Numerical controllability of the wave equation through primal methods and Carleman estimates

ESAIM: COCV 19 (2013) 1076–1108 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2013046 www.esaim-cocv.org NUMERICAL CONTROLLABILITY OF THE WAVE EQUATION THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES Nicolae Cˆ ındea1, Enrique Fern´ andez-Cara2and Arnaud M¨ unch1 Abstract. This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments. Mathematics Subject Classification. 35L10, 65M12, 93B40. Received February 8, 2012. Revised December 19, 2012. Published online August 13, 2013. 1. Introduction. The null controllability problem We are concerned in this work with the null controllability for the 1D wave equation with a potential. The state equation is the following: ⎧ ⎪ ⎨ ⎪ ⎩ ytt −(a(x)yx)x+b(x, t)y=0,(x, t)∈(0,1) ×(0,T) y(0,t)=0,y(1,t)=v(t),t∈(0,T) y(x, 0) = y0(x),y t(x, 0) = y1(x),x∈(0,1). (1.1) Here, T>0 and we assume that a∈C3([0,1]) with a(x)≥a0>0in[0,1], b∈L∞((0,1)×(0,T)), y0∈L2(0,1) and y1∈H−1(0,1); v=v(t)isthecontrol (a function in L2(0,T)) and y=y(x, t) is the associated state. Keywords and phrases. One-dimensional wave equation, null controllability, finite element methods, Carleman estimates. 1LaboratoiredeMath´ematiques, Universit´e Blaise Pascal (Clermont-Ferand 2), UMR CNRS 6620, Campus de C´ezeaux, 63177 Aubi`ere, France. [email protected]pclermont.fr; [email protected]rmont.fr 2Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2013 NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1077 In the sequel, for any τ>0wedenotebyQτand Στthe sets (0,1) ×(0,τ)and{0,1}×(0,τ), respectively. We will also use the following notation: Ly:= ytt −(a(x)yx)x+b(x, t)y. (1.2) For any (y0,y 1)∈Y:= L2(0,1) ×H−1(0,1) and any v∈L2(0,T), it is well known that there exists exactly one solution yto (1.1), with the following regularity: y∈C0([0,T]; L2(0,1)) ∩C1([0,T]; H−1(0,1)) (1.3) (see for instance [23]). On the other hand, for any T>0, the null controllability problem for (1.1)attimeTis the following: for each (y0,y 1)∈Y, find v∈L2(0,T) such that the corresponding solution to (1.1)satisfies y(·,T)=0,y t(·,T)=0 in (0,1).(1.4) In view of the linearity and reversibility of the wave equation, (1.1) is null-controllable at Tif and only if it is exactly controllable in Yat time T, i.e. if and only if for any (y0,y 1)∈Yand any (z0,z 1)∈Ythere exist controls v∈L2(0,T) such that the associated ysatisfies y(·,T)=z0,y t(·,T)=z1in (0,1). It is well known that (1.1) is null-controllable at any large time T>T for some Tthat depends on a(for instance, see [2,23]fora≡1andb≡0 leading to T=2andsee[32] for a general situation). As a consequence of the Hilbert Uniqueness Method of Lions [23], it is also known that the null controllability of (1.1)isequivalent to an observability inequality for the associated adjoint problem. The goal of this paper is to design and analyze a numerical method allowing to solve the previous null controllability problem. So far, the approximation of the minimal L2-norm control – the so-called HUM control – has focused most of the attention. The earlier contribution is due to Glowinski and Lions in [19](seealso[21] for an update) and relies on duality arguments. Duality allows to replace the original constrained minimization problem by an unconstrained and apriorieasier minimization (dual) problem. However, as observed in [19]andlaterin[34], depending on the approximation method that is used, this approach can lead to some numerical difficulties. Letusbemoreprecise.Itiseasilyseen that the HUM control is given by v(t)=a(1)φx(1,t), where φsolves the backwards wave system ⎧ ⎨ ⎩ Lφ =0 in QT φ=0 on ΣT (φ(·,T),φ t(·,T)) = (φ0,φ 1)in(0,1) (1.5) and (φ0,φ 1) minimizes the strictly convex and coercive functional I(φ0,φ 1)=1 2a(1)φx(1,·)2 L2(0,T)+1 0 y0(x)φt(x, 0) dx−y1,φ(·,0)H−1,H1 0(1.6) over H=H1 0(0,1) ×L2(0,1). Here ·,·H−1,H1 0denotes the duality product for H−1(0,1) and H1 0(0,1). The coercivity of Iover His a consequence of the observability inequality φ02 H1 0(0,1) +φ12 L2(0,1) ≤Cφx(1,·)2 L2(0,T)∀(φ0,φ 1)∈H,(1.7) that holds for some constant C=C(T). This inequality has been derived in [23]usingthemultipliers method. At the numerical level, for standard approximation schemes (based on finite difference or finite element methods), the discrete version of (1.7) may not hold uniformly with respect to the discretization parameter, 1078 N. Cˆ INDEA ET AL. say h. In other words, the constant C=C(h) may blow up as hgoes to zero. Consequently, in such cases the functional Ih(the discrete version of I) fails to be coercive uniformly with respect to hand the sequence {vh}h>0may not converge to vas h→0, but diverge exponentially. These pathologies, by now well-known and understood, are due to the spurious discrete high frequencies generated by the finite dimensional approximation; we refer to [34] for a review on that topic; see [24] for detailed examples of the behavior observed with finite difference methods. Several remedies based on more elaborated approximations have been proposed and analyzed in the last decade. Let us mention the