A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation

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1 A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation Arnaud Munch, Souza Diego A. To cite this version: Arnaud Munch, Souza Diego A.. A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation <hal v2> HAL Id: hal Submitted on 3 Jun 214 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A mixed formulation for the direct approximation of L 2 -weighted controls for the linear heat equation Arnaud Münch Diego A. Souza June 3, 214 Abstract This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 213], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension. Keywords: Linear heat equation; Null controllability; Finite element methods; Mixed formulation. Mathematics Subject Classification (21)- 35K35, 65M12, 93B4, 65K1 Contents 1 Introduction. The null controllability problem 2 2 Control of minimal weighted L 2 -norm: mixed reformulations The penalized case: Mixed formulation I Mixed formulation Dual problem of the extremal problem (15) The penalized case : Mixed formulation II (relaxing the condition L ϕ ε = in L 2 (Q T )) Third mixed formulation of the controllability problem : the limit case ε = Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 662, Campus de Cézeaux, 63177, Aubière, France. s: arnaud.munch@math.univ-bpclermont.fr. Dpto. EDAN, University of Sevilla, 418 Sevilla, Spain and Departamento de Matemática, Universidade Federal da Paraíba, , João Pessoa PB, Brazil. desouza@us.es. Partially supported by CAPES (Brazil) and grant MTM (DGI-MICINN, Spain). 1

3 1 INTRODUCTION. THE NULL CONTROLLABILITY PROBLEM 2 3 Numerical approximation and experiments Discretization of the mixed formulation (16) Normalization and discretization of the mixed formulation (32) The discrete inf-sup test Numerical experiments for the mixed formulation (16) Conjugate gradient for Jε,r Numerical experiments for the mixed formulation (32) - limit case ε = Concluding remarks and Perspectives 31 A Appendix 32 A.1 Appendix : Fourier expansion of the control of minimal L 2 (ρ, q T ) norm A.2 Appendix: Tables Introduction. The null controllability problem Let Ω R N be a bounded connected open set whose boundary Ω is regular enough (for instance of class C 2 ). Let ω Ω be a (small) nonempty open subset and assume that T >. In the sequel, for any τ > we denote by Q τ, q τ and Σ τ the sets Ω (, τ), ω (, τ) and Ω (, τ), respectively. This work is concerned with the null controllability problem for the heat equation y t (c(x) y) + d(x, t)y = v 1 ω, in Q T, y =, on Σ T, y(x,) = y (x), in Ω. Here, we assume that c := (c i,j ) C 1 (Ω; M N (R)) with (c(x)ξ, ξ) c ξ 2 in Ω (c > ), d L (Q T ) and y L 2 (Ω); v = v(x, t) is the control (a function in L 2 (q T )) and y = y(x, t) is the associated state. Moreover, 1 ω is the characteristic function associated to the set ω. In the sequel, we shall use the following notation : (1) L y := y t (c(x) y) + d(x, t)y, L ϕ := ϕ t (c(x) ϕ) + d(x, t)ϕ. For any y L 2 (Ω) and v L 2 (q T ), there exists exactly one solution y to (1), with the regularity y C ([, T];L 2 (Ω)) L 2 (, T;H 1 (Ω)) (see [3, 7]). Accordingly, for any final time T >, the associated null controllability problem at time T is the following : for each y L 2 (Ω), find v L 2 (q T ) such that the corresponding solution to (1) satisfies y(,t) = in Ω. (2) The controllability of PDEs is an important area of research and has been the subject of many papers in recent years. Some relevant references are [27, 29, 35] and [12]. In particular, we refer to [21] and [28] where the null controllability of (1) is proved. The numerical approximation is also a fundamental issue, since it is not in general possible to get explicit expression of controls. Due to the strong regularization property of the heat kernel, numerical approximation of controls is a rather delicate issue. The same holds in inverse problems theory when parabolic equations and systems are involved (see [15]). This have been exhibited numerically in [5] who made use of duality argument and focused on the control of minimal square integrable norm: the problem reads Minimize J 1 (y,v) := 1 v(x, t) 2 dx dt 2 q T (3) Subject to (y, v) C(y, T)

4 1 INTRODUCTION. THE NULL CONTROLLABILITY PROBLEM 3 where C(y ;T) denotes the linear manifold C(y ;T) := { (y,v) : v L 2 (q T ), y solves (1) and satisfies (2) }. The earlier contribution is due to Glowinski and Lions in [23] (updated in [24]) and relies on duality arguments. Duality allows to replace the original constrained minimization problem by an unconstrained and a priori easier minimization (dual) problem. The dual problem associated with (3) is : min ϕ T H J 1(ϕ T ) := 1 ϕ(x, t) 2 dxdt + y (x)ϕ(x,)dx (4) 2 q T Ω where the variable ϕ solves the backward heat equation : L ϕ = in Q T, ϕ = on Σ T ; ϕ(, T) = ϕ T in Ω, (5) and the Hilbert space H is defined as the completion of D(Ω) with respect to the norm ϕ T H := ϕ L 2 (q T ). In view of the unique continuation property to (5), the mapping ϕ T ϕ T H is a Hilbertian norm in D(Ω). Hence, we can certainly consider the completion of D(Ω) for this norm. The coercivity of the functional J 1 in H is a consequence of the so-called observability inequality ϕ(,) 2 L 2 (Ω) C q T ϕ(x, t) 2 dx dt ϕ T H, (6) where ϕ solves (5). This inequality holds for some constant C = C(ω,T) and, in turn, is a consequence of some appropriate global Carleman inequalities; see [21]. The minimization of J1 is numerically ill-posed, essentially because of the hugeness of the completed space H. The control of minimal square integrable norm highly oscillates near the final time T, property which is hard to capture numerically. We refer to [1, 25, 31, 34] where this phenomenon is highlighted under several perspectives. Moreover, at the level of the approximation, the minimization of J1 requires to find a finite dimensional and conformal approximation of H such that the corresponding discrete adjoint solution satisfies (5), which is in general impossible for polynomial piecewise approximations. In practice, the trick initially described in [23], consists first to introduce a discrete and consistent approximation of (1) and then to minimize the corresponding discrete conjugate functional. However, this requires to get some uniform discrete observability inequalities which is a delicate issue, strongly depend on the approximations used (we refer to [3, 16, 36] and the references therein) and is still open in the general case of the heat equation with non constant coefficients. This fact and the hugeness of H has raised many authors to relax the controllability problem: precisely, the constraint (2). We mention the references [3, 5, 36] and notably [2, 19, 26] for some numerical realizations. In [18] (see also [17] in a semi-linear case), a different - so-called primal approach - allowing more general results has been used and consists to solve directly optimality conditions : specifically, the following general extremal problem (initially introduced by Fursikov and Imanuvilov in [21]) is considered : Minimize J(y,v) := 1 ρ 2 y 2 dx dt + 1 ρ 2 2 Q T 2 v 2 dx dt q T (7) Subject to (y, v) C(y, T). The weights ρ = ρ(x, t) and ρ = ρ (x, t) are continuous, uniformly positive and are assumed to belong to L (Q T δ ) for any δ > (hence, they can blow up as t T ). Under those conditions, the extremal problem (7) is well-posed (see [18]).

