in [10] for a one-dimensional Burgers-particle system studied in [28]. Also, similar results to the one presented here has been simultaneously and independently obtained in [25]. Let Ω ⊂R2be a fixed bounded connected open subset with regular boundary. Let ΩS(t) and ΩF(t)=Ω\ΩS(t) be the domains occupied by the structure and by the fluid respectively and let nbe the unit exterior normal to ∂ΩS(t). The fluid is descibed in velocity-pressure (u, p) with σ(u, p) = ν(∇u+∇ut)−pId for ν > 0. The solid of mass 6 Variants of global Carleman weights m > 0 and inertia J > 0 is described by the velocity of its center of mass a(t)∈R2and by its angular velocity r(t)∈R. The system is                ∂tu+ (u· ∇)u−div σ(u, p) = f1ω,div u= 0 in ΩF(t) m¨a=Z∂ΩS(t) σ(u, p)ndσ, J ˙r=Z∂ΩS(t) (σ(u, p)n)·(x−a)⊥dσ, u= ˙a+r(x−a)⊥on ∂ΩS(t), u = 0 on ∂Ω, u(0,·) = u0in ΩF(0), a(0) = a0,˙a(0) = a1, r(0) = r0, (16) Here the function fis the control function which acts over a fixed small nonempty open subset ω(with characteristic function 1ω). We have used the notation x⊥= (x1, x2)⊥= (−x2, x1). The total angle θassociated to the angular velocity ris defined by θ(t) = θ0+Rt 0r(s)ds, where θ0∈Rcomplements the initial data. The existence of solutions and regularity for this system has been recently studied in several papers (see [24] and the references therein). The controllability result is the following, saying that it is possible to drive the structure and the fluid at rest and the immersed solid up to its reference position in arbitrarily small time with a localized control f, provided the initial conditions are sufficiently small. Theorem 5.1 ([8]) : Suppose that: i) the initial body solid shape satisfies ΩS(0) ⊂Ω\ω, d(ΩS(0), ∂(Ω \ω)) >0,R∂ΩS(0)(y−a0)dσ = 0, ii) u0∈H3(ΩF(0))2,a0∈R2,a1∈R2, θ0∈Rand r0∈Rsatisfy div u0= 0 in ΩF(0),u0=a1+r0(x−a0)⊥on ∂ΩS(0) and u0= 0 on ∂Ωiii) the accelerations u1of the fluid and a2and r1of the structure at t= 0 satisfy u1= 0 on ∂Ω,u1=a2+r1(x−a0)⊥−r2 0(x−a0)− ∇u0¡a1+r0(x−a0)⊥¢on ∂ΩS(0). Then for all T > 0there exists ε > 0and f∈L2((0, T)×ω)2such that if ku0kH3(ΩF(0))2+|a0|+|a1|+|θ0|+|r0| ≤ εthen u(T, ·) = 0 in ΩF(T),a(T) = 0,˙a(T) = 0, θ(T) = 0 and r(T) = 0. The first condition above is a symmetry restriction over the shape of the solid. The result only holds for small initial data because we want to keep the non-collision condition on the whole interval (0, T) inft∈(0,T )d(ΩS(t), ∂(Ω \ω)) >0. The proof follows ideas from [13] used to study the local exact controllability to trajectories of the Navier-Stokes equation and the ideas of [10] for a Burgers-mass model. 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