use of mixed finite elements [6], additional viscosity terms which have the effect to restore the uniform property [1,24] and also filtering technics [11]. Also, notice that some error estimates have been obtained recently, see [7,11]. In this paper, following the recent work [12] devoted to the heat equation, we consider a different approach. Specifically, we consider the following extremal problem: ⎧ ⎪ ⎨ ⎪ ⎩ Minimize J(y,v)=1 2QT ρ2|y|2dxdt+1 2T 0 ρ2 0|v|2dt Subject to (y,v)∈C(y0,y 1;T) (1.8) where C(y0,y 1;T) denotes the linear manifold C(y0,y 1;T)={(y,v):v∈L2(0,T),ysolves (1.1) and satisfies (1.4)}. Here, we assume that the weights ρand ρ0are strictly positive, continuous and uniformly bounded from below by a positive constant in QTand (0,T), respectively. As in the previous L2-norm situation (where we simply have ρ≡0andρ0≡1), we can apply duality arguments in order to find a solution to (1.8), by introducing the unconstrained dual problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Minimize J(μ, φ0,φ 1)=1 2QT ρ−2|μ|2dxdt+1 2T 0 ρ−2 0|a(1)φx(1,t)|2dt +1 0 y0(x)φt(x, 0) dx−y1,φ(·,0)H−1,H1 0 Subject to (μ, φ0,φ 1)∈L2(QT)×H, (1.9) where φsolves the nonhomogeneous backwards problem ⎧ ⎨ ⎩ Lφ =μin QT φ=0 on ΣT (φ(·,T),φ t(·,T)) = (φ0,φ 1)in(0,1). Here, Jis the conjugate function of Jin the sense of Fenchel and Rockafellar [10,28] and, if ρ∈L∞(QT) and ρ0∈L∞(0,T) (that is, ρ−2and ρ−2 0are positively bounded from below), Jis coercive in L2(QT)×H thanks to (1.7). Therefore, if (ˆμ, ˆ φ0,ˆ φ1) denotes the minimizer of J, the corresponding optimal pair for Jis given by v=−a(1)ρ−2 0φx(1,·)in(0,T)andy=−ρ−2μin QT. At the discrete level, at least for standard approximation schemes, we may suspect that the coercivity of J may not hold uniformly with respect to the discretization parameters, leading to the pathologies and the lack of convergence we have just mentioned. On the other hand, the fact that the state variable yappears explicitly in the cost Jmakes it possible to avoid dual methods. We can use instead suitable primal methods to get an optimal pair (y,v)∈C(y0,y 1;T). The formulation, analysis and practical implementation of these primal methods is the main goal of this paper. NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1079 More precisely, the optimality conditions for the functional Jallow to express explicitly the optimal pair (y,v) in terms of a new variable, the solution of a fourth-order elliptic problem in the space-time domain QT that is well-posed under some conditions on T, the coefficient aand the weights ρand ρ0. Sufficient conditions are deduced from an appropriate global Carleman estimate, an updated version of the inequalities established in [3]. From a numerical viewpoint, this elliptic formulation is appropriate for a standard finite element analysis. By introducing adequate finite dimensional spaces, we are thus able to deduce satisfactory convergence results for the control, something that does not seem easy to get in the framework of a dual approach. A similar primal approach, based on ideas by Fursikov and Imanuvilov [16], has been used in [12]forthe numerical null controllability of the heat equation. This paper is organized as follows. In Section 2, adapting the arguments and results in [12], we show that the solution to (1.8) can be expressed in terms of the unique solution pto the variational problem (2.14) in the Hilbert space P, defined as the completion of P0with respect to the inner product (2.9); see Proposition 2.5. The well-posedness is deduced from the application of Riesz’s Theorem: a suitable global Carleman inequality ensures the continuity of the linear form in (2.14)forTlarge enough when ρand ρ0are given by (2.10); see Theorem 2.2. In Section 3, we analyze the variational problem (2.14) from the viewpoint of the finite element theory. Thus, we replace Pby a conformal finite element space Phof C1(QT) functions defined by (3.5)andweshowthat the unique solution ˆph∈Phto the finite dimensional problem (3.10) converges (strongly) for the P-norm to p as hgoes to zero. Section 4contains some numerical experiments that illustrate and confirm the convergence of the sequence {ˆph}. Finally, we present some additional comments in Section 5and we provide some details of the proof of Theorem 2.2 in the Appendix. 2. A variational approach to the null controllability problem With the notation introduced in Section 1, the following result holds: Proposition 2.1. Let T>0be large enough. Let us assume that ρand ρ0are positive and satisfy ρ∈C0(QT), ρ0∈C0(0,T)and ρ, ρ0≥ρ > 0. Then, for any (y0,y 1)∈Y, there exists exactly one solution to the extremal problem (1.8). The proof is simple. Indeed, for T≥T, null controllability holds and C(y0,y 1;T) is non-empty. Furthermore, it is a closed convex set of L2(QT)×L2(0,T). On the other hand, (y,v)→ J(y,v) is strictly convex, proper and lower-semicontinuous in L2(QT)×L2(0,T)and J(y,v)→+∞as (y,v)L2(QT)×L2(ΣT)→+∞. Hence, the extremal problem (1.8) certainly possesses a unique solution. In this paper, it will be convenient to assume that the coefficient abelongs to the family A(x0,a 0)=a∈C3([0,1]) :a(x)≥a0>0, −min [0,1] (a(x)+(x−x0)ax(x)) <min [0,1] a(x)+1 2(x−x0)ax(x)(2.1) where x0<0anda0is a positive constant. It is easy to check that the constant function a(x)≡a0belongs to A(x0,a 0). Similarly, any non-decreasing smooth function bounded from below by a0belongs to A(x0,a 0). Roughly speaking, a∈A(x0,a 0)meansthat ais sufficiently smooth, strictly positive and not too decreasing in [0,1]. 