5 1 INTRODUCTION. THE NULL CONTROLLABILITY PROBLEM 4 Moreover, the explicit occurrence of the term y in the functional allow to solve directly the optimality conditions associated with (7): defining the Hilbert space P as the completion of the linear space P = {q C (Q T ) : q = on Σ T } with respect to the scalar product (p, q) P := ρ 2 L p L q dxdt + ρ 2 p q dx dt, (8) Q T q T the optimal pair (y, v) for J is characterized as follows y = ρ 2 L p in Q T, v = ρ 2 p 1 ω in Q T (9) in term of an additional variable p P unique solution to the following variational equality : (p, q) P = y (x)q(x,) dx, q P. (1) Ω The well-posedeness of this formulation is ensured as soon as the weights ρ, ρ are of Carleman type (in particular ρ and ρ blow up exponentially as t T ); this specific behavior near T reinforces the null controllability requirement and prevents the control of any oscillations near the final time. The search of a control v in the manifold C(y, T) is reduced to solve the (elliptic) variational formulation (1). In [18], the approximation of (1) is performed in the framework of the finite element theory through a discretization of the space-time domain Q T. In practice, an approximation p h of p is obtained in a direct way by inverting a symmetric positive definite matrix, in contrast with the iterative (and possibly divergent) methods used within dual methods. Moreover, a major advantage of this approach is that a conformal approximation, say P h of P, leads to the strong convergence of p h toward p in P, and consequently from (9), to a strong convergence in L 2 (q T ) of v h := ρ 2 p h1 ω toward v, a null control for (1). It is worth to mention that, for any h >, v h is not a priori an exact control for any finite dimensional system (which is not necessary at all in practice) but an approximation for the L 2 -norm of the control v. The variational formulation (1) derived from the optimality conditions (9) is obtained assuming that the weights ρ and ρ are both strictly positive in Q T and q T respectively. In particular, this approach does not apply for the control of minimal L 2 -norm, for which simply ρ := and ρ := 1. The main reason of the present work is to adapt this approach to cover the case ρ := and therefore obtain directly an approximation v h of the control of some minimal weighted L 2 -norm. To do so, we adapt the idea developed in [11] devoted to the wave equation. We also mention [33] where a different space-time variational approach (based on Least-squares principles) is used to approximate null controls for the heat equation. The paper is organized as follows. In Section 2, we associate to the dual problem (4) an equivalent mixed formulation which relies on the optimality conditions associated to the problem (7) with ρ =. In Section 2.1, we first address the penalization case and write the constraint L ϕ = as an equality in L 2 (Q T ). We then show the well-posedness of this mixed formulation, in particular we check the inf-sup condition (Theorem 2.1). The mixed formulation allows to approximate simultaneously the dual variable and the primal one, controlled solution of (1). Interestingly, we also derive an equivalent extremal problem in the primal variable y only (see Prop 2.2, Section 2.1.2). In Section 2.2, we reproduce the analysis relaxing the condition L ϕ = in the weaker space L 2 (, T, H 1 (Ω)). Then, in Section 2.3, by using the Global Carleman estimate (34), we show that a well-posed mixed formulation is also available for the limit and singular case for which ε = leading to Theorem 2.3. Section 3 is devoted to the numerical approximation of the mixed formulation (16) in the case ε > (Section 3.1) and of the mixed formulation (32) in the case ε = (section 3.2). Conformal approximations based on space-time finite elements are employed. In Section 3.3, we numerically check that the approximations used lead to discrete inf-sup properties, uniformly w.r.t. the discretization parameter h. Then the remaining of Section 3 is devoted to some experiments which emphasize the remarkable robuteness of the method. Section 4 concludes with some perspectives.

6 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 5 2 Control of minimal weighted L 2 -norm: mixed reformulations In order to avoid the minimization of the conjugate functional J with respect to the final state ϕ T by an iterative process, we now present a direct way to approximate the control of minimal square integrable norm in the spirit of the primal approach recalled in the introduction and developed in [18]. We adapt the case of the wave equation studied in [11]. 2.1 The penalized case: Mixed formulation I Let ρ R + and let ρ R defined by R := {w : w C(Q T );w ρ > in Q T ;w L (Q T δ ) δ > } (11) so that in particular, the weight ρ may blow up as t T. We first consider the approximate controllability case. For any ε >, the problem reads as follows: Minimize J ε (y, v) := 1 ρ 2 2 v 2 dt + 1 q T 2ε y(, T) 2 L 2 (Ω) (12) Subject to (y, v) A(y ;T) where A(y ;T) denotes the linear manifold A(y ;T) := { (y, v) : v L 2 (q T ), y solves (1) } and where ε denotes a penalty parameter (see [2, 5, 19]). The corresponding conjugate and well-posed problem is given by Minimize Jε (ϕ T ) := 1 ρ 2 2 ϕ(x, t) 2 dx dt + ε q T 2 ϕ T 2 L 2 (Ω) + (y, ϕ(,)) L 2 (Ω) (13) Subject to ϕ T L 2 (Ω) where ϕ solves (5). We recall that the penalized problem (12) is a consistent approximation of the original null controllability problem, in the sense that its unique solution converges to the solution of (7) with ρ = as ε. We refer for instance to [19], Prop. 3.3 for a proof of the following result, consequence of the null controllability for the heat equation. Proposition 2.1 Let (y ε, v ε ) be the solution of Problem (12) and let (y, v) be the solution of Problem (7) with ρ =. Then, one has as ε Mixed formulation y ε y strongly in L 2 (Q T ), v ε v strongly in L 2 (q T ) Since the variable ϕ, solution of (5), is completely and uniquely determined by the data ϕ T, the main idea of the reformulation is to keep ϕ as main variable. We introduce the linear space Φ := {ϕ C 2 (Q T ), ϕ = on Σ T }. For any η >, we define the bilinear form (ϕ, ϕ) Φ := ρ 2 ϕ ϕ dxdt + ε(ϕ(, T),ϕ(, T)) L 2 (Ω) + η q T L ϕ L ϕ dxdt, Q T ϕ, ϕ Φ.

7 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 6 From the unique continuation property for the heat equation, this bilinear form defines for any ε a scalar product. For any ε >, let Φ ε be the completion of Φ for this scalar product. We denote the norm over Φ ε by Φε such that ϕ 2 Φ ε := ρ 1 ϕ 2 L 2 (q T ) + ε ϕ(, T) 2 L 2 (Ω) + η L ϕ 2 L 2 (Q T ), ϕ Φ ε. (14) Finally, we defined the closed subset W ε of Φ ε by W ε = {ϕ Φ ε : L ϕ = in L 2 (Q T )} and we endow W ε with the same norm than Φ ε. Then, we define the following extremal problem : min Ĵε (ϕ) := 1 ρ 2 ϕ W ε 2 ϕ(x, t) 2 dx dt + ε q T 2 ϕ(, T) 2 L 2 (Ω) + (y, ϕ(,)) L 2 (Ω). (15) Standard energy estimates for the heat equation imply that, for any ϕ W ε, ϕ(,) L 2 (Ω) so that the functional Ĵε is well-defined over W ε. Moreover, since for any ϕ W ε, ϕ(, T) belongs to L 2 (Ω), Problem (15) is equivalent to the minimization problem (13). As announced, the main variable is now ϕ submitted to the constraint equality (in L 2 (Q T )) L ϕ =. This constraint equality is addressed by introducing a Lagrangian multiplier. We consider the following mixed formulation : find (ϕ ε, λ ε ) Φ ε L 2 (Q T ) solution of { a ε (ϕ ε,ϕ) + b(ϕ, λ ε ) = l(ϕ), ϕ Φ ε (16) b(ϕ ε,λ) =, λ L 2 (Q T ), where a ε : Φ ε Φ ε R, a ε (ϕ, ϕ) := ρ 2 ϕ ϕ dxdt + ε(ϕ(, T),ϕ(, T)) L 2 (Ω) q T b : Φ ε L 2 (Q T ) R, b(ϕ, λ) := L ϕ λdx dt Q T l : Φ ε R, l(ϕ) := (y, ϕ(,)) L 2 (Ω). We have the following result : Theorem 2.1 (i) The mixed formulation (16) is well-posed. (ii) The unique solution (ϕ ε, λ ε ) Φ ε L 2 (Q T ) is the unique saddle-point of the Lagrangian L ε : Φ ε L 2 (Q T ) R defined by L ε (ϕ, λ) := 1 2 a ε(ϕ, ϕ) + b(ϕ, λ) l(ϕ). (17) (iii) The optimal function ϕ ε is the minimizer of Ĵ ε over W ε while the optimal multiplier λ ε L 2 (Q T ) is the state of the heat equation (1) in the weak sense. Proof - We easily check that the bilinear form a ε is continuous over Φ ε Φ ε, symmetric and positive and that the bilinear form b ε is continuous over Φ ε L 2 (Q T ). Furthermore, for any fixed ε, the continuity of the linear form l over Φ ε can be viewed from the energy estimate : ) ϕ(,) 2 L 2 (Ω) (Q C L ϕ 2 dx dt + ϕ(, T) 2 L 2 (Ω), ϕ Φ ε, T for some C >, so that ϕ(,) 2 L 2 (Ω) C max(η 1, ε 1 ) ϕ 2 Φ ε. Therefore, the well-posedness of the mixed formulation is a consequence of the following two properties (see [4]):