1080 N. Cˆ INDEA ET AL. Under the assumption (2.1), there exists “good” weight functions ρand ρ0which provide a very suitable solution to the original null controllability problem. They can be deduced from global Carleman inequalities. The argument is the following. First, let us introduce a constant β,with −min [0,1] (a(x)+(x−x0)ax(x)) <β<min [0,1] a(x)+1 2(x−x0)ax(x)(2.2) and let us consider the function φ(x, t):=|x−x0|2−βt2+M0,(2.3) where M0is such that φ(x, t)≥1∀(x, t)∈(0,1) ×(−T,T),(2.4) i.e. M0≥1−|x0|2+βT2. Then, for any λ>0weset ϕ(x, t):=e λφ(x,t).(2.5) The Carleman estimates for the wave equation are given in the following result: Theorem 2.2. Let us assume that x0<0,a0>0and a∈A(x0,a 0).Letβand ϕbe given respectively by (2.2) and (2.5). Moreover, let us assume that T>1 βmax [0,1] a(x)1/2(x−x0).(2.6) Then there exist positive constants s0and M, only depending on x0,a0,aC3([0,1]),bL∞(QT)and T,such that, for all s>s 0, one has sT −T1 0 e2sϕ |wt|2+|wx|2dxdt+s3T −T1 0 e2sϕ|w|2dxdt ≤MT −T1 0 e2sϕ|Lw|2dxdt+MsT −T e2sϕ|wx(1,t)|2dt (2.7) for any w∈L2(−T,T;H1 0(0,1)) satisfying Lw ∈L2((0,1) ×(−T,T)) and wx(1,·)∈L2(−T,T). There exists an important literature related to (global) Carleman estimates for the wave equation. Almost all references deal with the particular case a≡1; we refer to [3,4,18,31,33]. The case where ais non-constant is less studied; we refer to [17]. The proof of Theorem 2.2 follows closely the ideas used in the proofs of Theorems 2.1 and 2.5 in [4]toobtain a global Carleman estimate for the wave equation when a≡1. The parts of the proof which become different for non-constant aare detailed in the Appendix of this paper. In the sequel, it is assumed that x0<0anda0>0aregiven,a∈A(x0,a 0)and T>2 βmax [0,1] a(x)1/2(x−x0),with βsatisfying (2.2).(2.8) Let us consider the linear space P0={q∈C∞(QT):q=0onΣT}. The bilinear form (p, q)P:= QT ρ−2Lp Lq dxdt+T 0 ρ−2 0a(1)2px(1,t)qx(1,t)dt(2.9) NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1081 is a scalar product in P0. Indeed, in view of (2.8), the unique continuation property for the wave equation holds. Accordingly, if q∈P0,Lq =0inQTand qx=0on{1}×(0,T), then q≡0. This shows that (·,·)Pis certainly a scalar product in P0. Let Pbe the completion of P0with respect to this scalar product. Then Pis a Hilbert space for (·,·)Pand we can deduce from Theorem 2.2 the following result, that indicates which are the appropriate weights ρand ρ0for our controllability problem: Lemma 2.3. Let us assume that s>s 0,letusset ρ(x, t):=e −sϕ(x,2t−T),ρ 0(t):=ρ(1,t) (2.10) and let us consider the corresponding Hilbert space P. Then there exists a constant C0>0, only depending on x0,a0,aC3([0,1]),bL∞(QT),λ,sand T, such that p(·,0)2 H1 0(0,1) +pt(·,0)2 L2(0,1) ≤C0(p, p)P∀p∈P. (2.11) Proof. For every p∈P,wedenotebyp∈L2((0,1) ×(−T,T)) the function defined by p(x, t)=px, t+T 2. It is easy to see that p∈L2(−T,T;H1 0(Ω)), Lp ∈L2((0,1) ×(−T,T)) and px(1,·)∈L2(−T,T), so that we can apply Theorem 2.2 to p. Accordingly, we have sT −T1 0 e2sϕ |pt|2+|px|2dxdt+s3T −T1 0 e2sϕ|p|2dxdt ≤CT −T1 0 e2sϕ|Lp|2dxdt+CsT −T e2sϕ(1,t)|px(1,t)|2dt (2.12) where Cdepends on x0,a0,aC3([0,1]),bL∞(QT)and T. Replacing pby its definition in (2.12) and changing the variable tby t=2t−Twe obtain the following for any Tsatisfying (2.8): sQT ρ−2(|pt|2+|px|2)dxdt+s3QT ρ−2|p|2dxdt ≤CQT ρ−2|Lp|2dxdt+CsT 0 ρ−2 0|px(1,t)|2dt, where Cis replaced by a slightly different constant. Finally, from Corollary 2.8 in [4], we obtain the estimate (2.11).  Remark 2.4. Theestimate(2.11)mustbeviewedasanobservability inequality. As expected, it holds if and only if Tis large enough. Notice that, when a(x)≡1, the assumption (2.8)reads T>2(1 −x0). This confirms that, in this case, whenever T>2, (2.11) holds (it suffices to choose x0appropriately and apply Lem. 2.3;see[23]).  1082 N. Cˆ INDEA ET AL. The previous results lead to a very useful characterization of the optimal pair (y,v)forJ: Proposition 2.5. Let us assume that s>s 0,letussetρand ρ0as in (2.10)and let us consider the corresponding Hilbert space P.Let(y,v)∈C(y0,y 1,T)be the solution to (1.8). Then there exists p∈Psuch that y=−ρ−2Lp, v =−(a(x)ρ−2 0px)x=1.(2.13) Moreover, pis the unique solution to the following variational equality: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ QT ρ−2Lp Lq dxdt+T 0 ρ−2 0a2(1)px(1,t)qx(1,t)dt =1 0 y0(x)qt(x, 0) dx−y1,q(·,0)H−1,H1 0∀q∈P;p∈P. (2.14) Here and in the sequel, we use the following duality pairing: y1,q(·,0)H−1,H1 0=1 0 ∂ ∂x((−Δ)−1y1)(x)qx(x, 0) dx, where −Δis the Dirichlet Laplacian in (0,1). Proof. From the definition of the scalar product in P,weseethatpsolves (2.14) if and only if (p, q)P=1 0 y0(x)qt(x, 0) dx−y1,q(·,0)H−1,H1 0∀q∈P;p∈P. In view of Lemma 2.3 and Riesz’s Representation Theorem, problem (2.14) possesses exactly one solution in P. Let us now introduce yand vaccording to (2.13) and let us check that (y,v)solves(1.8). First, notice that y∈L2(QT)andv∈L2(0,T). Then, by replacing yand vin (2.14), we obtain the following: QT yLqdxdt+T 0 a(1)v(t)qx(1,t)dt=1 0 y0(x)qt(x, 0) dx−y1,q(·,0)H−1,H1 0∀q∈P. (2.15) Hence, (y,v) is the solution of the controlled wave system (1.1) in the transposition sense. Since y∈L2(QT) and v∈L2(0,T) the couple (y,v) belongs to C(y0,y 1,T). It remains to check that (y,v) minimizes the cost function Jin (1.8). But this is easy. Indeed, for any (z,w)∈C(y0,y 1,T) such that J(z,w)<+∞, one has: J(z,w)≥J(y,v)+QT ρ2y(z−y)dxdt+T 0 ρ2 0v(w−v)dt =J(y,v)−QT Lp (z−y)dxdt+T 0 ρ2 0v(w−v)dt=J(y,v). The last equality follows from the fact that QT Lp (z−y)dxdt=QT pL(z−y)dxdt +1 0 [pt(z−y)]T 0dx−[<(z−y)t,p> H−1,H1 0]T 0 −T 0 [a(x)px(z−y)]1 0dt+T 0 [a(x)p(z−y)x]1 0dt, the boundary condition for p(see Rem. 2.6 below), the fact that both (y,v)and(z,w)belongtoC(y0,y 1;T) and (2.13).  NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1083 Remark 2.6. From (2.13)and(2.14), we see that the function pfurnished by Proposition 2.5 solves, at least in the distributional sense, the following differential problem, that is of the fourth-order in time and space: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ L(ρ−2Lp)=0,(x, t)∈QT p(0,t)=0,(ρ−2Lp)(0,t)=0,t∈(0,T) p(1,t)=0,(ρ−2Lp +aρ−2 0px)(1,t)=0,t∈(0,T) (ρ−2Lp)(x, 0) = y0(x),(ρ−2Lp)(x, T )=0,x∈(0,1) (ρ−2Lp)t(x, 0) = y1(x),(ρ−2Lp)t(x, T )=0,x∈(0,1). (2.16) Notice that the “boundary” conditions at t=0andt=Tare of the Neumann kind.  Remark 2.7. The weights ρ−1and ρ−1 0behave exponentially with respect to s. For instance, we have ρ(x, t)−1=expseλ(|x−x0|2−β(2t−T)2+M0). For large values of the parameter s(greater than s0>0, see the statement of Thm. 2.2), the weights ρ−2and ρ−2 0may lead in practice to numerical overflow. One may overcome this situation by introducing a suitable change of variable. More precisely, let us introduce the variable z=ρp and the Hilbert space M=ρP, so that the formulation (2.14) becomes: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ QT ρ−2L(ρz)L(ρz)dxdt+T 0 ρ−2 0a2(1)(ρz)x(1,t)(ρz)x(1,t)dt =1 0 y0(x)(ρz)t(x, 0) dx−y1,(ρz)(·,0)H−1,H1 0∀z∈M;z∈M. (2.17) The well-posedness of this formulation is a consequence of the well-posedness of (2.14). Then, after some computations, the following is found: ρL(ρ−1z)=ρ−1(ρz)t−(a(ρz)x)x+bρz =(ρ−1ρt)z+zt−ax((ρ−1ρx)z+zx)−a(2ρ−1ρxzx+ρ−1ρxxz+zxx)+bz with ρ−1ρx=−sϕx(x, 2t−T),ρ −1ρt=−2sϕt(x, 2t−T),ρ −1ρxx =−sϕxx +(sϕx)2. Similarly, (ρ−1 0(ρz)x)(1,t)=zx(1,t). Consequently, in the bilinear part of (2.17), there is no exponential (but only polynomial) function of s.Inthe right hand side (the linear part), the change of variable introduces negative exponentials in s. A similar trick has been used in [12] in the context of the heat equation, where we find weights that blow up exponentially as t→T−. Remark 2.8. The exponential form of the weights ρand ρ0is purely technical and is related to Carleman estimates. Actually, since for any sand λthese weights are uniformly bounded and uniformly positive in QT, the space Pis independent of ρand ρ0and one could apply the primal approach to the cost J(defined in (1.8)) for any bounded and positive weights. In particular, one could simply take ρ≡1andρ0≡1; the estimates (2.11) would then read as follows: p(·,0)2 H1 0(0,1) +pt(·,0)2 L2(0,1) ≤C0Lp2 L2(QT)+a(1) px(1,·)2 L2(0,T)∀p∈P(2.18) 1084 N. Cˆ INDEA ET AL. for some constant C0>0. This inequality can also be obtained directly by the multipliers method; we refer to [32] and references therein.  Remark 2.9. As remarked in [4](seeRem.2.7),theestimate(2.11)canbeprovenforaweightρ0which blows up at t=0andt=T. For this purpose, we consider a function θδ∈C2([0,T]) with θδ(0) = θδ(1) = 0 and θδ(x) = 1 for every x∈(δ, T −δ). Then, introducing again p(x, t):=θδ(t)p(x, (t+T)/2), it is not difficult to see that the proofs of Lemma 2.3 and Theorem 2.2 can be adapted to obtain (2.11)with ρ(x, t)=e −sϕ(x,2t−T),ρ 0(t)=θδ(t)−1ρ(1,t). Thanks to the properties of θδ,thecontrolvdefined by v=−θ2 δρ−2 0a(1)pxx=1 vanishes at t=0andalsoatt=T, a property which is very natural and useful in the boundary controllability context. In the sequel, we will use this modified weight ρ0, imposing in addition, for numerical purposes, the following behavior near t=0andt=T: lim t→0+ θδ(t) √t=O(1),lim t→T− θδ(t) √T−t=O(1).(2.19) 3. Numerical analysis of the variational approach We now highlight that the variational formulation (2.14) allows to obtain a sequence of approximations {vh} that converge strongly towards the null control vfurnished by the solution to (1.8). 3.1. A conformal finite dimensional approximation Let us introduce the bilinear form m(·,·)overP×P m(p, q):=(p, q)P=QT ρ−2Lp Lq dxdt+T 0 a(1)2ρ−2 0px(1,t)qx(1,t)dt and the linear form ,with , q:= 1 0 y0(x)qt(x, 0) dx−y1,q(·,0)H−1,H1 0. Then (2.14) reads as follows: m(p, q)=, q,∀q∈P;p∈P. (3.1) Let us assume that a finite dimensional space Ph⊂Pis given for each h∈R2 +. Then we can introduce the following approximated problems: m(ph,q h)=, qh,∀qh∈Ph;ph∈Ph.