8 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 7 a ε is coercive on N(b), where N(b) denotes the kernel of b : N(b) := {ϕ Φ ε : b(ϕ, λ) = for every λ L 2 (Q T )}; b satisfies the usual inf-sup condition over Φ ε L 2 (Q T ): there exists δ > such that inf sup b(ϕ, λ) δ. (18) λ L 2 (Q T ) ϕ Φ ε ϕ Φε λ L 2 (Q T ) From the definition of a ε, the first point is clear : for all ϕ N(b) = W ε, a ε (ϕ, ϕ) = ϕ 2 Φ ε. Let us check the inf-sup condition. For any fixed λ L 2 (Q T ), we define the (unique) element ϕ of L ϕ = λ in Q T, ϕ = on Σ T ; ϕ (, T) = in Ω, so that ϕ solves the backward heat equation with source term λ L 2 (Q T ), null Dirichlet boundary condition and zero initial state. Since λ L 2 (Q T ), then using energy estimates, there exists a constant C Ω,T > such that the solution ϕ of the backward heat equation with source term λ satisfies the inequality ρ 2 ϕ 2 dx dt ρ 2 ϕ 2 dx dt ρ 2 C Ω,T λ 2 L 2 (Q T ). q T q T Consequently, ϕ Φ ε. In particular, we have b(ϕ, λ ) = λ 2 L 2 (Q T ) and b(ϕ, λ ) sup ϕ Φ ε ϕ Φε λ L2 (Q T ) b(ϕ, λ ) ϕ Φε λ L2 (Q T ) = λ 2 L 2 (Q T ) ( ) 1. ρ 1 ϕ 2 L 2 (q T ) + η λ 2 2 L 2 (Q T ) λ L 2 (Q T ) Combining the above two inequalities, we obtain b(ϕ, λ ) 1 sup ϕ Φ ε ϕ Φε λ L2 (Q T ) ρ 2 C Ω,T + η and, hence, (18) holds with δ = ( ρ 2 C Ω,T + η ) 1/2. The point (ii) is due to the symmetry and to the positivity of the bilinear form a ε. (iii) Concerning the third point, the equality b(ϕ ε,λ) = for all λ L 2 (Q T ) implies that L ϕ ε = as an L 2 (Q T ) function, so that if (ϕ ε, λ ε ) Φ ε L 2 (Q T ) solves the mixed formulation, then ϕ ε W ε and L ε (ϕ ε, λ ε ) = Ĵ ε (ϕ ε ). Finally, the first equation of the mixed formulation reads as follows: ρ 2 ϕ ε ϕ dxdt + ε(ϕ ε (, T),ϕ(, T)) L ϕ(x, t)λ ε (x, t) dx dt = l(ϕ), ϕ Φ ε, q T Q T or equivalently, since the control is given by v ε := ρ 2 ϕ ε 1 ω, v ε ϕ dxdt + (εϕ ε (, T),ϕ(, T)) q T L ϕ(x, t) λ ε (x, t) dx dt = l(ϕ), Q T ϕ Φ ε. But this means that λ ε L 2 (Q T ) is solution of the heat equation in the transposition sense. Since y L 2 (Ω) and v ε L 2 (q T ), λ ε must coincide with the unique weak solution to (1) (y ε = λ ε ) such that λ ε (, T) = εϕ ε (, T). As a conclusion, the optimal pair (y ε, v ε ) to (12) is characterized in term of the adjoint variable ϕ ε solution of (16) by v ε = ρ 2 ϕ ε 1 ω and y ε (, T) = εϕ ε (, T). Theorem 2.1 reduces the search of the approximated control to the resolution of the mixed formulation (16), or equivalently the search of the saddle point for L ε. In general, it is convenient

9 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 8 to augment the Lagrangian (see [2]), and consider instead the Lagrangian L ε,r defined for any r > by L ε,r (ϕ, λ) := 1 2 a ε,r(ϕ, ϕ) + b(ϕ, λ) l(ϕ), a ε,r (ϕ, ϕ) := a ε (ϕ, ϕ) + r L ϕ 2 dx dt. Q T Since a ε (ϕ, ϕ) = a ε,r (ϕ, ϕ) on W ε and since the function ϕ ε such that (ϕ ε, λ ε ) is the saddle point of L ε verifies ϕ ε W ε, the lagrangian L ε and L ε,r share the same saddle-point Dual problem of the extremal problem (15) The mixed formulation allows to solve simultaneously the dual variable ϕ ε, argument of the conjugate functional (15), and the Lagrange multiplier λ ε. Since λ ε turns out to be the (approximate) controlled state of (1), we may qualify λ ε as the primal variable of the problem. We derive in this section the corresponding extremal problem involving only that variable λ ε. For any r >, let us define the linear operator A ε,r from L 2 (Q T ) into L 2 (Q T ) by where ϕ Φ ε is the unique solution to A ε,r λ := L ϕ, λ L 2 (Q T ) a ε,r (ϕ, ϕ) = b(ϕ, λ), ϕ Φ ε. (19) Note that the assumption r > is necessary here in order to guarantee the well-posedness of (19). Precisely, for any r >, the form a ε,r defines a norm equivalent to the norm on Φ ε (see (14)). We have the following crucial lemma : Lemma 2.1 For any r >, the operator A ε,r is a strongly elliptic, symmetric isomorphism from L 2 (Q T ) into L 2 (Q T ). Proof- From the definition of a ε,r, we easily get that A ε,r λ L 2 (Q T ) r 1 λ L 2 (Q T ) and the continuity of A ε,r. Next, consider any λ L 2 (Q T ) and denote by ϕ the corresponding unique solution of (19) so that A ε,r λ := L ϕ. Relation (19) with ϕ = ϕ then implies that Q T (A ε,r λ )λ dxdt = a ε,r (ϕ, ϕ ) (2) and therefore the symmetry and positivity of A ε,r. The last relation with λ = λ implies that A ε,r is also positive definite. Finally, let us check the strong ellipticity of A ε,r, equivalently that the bilinear functional (λ, λ ) (A ε,r λ)λ dx dt Q T is L 2 (Q T )-elliptic. Thus we want to show that Q T (A ε,r λ)λ dxdt C λ 2 L 2 (Q T ), λ L2 (Q T ) (21) for some positive constant C. Suppose that (21) does not hold; there exists then a sequence {λ n } n of L 2 (Q T ) such that λ n L 2 (Q T ) = 1, n, lim (A ε,r λ n )λ n dx dt =. n Q T

10 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 9 Let us denote by ϕ n the solution of (19) corresponding to λ n. From (2), we then obtain that lim n L ϕ n L2 (Q T ) =, lim n ρ 1 ϕ n L2 (q T ) =, lim n ϕ n(, T) L2 (Ω) =. (22) From (19) with λ = λ n and ϕ = ϕ n, we have 1 ρ 2 ϕ n ϕ dxdt + ε ϕ n (, T)ϕ(, T)dx + (rl ϕ n λ n )L ϕ dxdt =, ϕ Φ ε. (23) q T Q T We define the sequence {ϕ n } n as follows : L ϕ n = r L ϕ n λ n in Q T, ϕ n = on Σ T ; ϕ n (, T) = in Ω, so that, for all n, ϕ n is the solution of the backward heat equation with zero initial datum and source term r L ϕ n λ n in L 2 (Q T ). Using again energy type estimates, we get ρ 1 ϕ n L 2 (q T ) ρ 1 ϕ n L 2 (q T ) ρ 1 C Ω,T rl ϕ n λ n L 2 (Q T ), so that ϕ n Φ ε. Then, using (23) with ϕ = ϕ n, we get rl ϕ n λ n L 2 (Q T ) ρ 1 C Ω,T ρ 1 ϕ n L 2 (q T ). Then, from (22), we conclude that lim n + λ n L2 (Q T ) = leading to a contradiction and to the strong ellipticity of the operator A ε,r. The introduction of the operator A ε,r is motivated by the following proposition: Proposition 2.2 For any r >, let ϕ Φ ε be the unique solution of a ε,r (ϕ,ϕ) = l(ϕ), ϕ Φ ε and let Jε,r : L 2 (Q T ) L 2 (Q T ) be the functional defined by Jε,r(λ) := 1 (A ε,r λ)λdxdt b(ϕ, λ). 2 Q T The following equality holds : sup λ L 2 (Q T ) inf L ε,r (ϕ, λ) = ϕ Φ ε inf λ L 2 (Q T ) J ε,r(λ) + L ε,r (ϕ,). Proof- For any λ L 2 (Q T ), let us denote by ϕ λ Φ ε the minimizer of ϕ L ε,r (ϕ, λ); ϕ λ satisfies the equation a ε,r (ϕ λ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ ε and can be decomposed as follows : ϕ λ = ψ λ + ϕ where ψ λ Φ ε solves We then have a ε,r (ψ λ,ϕ) + b(ϕ, λ) =, ϕ Φ ε. inf L ε,r (ϕ, λ) = L ε,r (ϕ λ, λ) = L ε,r (ψ λ + ϕ, λ) ϕ Φ ε = 1 2 a ε,r(ψ λ + ϕ, ψ λ + ϕ ) + b(ψ λ + ϕ, λ) l(ψ λ + ϕ ) := X 1 + X 2 + X 3 with X 1 := 1 2 a ε,r(ψ λ, ψ λ ) + b(ψ λ, λ) + b(ϕ, λ) X 2 := a ε,r (ϕ, ψ λ ) l(ψ λ ), X 3 := 1 2 a ε,r(ϕ, ϕ ) l(ϕ ).

11 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 1 From the definition of ϕ, X 2 = while X 3 = L ε,r (ϕ,). Eventually, from the definition of ψ λ, X 1 = 1 2 b(ψ λ, λ) + b(ϕ, λ) = 1 (A ε,r λ)λdxdt + b(ϕ, λ) = J 2 ε,r(λ) Q T and the result follows. From the ellipticity of the operator A ε,r, the minimization of the functional Jε,r over L 2 (Q T ) is well-posed. It is interesting to note that with this extremal problem involving only λ, we are coming to the primal variable, controlled solution of (1) (see Theorem 2.1, (iii)). This argument allows notably to avoid the direct minimization of J ε (introduced in Problem (12)) with respect to the state y (ill-conditioned due to the term ε 1 for ε small). Here, any constraint equality is assigned to the variable λ. 2.2 The penalized case : Mixed formulation II (relaxing the condition L ϕ ε = in L 2 (Q T )) The previous mixed formulation amounts to find a backward solution ϕ ε satisfying the condition L ϕ ε = in L 2 (Q T ). For numerical purposes, it may be interesting to relax this condition, which typically leads to the use of C 1 type approximations in the space variable (see Section 3). In order to circumvent this difficulty, we introduce and analyze in this section a second penalized mixed formulation where the condition on ϕ ε is relaxed, namely we impose the constraint L ϕ ε = in L 2 (, T;H 1 (Ω)). Considering as before the full adjoint variable ϕ as the main variable, we associated to (13) the following extremal problem : min ϕ c W ε Ĵ ε (ϕ) = 1 2 over the space Ŵε = Q T ρ 2 ϕ(x, t) 2 dx dt + ε 2 ϕ(, T) 2 L 2 (Ω) + Ω y (x)ϕ(x,)dx, (24) { ϕ Φ } ε : L ϕ = in L 2 (, T;H 1 (Ω)). The space Φ ε is again defined as the completion of Φ with respect to the inner product ( (ϕ, ϕ) bφε := ρ 2 ϕ ϕ dxdt + ε(ϕ(, T),ϕ(, T)) + η ϕ ϕ dxdt + q T Q T defined over Φ. We denote by bφε the associated norm such that T (ϕ t,ϕ t ) H 1dt ϕ 2 b Φε := ρ 1 ϕ 2 L 2 (q T ) + ε ϕ(, T) 2 L 2 (Ω) + η( ϕ 2 L 2 (Q T ) + ϕ t 2 L 2 (,T;H 1 ) ), ϕ Φ ε. (25) Lemma 2.2 The equality Ŵε = W ε holds. Therefore, the minimization problem (24) is equivalent to the minimization (15). Proof - First, let us see that W ε Ŵε. To do this, it is enough see that Φ ε Φ ε. In fact, if ϕ Φ ε then there exists a sequence (ϕ n ) n=1 in Φ such that ϕ n ϕ in Φ ε. So, we can conclude that ϕ n ϕ in L 2 (, T;H(Ω)) 1 and ϕ n t ϕ t in L 2 (, T;H 1 (Ω)). Hence, ϕ n ϕ in Φ ε. Secondly, let us see that Ŵε W ε. Indeed, if ϕ Ŵε then ϕ Φ ε and L ϕ =. Let us denote ϕ T := ϕ(, T), so there exists a sequence (ϕ n T ) n=1 in C (Ω) such that ϕ n T ϕ T in L 2 (Ω). Now, if (ϕ n ) n=1 is a sequence such that L ϕ n =, ϕ n = on Σ T and ϕ n (, T) := ϕ n T then this sequence belongs to Φ. Hence, ϕ n ϕ in Φ ε and ϕ n ϕ in Φ ε. Therefore, ϕ belongs to W ε. The main variable is now ϕ submitted to the constraint equality L ϕ = L 2 (, T;H 1 ). As before, this constraint is addressed by introducing a mixed formulation given as follows : find ),