(3.2) Obviously, each (3.2) is well-posed. Furthermore, we have the following classical result: Lemma 3.1. Let p∈Pbe the unique solution to (3.1)and let ph∈Phbe the unique solution to (3.2).Then we have: p−phP≤inf qh∈Php−qhP. NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1091 4. Numerical experiments We now present some numerical experiments concerning the solution of (3.10), which can in fact be viewed as a linear system involving a banded sparse, definite positive, symmetric matrix of order 4NxNt. We will denote by Mhthis matrix. If {ˆph}stands for the corresponding vector solution of size 4NxNt,wemaywrite (ˆph,q h)Ph=(Mh{ˆph},{qh}) for any qh∈Ph. We will use an exact integration method in order to compute the components of Mhand the (direct) Cholesky method with reordering to solve the linear system. After the computation of ˆph,thecontrolˆvhis given by (3.12). Observe that, in view of the definition of the space Ph, the derivative with respect to xof ˆphis a degree of freedom of {ˆph}; hence, the computation of ˆvh does not require any additional calculus. The corresponding controlled state ˆyhmay be obtained by using the pointwise first equality (3.12)or,equivalently, by solving (2.15). However, in order to check the action of the control function ˆvhproperly, we have computed ˆyhby solving (1.1)withaC1finite element method in space and a standard centered scheme of second order in time. Thus, let us introduce the finite dimensional spaces Zh={zh∈C1([0,1]) : zh[xi,xi+Δx]∈P3,x ∀i=1,...,N x} and Z0h={zh∈Zh:zh(0) = zh(1) = 0 }. Then, a suitable approximation ˆyhof the controlled state yis defined in the following standard way: •At time t=0,ˆyhis given by yh(·,0) = PZh(y0), the projection of y0on Zh; •At time t1=Δt,ˆyh(·,t 1)∈Zhis given by the solution to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 21 0 (ˆyh(x, t1)−ˆyh(x, t0)−Δt y1(x)) (Δt)2φdx +1 0 [a(x)ˆyh,x(x, t0)φx+b(x, t0)ˆyh(x, t0)φ]dx=0 ∀φ∈Zh0;ˆyh(0,t 1)∈Zh,ˆyh(0,t 1)=0,ˆyh(1,t 1)=ˆvh(t1). (4.1) •At time t=tn=nΔt,n=2,···,N t,ˆyh(·,t n) solves the following linear problem: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 21 0 (ˆyh(x, tn)−2ˆyh(x, tn−1)+ˆyh(·,t n−2) (Δt)2φdx +1 0 [a(x)ˆyh,x(x, tn−1)φx+b(x, tn)ˆyh(x, tn−1)φ]dx=0 ∀φ∈Zh0;ˆyh(0,t n)∈Zh,ˆyh(0,t n)=0,ˆyh(1,t n)=ˆvh(tn). (4.2) This requires a preliminary projection of ˆvhon a grid on (0,T) fine enough in order to fulfill the underlying CFL condition. To this end, we use the following interpolation formula: for any ph∈Phand any θ∈[0,1], we have: ph,x(1,t j+θΔt)=(2θ+1)(θ−1)2ph,x(1,t j)+Δt θ(1 −θ)2ph,xt(1,t j) +θ2(3 −2θ)ph,x(1,t j+1)+Δt θ2(θ−1) ph,xt(1,t j+1)(4.3) for all t∈[tj,t j+1]. We will consider a constant coefficient a(x)≡a0= 1 and a constant potential b(x, t)≡1inQT. We will take T=2.2, x0=−1/20, β=0.99 and M0=1−x2 0+βT2,sothat(2.8) holds. Finally, concerning the parameters λand s(which appear in (2.12)), we will take λ=0.1ands=1. Remark 4.1. Let us emphasize that our approach does not require in any way the discretization meshes to be uniform.  1092 N. Cˆ INDEA ET AL. Table 1. The constant C0hwith respect to h. Δx, Δt 1/10 1/20 1/40 1/80 T=2.26.60 ×10−27.61 ×10−28.56 ×10−29.05 ×10−2 T=1.50.565 2.672 17.02 96.02 4.1. Estimating the Carleman constant Before prescribing the initial data, let us check that the finite dimensional analog of the observability constant C0in (2.11) is uniformly bounded with respect to hwhen (2.8) is satisfied. We consider here the case a≡1and b≡1. In the space Ph, the approximate version of (2.11)is (Ah{ph},{ph})≤C0h(Mh{ph},{ph})∀{ph}∈Ph, where Ahis the square matrix of order 4NtNxdefined by the identities (Ah{ph},{qh}):=1 0 (ph,x(x, 0) qh,x(x, 0) + ph,t(x, 0) qh,t(x, 0)) dx. Therefore, C0his the solution of a generalized eigenvalue problem: C0h=max{λ:∃ph∈Ph,p h=0,such that Ah{ph}=λMh{ph}}.(4.4) We can easily solve (4.4) by the power iteration algorithm. Table 1collects the values of C0hfor various h=(Δx, Δt)forT=2.2andT=1.5, with Δt =Δx. As expected, C0his bounded in the first case only. The same results are obtained for Δt =Δx. In agreement with Remark 2.8, we obtain the same behavior of the constant with respect to Tfor any s,in particular for s= 0 leading to ρ≡1andρ0≡1. 4.2. Smooth initial data and constant speed of propagation We now solve (2.14)witha≡1 and smooth initial data. For simplicity, we also take a constant potential b≡1. For (y0,y 1)=(sin(πx),0), Table 2collects relevant numerical values with respect to h=(Δx, Δt). We have taken Δt =Δx for simplicity but, in this finite element framework, any other choice is possible. In particular, we have reported the condition number κ(Mh)ofthematrixMh, defined by κ(Mh)=|||Mh|||2|||M−1 h|||2 (the norm |||Mh|||2stands for the largest singular value of Mh). We observe that this number behaves polynomially with respect to h. Table 2clearly exhibits the convergence of the variables ˆphand ˆvhas hgoes to zero. Assuming that h= (1/160,1/160) provides a reference solution, we have also reported in Table 2the