12 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 11 (ϕ ε, λ ε ) Φ ε Λ ε solution of â ε (ϕ ε,ϕ) + ˆb(ϕ, ˆλ ε ) = ˆl(ϕ), ϕ Φ ε ˆb(ˆϕε,λ) =, λ Λ ε, (26) where Λ ε := L 2 (, T;H(Ω)) 1 and â ε : Φ ε Φ ε R, â ε (ϕ, ϕ) := ρ 2 ϕ ϕ dxdt + ε(ϕ(, T),ϕ(, T)) L 2 (Ω) q T ˆb : Φε Λ ε R, T ˆb(ϕ, λ) := < L ϕ, λ > H 1 (Ω),H 1(Ω) dt = ˆl : Φε R, T ϕ t (t), λ(t) H 1,H 1 dt ˆl(ϕ) := Ω Q T y (x)ϕ(x,)dx. Similarly to Theorem 2.1, the following holds : ( ) (c(x) ϕ, λ) + d(x, t)ϕλ dx dt Theorem 2.2 (i) The mixed formulation (26) is well-posed. (ii) The unique solution (ϕ ε, λ ε ) Φ ε Λ ε is the unique saddle-point of the Lagrangian operator L ε : Φ ε Λ ε R defined by L ε (ϕ, λ) := 1 2âε(ϕ, ϕ) + ˆb(ϕ, λ) ˆl(ϕ). (27) (iii) The optimal function ϕ ε is the minimizer of Ĵ ε over Ŵε while the optimal multiplier λ ε ˆΛ ε is the weak solution of the heat equation (1). Proof - We easily check that the bilinear form â ε is continuous over Φ ε Φ ε, symmetric and positive and that the bilinear form ˆb is continuous over Φ ε Λ ε. Furthermore, the continuity of the linear form ˆl over Φ ε is a direct by the continuous embedding Φ ε C ([, T];L 2 (Ω)). Therefore, the well-posedness of the mixed formulation is a consequence of the following two properties (see [4]): â ε is coercive on N(ˆb), where N(ˆb) denotes the kernel of ˆb : N(ˆb) = {ϕ Φ ε such that ˆb(ϕ, λ) = for every λ Λ } ε. ˆb satisfies the usual inf-sup condition over Φ ε Λ ε : there exists δ > such that inf λ b Λ ε sup ϕ b Φ ε ˆb(ϕ, λ) ϕ bφε λ bλε δ. (28) From the definition of â ε, the first point is clear : for all ϕ N(ˆb ε ) = Ŵε, thanks to classical energy estimates, we have â ε (ϕ, ϕ) = ρ 1 ϕ 2 L 2 (q T ) + ε 2 ϕ(, T) 2 L 2 (Ω) + ε 2 ϕ(, T) 2 L 2 (Ω) ρ 1 ϕ 2 L 2 (q T ) + ε 2 ϕ(, T) 2 L 2 (Ω) + ε C( ϕ 2 L 2 (Q T ) + ϕ t 2 L 2 (,T;H 1 ) ) C ε,η ϕ 2 b Φε,

13 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 12 where C = C(T, c, d ) > and C ε,η := min(2 1, C ε η 1 ). Let us check the inf-sup condition. For any fixed λ Λ ε, we define the (unique) element ϕ of L ϕ = λ in Q T, ϕ = on Σ T ; ϕ (, T) = in Ω, so that ϕ solves the backward heat equation with source term λ, null Dirichlet boundary condition and zero initial state. Since λ L 2 (, T;H 1 ), then ϕ Φ ε : precisely, using energy estimates, there exists a constant C > such that ϕ satisfies the inequalities and ϕ 2 L 2 (Q T ) C λ 2 L 2 (Q T ) ϕ 2 b Φε = ρ 1 ϕ 2 L 2 (q T ) + ε ϕ (, T) 2 L 2 (Ω) + η( ϕ 2 L 2 (Q T ) + ϕ t 2 L 2 (,T;H 1 ) ) = ρ 1 ϕ 2 L 2 (q T ) + η( ϕ 2 L 2 (Q T ) + ϕ t 2 L 2 (,T;H 1 ) ) C η λ 2 L 2 (Q T ). where C = C(T, c, d ) > and C η := C(1 + η). Consequently, ϕ Φ ε. In particular, we have ˆb ε (ϕ, λ ) = λ 2 L 2 (Q T ) and b ε (ϕ, λ ) sup ϕ Φ ε ϕ cwε λ b ε(ϕ, λ ) λ 2 L bλε ϕ cwε λ = 2 (Q T ). bλε Cη 1/2 λ L 2 (Q T ) λ L 2 (Q T ) Combining the above two inequalities, we obtain sup ϕ Φ b(ϕ, λ ) ϕ Φ λ L 2 (Q T ) 1 Cη and, hence, (28) holds with δ = C 1 2 η. The point (ii) is due to the symmetry and to the positivity of the bilinear form â ε. Concerning the third assertion, the equality b(ˆϕ ǫ,λ) = for all λ Λ ε implies that L ϕ ǫ = as an L 2 (, T;H 1 ) function, so that if (ϕ ε, λ ε ) Φ ε Λ ε solves the mixed formulation (26), then ϕ ε Ŵε and L ε (ϕ ε, λ ε ) = Ĵ ε (ϕ ε ). This implies that ϕ ε of the two mixed formulations coincide. Assuming y L 2 (Ω) and v L 2 (q T ), it is said here that y L 2 (, T;H 1 (Ω)) is the (unique) solution by transposition of the heat equation (1) if and only if, for every g L 2 (, T;H 1 ), we have where ϕ solves T g, y H 1,H 1 dt = q T v ϕ dxdt + (ϕ(,), y ) L 2 (Ω), L ϕ = g in Q T, ϕ = on Σ T, ϕ(, T) = in Ω. As g (v,ϕ) L2 (q T ) + (ϕ(,), y ) L2 (Ω) is linear and continuous on L 2 (, T;H 1 ) the Riesz representation theorem guarantees that this definition makes sense. Finally, the first equation of the mixed formulation (26) reads as follows: T ρ 2 ϕ ε ϕ dxdt + ε(ϕ ε (, T),ϕ(, T)) + ϕ t, λ ε H 1,H 1 q T (c(x) ϕ, λ ε ) + d(x, t)ϕλ ε dx dt = ˆl(ϕ), ϕ Φ ε, Q T or equivalently, since the control is given by v ε = ρ 2 ϕ ε (recall that the formulations (15) and (24) are equivalent), v ε ϕ dxdt + (εϕ ε (, T),ϕ(, T)) + q T T ϕ t, λ ε H 1,H 1 dt (c(x) ϕ, λ ε ) + d(x, t)ϕλ ε dx dt = ˆl(ϕ), Q T ϕ Φ ε.

14 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 13 But this means that λ ε Λ ε is solution of the heat equation in the transposition sense. Since y L 2 (Ω) and v ε L 2 (q T ), λ ε must coincide with the unique weak solution to (1) (y ε = λ ε ) and, in particular, we can conclude that y ε (, T) = εϕ ε (, T). So from the unique of the weak solution, the solution (ϕ ε, λ ε ) of the two mixed formulation coincides. The equivalence of the mixed formulation (26) with the mixed formulation (16) is related to the regularizing property of the heat kernel. At the numerical level, the advantage is that this formulation leads naturally to continuous spaces of approximation both in time and space. 2.3 Third mixed formulation of the controllability problem : the limit case ε =. We consider in this section the limit case of Section 2.1 corresponding to ε =, i.e. to the null controllability. The conjugate functional J corresponding to this case is given in the introduction, see (4), with a weight ρ 2 (recall that ρ R defined by (11)) in the first term, precisely min ϕ T H J (ϕ T ) := 1 2 q T ρ 2 (x, t) ϕ(x, t) 2 dxdt + (y, ϕ(,)) L 2 (Ω) (29) where the variable ϕ solves the backward heat equation (5) and H is again defined as the completion of the L 2 (Ω) space with respect to the norm ϕ T H := ρ 1 ϕ L 2 (q T ). As explained in the introduction, the limit case is much more singular due to the hugeness of the space H. At the limit ε =, the control of the terminal state ϕ(, T) is lost in L 2 (Ω). Let ρ R. Proceeding as before, we consider again the space Φ = {ϕ C 2 (Q T ) : ϕ = on Σ T } and then, for any η >, we define the bilinear form (ϕ, ϕ) := eφρ ρ 2,ρ ϕ ϕ dxdt + η ρ q T Q 2 L ϕ L ϕ dxdt, ϕ, ϕ Φ. T The introduction of the weight ρ, which does not appear in the original problem (29) will be motivated at the end of this Section. From the unique continuation property for the heat equation, this bilinear form defines for any η > a scalar product. Let then Φ ρ,ρ be the completion of Φ for this scalar product. We denote the norm over Φ ρ,ρ by eφρ such that,ρ ϕ 2 e Φρ,ρ Finally, we defined the closed subset W ρ,ρ of Φ ρ,ρ by := ρ 1 ϕ 2 L 2 (q T ) + η ρ 1 L ϕ 2 L 2 (Q T ), ϕ Φ ρ,ρ. (3) W ρ,ρ = {ϕ Φ ρ,ρ : ρ 1 L ϕ = in L 2 (Q T )} and we endow W ρ,ρ with the same norm than Φ ρ,ρ. We then define the following extremal problem : min Ĵ (ϕ) = 1 ρ 2 ϕ W f ρ 2 ϕ(x, t) 2 dx dt + (y, ϕ(,)) L 2 (Ω). (31),ρ q T For any ϕ W ρ,ρ, L ϕ = a.e. in Q T and ϕ = ρ 1 fwρ,ρ ϕ L 2 (q T ) so that ϕ(, T) belongs by definition to the abstract space H: consequently, extremal problems (31) and (29) are equivalent. In particular, from the regularizing property of the heat kernel, ϕ(,) belongs to L 2 (Ω) and the linear term in ϕ in Ĵ is well defined. Then, we consider the following mixed formulation : find (ϕ, λ) Φ ρ,ρ L 2 (Q T ) solution of { ã(ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ ρ,ρ (32) b(ϕ, λ) =, λ L 2 (Q T ),

15 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 14 where ã : Φ ρ,ρ Φ ρ,ρ R, ã(ϕ, ϕ) = ρ 2 ϕ ϕ dxdt q T b : Φρ,ρ L 2 (Q T ) R, b(ϕ, λ) = Q T ρ 1 L ϕ λdx dt l : Φρ,ρ R, l(ϕ) = (y, ϕ(,)) L 2 (Ω). Before to study this mixed formulation, let us do the following comment. The continuity of l over the space Φ ρ,ρ holds true for a precise choice of the weights which appear in Carleman type estimates for parabolic equations (see [21]): we recall the following important result. Proposition 2.3 ( [21]) Let the weights ρ c, ρ c R (see (11)) be defined as follows : ( ) β(x) ρ c (x, t) := exp, β(x) := K 1 (e K2 e β(x)), T t ρ c (x, t) := (T t) 3/2 ρ c (x, t), (33) where the K i are sufficiently large positive constants (depending on T, c and c C1 (Ω)) such that β C (Ω), β > in Ω, β = on Ω, Supp( β) Ω \ ω. Then, there exists a constant C >, depending only on ω,t, such that ϕ(,) L 2 (Ω) C ϕ eφρ c c, ϕ Φ,ρ ρ c,ρc. (34) The estimate (34) is a consequence of the celebrated global Carleman inequality satisfied by the solution of (5), introduced and popularized in [21]. It allows to obtain the following existence and uniqueness result : Theorem 2.3 Let ρ R and ρ R L (Q T ) and assume that there exists a positive constant K such that ρ Kρ c, ρ Kρ c in Q T. (35) Then, we have : (i) The mixed formulation (32) defined over Φ ρ,ρ L 2 (Q T ) is well-posed. (ii) The unique solution (ϕ, λ) Φ ρ,ρ L 2 (Q T ) is the unique saddle-point of the Lagrangian L : Φ L 2 (Q T ) R defined by L(ϕ, λ) = 1 2ã(ϕ, ϕ) + b(ϕ, λ) l(ϕ). (36) (iii) The optimal function ϕ is the minimizer of Ĵ over Φ ρ,ρ while ρ 1 λ L 2 (Q T ) is the state of the heat equation (1) in the weak sense. Proof- The proof is similar to the proof of Theorem 2.1. From the definition, the bilinear form ã is continuous over Φ ρ,ρ Φ ρ,ρ, symmetric and positive and the bilinear form b is continuous over Φ ρ,ρ 1 L 2 (Q T ). Furthermore, the continuity of the linear form l over Φ ρ,ρ is the consequence of the estimate (34): precisely, from the assumptions (35), the inclusion Φ ρ,ρ Φ ρ c,ρc hold true. Therefore, estimate (34) implies ϕ(,) L 2 (Ω) C ϕ eφρ c,ρ c CK 1 ϕ eφρ,ρ, ϕ Φ ρ,ρ. (37)

16 2 CONTROL OF MINIMAL WEIGHTED L 2 -NORM: MIXED REFORMULATIONS 15 Therefore, the well-posedeness of the formulation (32) is the consequence of two properties: first, the coercivity of the form ã on the kernel N( b) := {ϕ Φ ρ,ρ : b(ϕ, λ) = λ L 2 (Q T )}: again, this holds true since the kernel coincides with the space W ρ,ρ. Second, the inf-sup property which reads as : b(ϕ, λ) inf sup λ L 2 (Q T ) ϕ λ δ (38) eφρ L,ρ 2 (Q T ) ϕ e Φ ρ,ρ for some δ >. For any fixed λ L 2 (Q T ), we define the unique element ϕ solution of Using energy estimates, we have ρ 1 L ϕ = λ inq T, ϕ = on Σ T, ϕ(, T) = in Ω. ρ 1 ϕ L2 (q T ) ρ 1 ϕ L2 (Q T ) ρ 1 ρλ L2 (Q T ) ρ 1 ρ L (Q T ) λ L2 (Q T ) (39) which proves that ϕ Φ ρ,ρ and that sup ϕ e Φ ρ,ρ b(ϕ, λ ) ϕ λ eφρ L2 (Q,ρ T ) b(ϕ, λ ) ϕ λ eφρ L2 (Q,ρ T ) = λ L 2 (Q T ) ( ) 1. ρ 1 ϕ 2 L 2 (q T ) + η λ 2 2 L 2 (Q T ) Combining the above two inequalities, we obtain sup ϕ e Φ ρ,ρ b(ϕ, λ ) ϕ λ eφρ,ρ L 2 (Q T ) 1 ρ 2 ρ 2 L (Q T ) + η ( ) 1/2. and, hence, (38) holds with δ = ρ 2 ρ 2 L (Q T ) + η The point (ii) is again due to the positivity and symmetry of the form ã. Concerning the last point of the Theorem, the equality b(ϕ, λ) = for all λ L 2 (Q T ) implies that ρ 1 L ϕ = as an L 2 (Q T ) function, so that if (ϕ, λ) Φ ρ,ρ L 2 (Q T ) solves the mixed formulation (32), then ϕ W ρ,ρ and L(ϕ, λ) = Ĵ (ϕ). Finally, the first equation of the mixed formulation (32) reads as follows: ρ 2 ϕ ϕ dxdt ρ q T Q 1 L ϕλ dxdt = l(ϕ), ϕ Φ, T or equivalently, since the control is given by v := ρ 2 ϕ 1 ω, v ϕ dxdt L ϕ(ρ 1 λ) dx dt = l(ϕ), q T Q T ϕ Φ, This means that ρ 1 λ L 2 (Q T ) is solution of the heat equation with source term v 1 ω in the transposition sense and such that (ρ 1 λ)(, T) =. Since y L 2 (Ω) and v L 2 (q T ), ρ 1 λ must coincide with the unique weak solution to (1) (y = ρ 1 λ) and, in particular, y(, T) =. Remark 1 The well-posedness of the mixed formulation (32), precisely the inf-sup property (38), is open in the case where the weight ρ is simply in R (instead of R L (Q T )): in this case, the weight ρ may blow up at t T. In order to get (38), it suffices to prove that the function ψ := ρ 1 ϕ solution of the boundary value problem ρ 1 L (ρ ψ) = λ in Q T, ψ = on Σ T, ψ(, T) = in Ω for any λ L 2 (Q T ) satisfies the following estimate for some positive constant C ψ L2 (q T ) C ρ 1 L (ρ ψ) L 2 (Q T ). In the cases of interest for which the weights ρ and ρ blow up at t T (for instance given by ρ c and ρ c ), this estimates is open and does not seem to be a consequence of the estimate (34).