estimates p−ˆphPand v−ˆvhL2(0,T). We observe then that p−ˆphP=O(h1.91),v−ˆvhL2(0,T )=O(h1.56). The corresponding state ˆyhis computed from the main equation (1.1), as explained above, taking Δt =Δx/4. That is, we use (4.3)withθ=0,1/4,1/2and3/4oneachinterval[tj,t j+1]. We observe the following behavior with respect to h: ˆyh(·,T)L2(0,1) =O(h1.71),ˆyt,h(·,T)H−1(0,T )=O(h1.31), NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1093 Table 2. (y0(x),y 1(x)) ≡(sin(πx),0), a≡1,b≡1–T=2.2. Δx, Δt 1/10 1/20 1/40 1/80 κ(Mh)3.06 ×1081.57 ×1010 6.10 ×1011 2.47 ×1013 ˆphP1.541 ×10−11.548 ×10−11.550 ×10−11.550 ×10−1 ˆph−pP4.46 ×10−21.45 ×10−24.01 ×10−38.38 ×10−4 ˆvhL2(0,T )5.421 ×10−15.431 ×10−15.434 ×10−15.434 ×10−3 ˆvh−vL2(0,T )2.39 ×10−28.12 ×10−32.48 ×10−39.57 ×10−4 ˆyh(·,T)L2(0,1) 1.80 ×10−28.18 ×10−31.64 ×10−35.85 ×10−4 ˆyt,h(·,T)H−1(0,1) 3.06 ×10−28.25 ×10−33.59 ×10−31.93 ×10−3 Table 3. (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and a≡1,b≡1–T=2.2. Δx, Δt 1/10 1/20 1/40 1/80 1/160 ˆphP4.38 ×10−23.95 ×10−24.20 ×10−24.31 ×10−24.33 ×10−2 ˆph−pP1.80 ×10−16.30 ×10−21.66 ×10−22.78 ×10−3– ˆvhL2(0,T )1.48 ×10−11.33 ×10−11.53 ×10−11.64 ×10−11.67 ×10−1 ˆvh−vL2(0,T )9.81 ×10−26.28 ×10−23.80 ×10−21.11 ×10−2– ˆyh(·,T)L2(0,1) 1.09 ×10−17.67 ×10−23.70 ×10−21.11 ×10−21.87 ×10−3 ˆyt,h(·,T)H−1(0,1) 1.36 ×10−18.82 ×10−25.16 ×10−21.76 ×10−22.82 ×10−3 which shows that the control ˆvhgiven by the second equality in (3.12) is a good approximation of a null control for (1.1). Figure 2-Left displays the function ˆph∈P(the unique solution to (3.10)) for h=(1/80,1/80). Figure 2-right displays the associated control ˆvh. As a consequence of the introduction of the function θδin the weight, we see that ˆvhvanishes at times t=0andt=T. Finally, Figure 3displays the corresponding controlled state ˆyh. Table 3and Figures 4and 5provide the results for y0(x)≡e−500(x−0.2)2and y1(x)≡0. We still observe the convergence of the variables ˆph,ˆvhand ˆyh, with a lower rate. This is due in part to the shape of the initial condition y0. Precisely, we get p−ˆphP=O(h1.74), ˆvh−vL2(0,T)=O(h0.68), ˆyh(·,T)L2(0,1) =O(h1.35) and ˆyt,h(·,T)H−1(0,T )=O(h1.11). 4.3. Initial data (y0,y 1)∈H1(0,1) ×L2(0,1) and constant speed of propagation Let us enhance that our approach, in agreement with the theoretical results, also provides convergent results for irregular initial data. We take a continuous but not differentiable initial state y0and a piecewise constant initial speed y1: y0(x)≡x1[0,1/2](x)+(1−x)1 ]1/2,1](x),y 1(x)≡10 ×1[1/5,1/2](x).(4.5) The other data are unchanged, except b, that is taken equal to zero. Observe that these functions remain compatible with the C1finite element used to approximate p,sincey0 and y1only appear in the right hand side of the variational formulation and πΔxy0and πΔxy1make sense; see (3.10). The unique difference is that, once ˆphand ˆvhare known, ˆyhmust be computed from (4.1)–(4.2)using aC0(and not C1) spatial finite element method. Recall however that these initial data typically generate pathological numerical behavior when the usual dual approach, based on the minimization of (1.6), is used. Some numerical results are given in Table 4and Figures 6and 7. As before, we observe the convergence of the variable ˆphand therefore ˆvhand ˆyhas h→0. We see that ˆph−pP=O(h1.48)andˆvh−vL2(0,1) =O(h1.23). In particular, we do not observe oscillations for the control or the functions ˆphand ˆph,t at the initial time. 1094 N. Cˆ INDEA ET AL. Figure 1. log10 p−ˆphP()andlog 10 v−ˆvhL2(0,T)(◦)vs. log10(h). Figure 2. (y0(x),y 1(x)) ≡(sin(πx),0) and a≡1–Thesolutionˆphover QT(Left) and the corresponding variable ˆvhon (0,T)(Right)–h=(1/80,1/80). Table 4. (y0,y 1)givenby(4.5)anda≡1–T=2.2. Δx, Δt 1/10 1/20 1/40 1/80 1/160 ˆphP3.16 ×10−12.89 ×10−22.73 ×10−22.65 ×10−22.61 ×10−1 ˆph−pP1.12 ×10−14.62 ×10−21.70 ×10−25.12 ×10−3- ˆvhL2(0,T )1.23 1.11 1.05 1.02 1.004 ˆvh−vL2(0,T )2.52 ×10−11.25 ×10−15.57 ×10−21.90 ×10−2- ˆyh(·,T)L2(0,1) 1.09 ×10−15.40 ×10−22.20 ×10−21.09 ×10−26.20 ×10−3 ˆyt,h(·,T)H−1(0,1) 7.25 ×10−24.62 ×10−22.85 ×10−25.12 ×10−36.75 ×10−3 NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1095 Figure 3. (y0(x),y 1(x)) ≡(sin(πx),0) and a≡1–Thesolutionˆyhover QT–h=(1/80,1/80). Figure 4. (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and a≡1–Thesolutionˆphover QT(left) and the corresponding variable ˆvhon (0,T)(right)–h=(1/80,1/80). 4.4. Discontinuous initial data y0and constant speed of propagation The method also provides convergent results for data y0only in L2(0,1). We consider the following initial condition: y0(x)≡1[0.5,0.7](x),y 1(x)≡0.(4.6) The other data are unchanged with respect to Section 4.3. This leads to pathological numerical behavior when other frequently used dual methods are employed (we refer to [24]). Some numerical results are given in Table 5 and Figure 8. Once again, the convergence of the variable ˆphand therefore ˆvhand ˆyhas h→0 is observed. 