17 3 NUMERICAL APPROXIMATION AND EXPERIMENTS 16 Let us now comment the introduction of the weight ρ. The solution ϕ of the mixed formulation (32) belongs to W ρ,ρ and therefore does not depend on the weight ρ (recall that ρ is strictly positive); this is in agreement with the fact that ρ does not appear in the original formulation formulation (29). Therefore, this weight may be seen as a parameter to improve some specific properties of the mixed formulation, specifically at the numerical level. Precisely, in the limit case ε =, we recall that the trace ϕ t=t of the solution does not belong to L 2 (Ω) but to a much larger and singular space. Very likely, a similar behavior occurs for the function L ϕ near Ω {T } so that the constraint L ϕ = in L 2 (Q T ) introduced in Section 2.1 is too strong and must be replaced at the limit in ε by the relaxed one ρ 1 L ϕ = in L 2 (Q T ) with ρ 1 small near Ω {T }. Remark that this is actually the effect and the role of the Carleman type weights ρ c defined by (33) and initially introduced in [21]. As in Section 2.1, it is convenient to augment the Lagrangian and consider instead the Lagrangian L r defined for any r > by L r (ϕ, λ) := 1 2ãr(ϕ, ϕ) + b(ϕ, λ) l(ϕ), ã r (ϕ, ϕ) := ã(ϕ, ϕ) + r ρ 1 L ϕ 2 dx dt. Q T Finally, similarly to Lemma 2.1 and Proposition 2.2, we have the following result. Let ρ R and ρ R L (Q T ) Proposition 2.4 For any r >, let ρ R and ρ R L (Q T ) verifying (35). Let us define the linear operator A r from L 2 (Q T ) into L 2 (Q T ) by where ϕ Φ ρ,ρ is the unique solution to A r λ := ρ 1 L ϕ, λ L 2 (Q T ), a r (ϕ, ϕ) = b(ϕ, λ), ϕ Φ ρ,ρ. A r is a strongly elliptic, symmetric isomorphism from L 2 (Q T ) into L 2 (Q T ). Let Ĵ r L 2 (Q T ) be the functional defined by Ĵr (λ) := 1 (A r λ)λdxdt 2 b(ϕ, λ). Q T where ϕ Φ ρ,ρ is the unique solution of : L 2 (Q T ) ã r (ϕ, ϕ) = l(ϕ), ϕ Φ ρ,ρ. The following equality holds : sup inf λ L 2 (Q T ) ϕ Φ e ρ,ρ L r (ϕ, λ) = inf λ L 2 (Q T ) Ĵ r (λ) + L r (ϕ,). 3 Numerical approximation and experiments 3.1 Discretization of the mixed formulation (16) We now turn to the discretization of the mixed formulation (16) assuming r >. Let then Φ ε,h and M ε,h be two finite dimensional spaces parametrized by the variable h such that, for any ε >, Φ ε,h Φ ε, M ε,h L 2 (Q T ), h >.

18 3 NUMERICAL APPROXIMATION AND EXPERIMENTS 17 Then, we can introduce the following approximated problems : find (ϕ h, λ h ) Φ ε,h M ε,h solution of { a ε,r (ϕ h,ϕ h ) + b(ϕ h, λ h ) = l(ϕ h ), ϕ h Φ ε,h (4) b(ϕ h,λ h ) =, λ h M ε,h. The well-posedness of this mixed formulation is again a consequence of two properties : the coercivity of the bilinear form a ε,r on the subset N h (b) = {ϕ h Φ ε,h ;b(ϕ h, λ h ) = λ h M ε,h }. Actually, from the relation a ε,r (ϕ, ϕ) C r,η ϕ 2 Φ ε, ϕ Φ ε, where C r,η = min{1, r/η}, the form a ε,r is coercive on the full space Φ ε, and so a fortiori on N h (b) Φ ε,h Φ ε. The second property is a discrete inf-sup condition : there exists δ h > such that b(ϕ h, λ h ) inf sup δ h. (41) λ h M ε,h ϕ h Φ ε,h ϕ h Φε,h λ h Mε,h For any fixed h, the spaces M ε,h and Φ ε,h are of finite dimension so that the infimum and supremum in (41) are reached: moreover, from the property of the bilinear form a ε,r, it is standard to prove that δ h is strictly positive (see Section 3.3). Consequently, for any fixed h >, there exists a unique couple (ϕ h, λ h ) solution of (4). On the other hand, the property inf h δ h > is in general difficult to prove and depends strongly on the choice made for the approximated spaces M ε,h and Φ ε,h. We shall analyze numerically this property in Section 3.3. Remark 2 For r =, the discrete mixed formulation (4) is not well-posed over Φ ε,h M ε,h because the bilinear form a ε,r= is not coercive over the discrete kernel of b: the equality b(λ h, ϕ h ) = for all λ h M ε,h does not imply that L ϕ h vanishes. Therefore, the term r L ϕ h 2 L 2 (Q T ) may be understood as a numerical stabilization term: for any h >, it ensures the uniform coercivity of the form a ε,r (and so the well-posedness) and vanishes at the limit in h. We also emphasize that this term is not a regularization term as it does not add any regularity to the solution ϕ h. As in [1], the finite dimensional and conformal space Φ ε,h must be chosen such that L ϕ h belongs to L 2 (Q T ) for any ϕ h Φ ε,h. This is guaranteed as soon as ϕ h possesses second-order derivatives in L 2 loc (Q T). Any conformal approximation based on standard triangulation of Q T achieves this sufficient property as soon as it is generated by spaces of functions continuously differentiable with respect to the variable x and spaces of continuous functions with respect to the variable t. We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. Then, we introduce the space Φ ε,h as follows : Φ ε,h = {ϕ h C 1 (Q T ) : ϕ h K P(K) K T h, ϕ h = on Σ T } where P(K) denotes an appropriate space of polynomial functions in x and t. In this work, we consider for P(K) the so-called Bogner-Fox-Schmit (BFS for short) C 1 -element defined for rectangles. In the one dimensional setting considered in the sequel, it involves 16 degrees of freedom, namely the values of ϕ h, ϕ h,x, ϕ h,t, ϕ h,xt on the four vertices of each rectangle K. Therefore P(K) = P 3,x P 3,t where P r,ξ is by definition the space of polynomial functions of order r in the variable ξ. We refer to [9] page 76. We also define the finite dimensional space M ε,h = {λ h C (Q T ) : λ h K Q(K) K T h }, where Q(K) denotes the space of affine functions both in x and t on the element K.

19 3 NUMERICAL APPROXIMATION AND EXPERIMENTS 18 Again, in the one dimensional setting, for rectangle, we simply have Q(K) = P 1,x P 1,t. We also mention that the approximation is conformal : for any h >, we have Φ ε,h Φ ε and M ε,h L 2 (Q T ). Let n h = dim Φ ε,h, m h = dim M ε,h and let the real matrices A ε,r,h R n h,n h, B h R m h,n h, J h R m h,m h and L h R n h be defined by a ε,r (ϕ h,ϕ h ) =< A ε,r,h {ϕ h }, {ϕ h } > R n h,r n h, ϕ h,ϕ h Φ ε,h, b(ϕ h, λ h ) =< B h {ϕ h }, {λ h } > R m h,r m h, ϕ h Φ ε,h, λ h M ε,h, λ h λ h dx dt =< J h {λ h }, {λ h } > R m h,r m h, λ h,λ h M ε,h, Q T l(ϕ h ) =< L h, {ϕ h } >, ϕ h Φ ε,h where {ϕ h } R n h denotes the vector associated to ϕ h and <, > R n h,r n h the usual scalar product over R n h. With these notations, Problem (4) reads as follows : find {ϕ h } R n h and {λ h } R m h such that ( Aε,r,h Bh T B h ) R n h +m h,n h +m h ( {ϕh } {λ h } ) R n h +m h = ( Lh ) R n h +m h. (42) The matrix A ε,r,h as well as the mass matrix J h are symmetric and positive definite for any h > and any r >. On the other hand, the matrix of order m h + n h in (42) is symmetric but not positive definite. We use exact integration methods developed in [14] for the evaluation of the coefficients of the matrices. The system (42) is solved using the direct LU decomposition method. Let us also mention that for r =, although the formulation (16) is well-posed, numerically, the corresponding matrix A ε,,h is not invertible in agreement with Remark 2. In the sequel, we shall consider strictly positive values for r. Once an approximation ϕ h is obtained, an approximation v ε,h of the control v ε is given by v ε,h = ρ 2 ϕ ε,h 1 ω. The corresponding controlled state y ε,h may be obtained by solving (1) with standard forward approximation (we refer to [1], Section 4 where this is detailed). Here, since the controlled state is directly given by the multiplier λ, we simply use λ h as an approximation of y and we do not report here the computation of y h. In the sequel, we only report numerical experiments in the one dimensional setting. We use uniform rectangular meshes. Each element is a rectangle of lengths x and t; x > and t > denote as usual the discretization parameters in space and time, respectively. We note where diam(k) denotes the diameter of K. h := max{diam(k), K T h } 3.2 Normalization and discretization of the mixed formulation (32) The same approximation may be used for the mixed formulation (32). In particular, we easily check that the finite dimensional spaces M ε,h and Φ ε,h (which actually do not depend on ε) are conformal approximation of L 2 (Q T ) and Φ ρ,ρ respectively. However, in the limit case ε =, a normalization of the variable ϕ, which is singular and takes arbitrarily large amplitude in the neighborhood of Ω {T } is very convenient and suitable in practice. Following [18], we introduce the variable ψ := ρ 1 ϕ ρ 1 Φ ρ,ρ and replace the mixed formulation (32) by the equivalent one: find (ψ,λ) ρ 1 Φ ρ,ρ L 2 (Q T ) solution of { â(ψ, ψ) + ˆb(ψ, λ) = ˆl(ψ), ψ ρ 1 Φ ρ,ρ (43) ˆb(ψ, λ) =, λ L 2 (Q T ),

20 3 NUMERICAL APPROXIMATION AND EXPERIMENTS 19 where â : ρ 1 Φ ρ,ρ ρ 1 Φ ρ,ρ R, â(ψ, ψ) = ψ ψ dxdt q T ˆb : ρ 1 Φ ρ,ρ L 2 (Q T ) R, ˆb(ψ, λ) = Q T ρ 1 L (ρ ψ) λ dxdt ˆl : ρ 1 Φ ρ,ρ R, ˆl(ϕ) = (y, ρ (,)ψ(,)) L2 (Ω). Well-posedness of this formulation is the consequence of Theorem 2.3. Moreover, the optimal controlled state is still given by ρ 1 λ while the optimal control is expressed in term of the variable ψ as v = ρ 1 ψ 1 ω. The corresponding discretization approximation (augmented with the term r ρ 1 L (ρ ψ) L 2 (Q T )) reads as follows: find (ψ h, λ h ) Φ h M h solution of with for any r >. { â r (ψ h,ψ h ) + ˆb(ψ h, λ h ) = ˆl(ϕ h ), ψ h Φ h ˆb(ψh,λ h ) =, λ h M h. a r (ψ h,ψ h ) :=a(ψ h,ψ h ) + r(ρ 1 L (ρ ψ h ), ρ 1 L (ρ ψ h )) L 2 (Q T ) =(ψ h,ψ h ) L2 (q T ) + r(ρ 1 L (ρ ψ h ), ρ 1 L (ρ ψ h )) L2 (Q T ). Remark 3 When the weights ρ and ρ are chosen in such a way that they are compensated each other in the term ρ 1 L (ρ ψ), the change of variable has the effect to reduced the amplitude (with respect to the time variable) of the coefficients in the integrals of â r and ˆb, and therefore, at the discrete level, to improve significantly the condition number of square matrix Âr,h so that â r (ψ h,ψ h ) =< Âr,h{ψ h }, {ψ h } > R n h,r n h. In this respect, the change of variable, can be seen as a preconditioner for the mixed formulation (32). Similarly to (42), we note the matrix form of (44) as follows : ( Âr,h ) ˆBT h ( ) {ψh } ˆB h {λ h } R n h +m h,n h +m h R n h +m h = ( ˆLh ) R n h +m h (44), (45) where ˆB h is the matrix so that ˆb(ψ h, λ h ) =< ˆB h {ψ h }, {λ h } > R m h,r m h and ˆL h is the matrix so that ˆl(ψ h ) =< ˆL h, {ψ h } >. 3.3 The discrete inf-sup test Before to give and discuss some numerical experiments, we first test numerically the discrete infsup condition (41). Taking η = r > so that a ε,r (ϕ, ϕ) = (ϕ, ϕ) Φε exactly for all ϕ, ϕ Φ ε, it is readily seen (see for instance [8]) that the discrete inf-sup constant satisfies { } δ δ ε,r,h = inf : Bh A 1 ε,r,h BT h {λ h } = δ J h {λ h }, {λ h } R m h \ {}. (46) The matrix B h A 1 ε,r,h BT h enjoys the same properties than the matrix A ε,r,h: it is symmetric and positive definite so that the scalar δ ε,h defined in term of the (generalized) eigenvalue problem (46) is strictly positive. This eigenvalue problem is solved using the power iteration algorithm (assuming that the lowest eigenvalue is simple): for any {vh } Rn h such that {vh } 2 = 1, compute for any n, {ϕ n h } Rn h, {λ n h } Rm h and {v n+1 h } R m h iteratively as follows : { Aε,r,h {ϕ n h} + Bh T {λ n h} =, {v n+1 B h {ϕ n h} = J h {vh} n h } = {λn h } {λ n h }. 2