1096 N. Cˆ INDEA ET AL. Figure 5. (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and a≡1–Thesolutionˆyhover QT–h= (1/80,1/80). Figure 6. (y0,y 1)givenby(4.5)anda≡1–Thesolutionˆphover QT(left) and the corresponding variable ˆvhon (0,T)(right)–h=(1/80,1/80). Table 5. (y0,y 1)givenby(4.6)anda≡1–T=2.2. Δx, Δt 1/10 1/20 1/40 1/80 1/160 ˆphP1.01 ×10−11.00 ×10−19.71 ×10−29.53 ×10−29.47 ×10−2 ˆvhL2(0,T )3.42 ×10−13.27 ×10−13.19 ×10−13.14 ×10−13.14 ×10−1 ˆyh(·,T)L2(0,1) 1.24 ×10−19.27 ×10−27.26 ×10−25.88 ×10−23.12 ×10−2 ˆyt,h(·,T)H−1(0,1) 1.55 ×10−11.16 ×10−11.06 ×10−17.13 ×10−26.02 ×10−2 NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1097 Figure 7. (y0,y 1)givenby(4.5)anda≡1–Thesolutionˆyhover QT–h=(1/80,1/80). Figure 8. (y0,y 1)givenby(4.6)anda≡1–Thesolutionˆphover QT(left) and the corresponding variable ˆvhon (0,T)(right)–h=(1/80,1/80). 4.5. Non constant smooth speed of propagation Finally, let us consider a non-constant function a=a(x)(wereferto[20] for the dual approach in this case). In order to illustrate the robustness of our method, we will take a coefficient a∈C1([0,1]) with a(x)=⎧ ⎪ ⎨ ⎪ ⎩ 1x∈[0,0.45] ∈[1., 5.](a(x)>0),x∈(0.45,0.55) 5x∈[0.55,1] (4.7) so that condition (2.8)isequivalenttoT>2(1 + 1/20)√5≈4.69 (taking again x0=−1/20). In order to reduce the computational cost, we take as before T=2.2 and we still observe that the constant C0hin (4.4)is uniformly bounded. 1098 N. Cˆ INDEA ET AL. Figure 9. (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and agiven by (4.7)–Thesolutionˆphover QT (left) and the corresponding variable ˆvhon (0,T)(right)–h=(1/80,1/80). Table 6. (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and agiven by (4.7)–T=2.2. Δx, Δt 1/10 1/20 1/40 1/80 1/160 ˆphPh3.87 ×10−23.44 ×10−23.75 ×10−23.85 ×10−23.86 ×10−2 ˆph−pPh1.25 ×10−15.75 ×10−22.64 ×10−21.01 ×10−2– ˆvhL2(0,T )7.74 ×10−26.53 ×10−29.16 ×10−21.01 ×10−11.03 ×10−1 ˆvh−vL2(0,T )5.07 ×10−14.17 ×10−22.03 ×10−24.86 ×10−3– ˆyh(·,T)L2(0,1) 1.09 ×10−17.89 ×10−21.81 ×10−21.16 ×10−21.71 ×10−3 ˆyt,h(·,T)H−1(0,1) 1.01 ×10−18.39 ×10−24.81 ×10−27.52 ×10−31.55 ×10−3 We take again (y0(x),y 1(x)) ≡(e−500(x−0.2)2,0) and b≡0. Table 6illustrates the convergence of the approximations with respect to h.Figures9and 10 depict for h=(1/80,1/80) the functions ˆph,ˆvhand ˆyh.Inparticular, in Figure 10, we can observe the diffraction of the wave when crossing the transitional zone (0.45,0.55). 5. Further comments and concluding remarks Let us begin this section with some general considerations on the use of Carleman weights that serve to justify our approach: (i) The search of a control minimizing Jin (1.8), where yis involved, is very appropriate from the numerical viewpoint. As shown in Section 2, the explicit occurrence of the state variable yleads to an elliptic problem in QT, that is easy to analyze and solve (at this level, the particular choice of the weight is less important). This approach does not require the discretization of the wave operator, as for usual dual approachs; therefore, it does not generate any spurious oscillations and leads to numerical well-posedness. This is an important feature of the approach. (ii) The Carleman weights provide regularity of the solution to (2.14) and therefore allows to derive estimates of the errors p−phPin term of h=(Δx, Δt). This will be detailed in a forthcoming work. (iii) The process can be viewed as a first step for the numerical controllability of semi-linear problems: if we just apply a fixed-point argument, we will find at each iterate a linear equation with non-regular coefficients depending on xand tfor which the present approach is adequate. (iv) In our numerical experiments we have not found any essential difference for small or large sor λ:thisisin full agreement with Remark 2.8 and Section 4.1. NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1099 Figure 10. y0(x)≡e−500(x−0.2)2and agiven by (4.7)–Thesolutionˆyhover QT–h= (1/80,1/80). (v) Furthermore, we mention that the approach has been considered by analogy with the similar analysis in references [12]to[15], dealing with heat equations. 5.1. Primal versus dual approach (I): analogies The solution to the variational formulation (2.14) is also the unique minimizer of the functional I,with I(p):=1 2QT ρ−2|Lp|2dxdt+1 2T 0 ρ−2 0a(1)2|px(1,·)|2dt −1 0 y0(x)pt(x, 0) dx+y1,p(·,0)H−1,H1 0. (5.1) This is similar to the conjugate functional Jin (1.9). Actually, we notice that J(μ, φ0,φ 1)=I(−φ) for all (μ, φ0,φ 1)∈L2(QT)×H. Therefore, the extremal problems (1.9)and(5.1) are connected to each other having (1.8)asstartingpoint. The problem (2.14), deduced from the primal approach belongs to the framework of elliptic variational problems in two dimensions and is well tailored for a resolution with finite elements. The dual problem (1.9) is of hyperbolic nature: the time variable is kept explicitly and time integration is required. Note that we may also derive the optimality conditions for J(as we did in Sect. 2for J): this leads, at least formally, to the problem (2.14). We also mention [26] where a (different) variational approach is introduced. 5.2. Primal versus dual approaches (II): discrete properties The variational approach used here leads to satisfactory convergence results, in particular the strong convergence of the approximate controls ˆvhtowards a null control of the wave equation. This relies in a fundamental way on the fact that we work in a subspace Phof P. Indeed, this allows to write directly the Carleman estimate in Phand get that the function I(given by (5.1)) is uniformly coercive with respect to the discretization parameter h. On the other hand, notice that no wave equation has to be solved in order to compute the approximations ˆvh. For each h,onceˆvhis known, we must solve the wave equation, in a post-treatment process, to compute the 1100 N. Cˆ INDEA ET AL. corresponding state ˆyh(recall that, actually, this may be avoided by using directly the optimality condition y=−ρ−2Lp). This is in contrast with the dual approach. Indeed, the minimization of Jby an iterative process requires the resolution of wave equations, through a decoupled space and time discretization. As recalled in the introduction, this may lea to numerical pathologies (the occurrence of spurious high frequency solutions) and, therefore, needs some specific numerical approximations and techniques. We mention the work [5], where the authors prove, in a close context and within a dual approach, a weaker uniform semi-discrete Carleman estimate with an additional term in the right hand side, necessary to absorb these possibly spurious high frequencies (see [5], Thm. 2.3). Notice that the computed ˆvhare not apriorinull controls for discrete systems (associated to the wave equation (1.1)), but simply approximations of the control vfurnished by the solution to (1.8). If one wants to go further in the comparison, it can be said that the primal approach aims to first compute the control for (1.1) and then approximate it, while the dual classical method aims first to discretize (1.1) and then control the corresponding finite dimensional system. Let us also observe that the (primal) approach in this paper is relatively easy to implement. In practice, the resolution is reduced to solve a linear system, with a banded sparse, symmetric and definite positive matrix, for which efficient direct LU type solvers are known and available. Furthermore, we may want to adapt (and refine locally) the mesh of QTin order to improve convergence and such adaptation is much simpler than in the dual approach, where tis ”conserved” as a time variable. For additional considerations, see also [8]and[9]. 5.3. Mixed formulation and C0-approximation The approach can be extended to the higher dimensional case of the wave equation in a bounded set Ω⊂RN, with N≥2. However, the use of C1-finite element is a bit more involved. Arguing as in [12], we may avoid this difficulty by introducing a mixed formulation equivalent to (2.14). The idea is to keep explicit the variable yin the formulation and to introduce a Lagrange multiplier, associated to the constraint ρ2y+Lp =0(see(2.13)). We obtain the following mixed formulation: find (y,p,λ)∈Z×P×Z such that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ QT ρ2y y dxdt+T 0 ρ−2 0a(1)2px(1,t)px(1,t)dt+QT λ(ρ2y+Lp)dxdt =1 0 y0(x)pt(x, 0) dx−y1, p(·,0)H−1,H1 0∀(y,p)∈Z×P, QT λ(ρ2y+Lp)dxdt=0 ∀λ∈Z, (5.2) where Z=L2(ρ2;QT):=z∈L1 loc(QT):QT ρ2|z|2dxdt<+∞. Taking advantage of the global estimate (2.7), we may show, through an appropriate inf-sup condition, that (5.2) is well-posed in Z×P×Z. Moreover, the approximation of this formulation may be addressed using C0-finite element, which is very convenient. The approximation is non-conformal. More precisely, the variable pis now sought in a space Rhof C0-functions that is not included in P. At the discrete level, (5.2) reduces the controllability problem to the inversion of a square, banded and symmetric matrix. Moreover, as before, no wave equation has to be solved, whence the numerical pathology described above is not expected. However, since the underlying approximation is not conformal (this is the price to pay to avoid C1finite elements), a careful (and apriorinot straightforward) choice for Rhhas to be done in order to guarantee a uniform discrete inf-sup condition. The analysis of this point, as well as the use of stabilized finite elements, will be detailed in a future work. NUMERICAL CONTROLLABILITY OF THE WE THROUGH PRIMAL METHODS AND CARLEMAN ESTIMATES 1107 Moreover, simple computations show that, for every t∈(−T,−T+η)∪(T−η,T), the function ψ(·,t)satisfies: ⎧ ⎪ ⎨ ⎪ ⎩ (1 −ε)min [0,1] |ψt(x, t)|≥max [0,1] a(x)1/2|ψx(x, t)| max [0,1] ψ(x, t)<min [0,1] ψ(x, 0).(A.8) Let χ∈C∞ c(R) a cut-off function such that 0 ≤χ≤1and χ(t)=1,if |t|≤T−η 0,if |t|≥T Then we can apply Theorem A.1 to the function ˜w:= χw, whence the following Carleman estimate holds sT −T1 0 e2sϕ |˜wt|2+|˜wx|2dxdt+s3T −T1 0 e2sϕ|˜w|2dxdt ≤MT −T1 0 e2sϕ|L˜w|2dxdt+MsT −T e2sϕ|˜wx(1,t)|2dt. Since L˜w=χLw +χttw+2χtwt, we deduce from (A.8)that sT−η −T+η1 0 e2sϕ |wt|2+|wx|2dxdt+s3T−η −T+η1 0 e2sϕ|w|2dxdt ≤MT −T1 0 e2sϕ|Lw|2dxdt+MsT −T e2sϕ|wx(1,t)|2dt +M−T+η −T1 0 e2sϕ(|wt|2+|w|2)dxdt+MT T−η1 0 e2sϕ(|wt|2+|w|2)dxdt. (A.9) Let us denote by Es=Es(t)theenergy associated to the operator L,thatis, Es(t):=1 21 0 e2sϕ |wt|2+a|wx|2dx. 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