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, 2013], 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.

This paper deals with the boundary controllability of inviscid incompressible fluids for which thermal effects are important and are modeled through the Boussinesq approximation. Almost all our results deal with zero heat diffusion. By adapting and extending some ideas from Coron and Glass, we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows.

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in Ω × (0, T ) -Ω a bounded subset of R N -from a partial distributed observation. We employ a least-squares technique and minimize the L 2 -norm of the distance from the observation to any solution. Taking the parabolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. The well-posedness of this mixed formulation -in particular the inf-sup property -is a consequence of classical energy estimates. We then reproduce the arguments to a linear first order system, involving the normal flux, equivalent to the linear parabolic equation. The method, valid in any dimension spatial dimension N , may also be employed to reconstruct solution for boundary observations.With respect to the hyperbolic situation considered in [10] by the first author, the parabolic situation requires -due to regularization properties -the introduction of appropriate weights function so as to make the problem numerically stable.• First, it is in general not possible to minimize over a discrete subspace of the set {y; Ly − f = 0} subject to the equality (in L 2 (Q T )) Ly − f = 0. Therefore, the minimization procedure first requires the discretization of the functional J and of the system (1); this raises the issue, when one wants to prove some convergence result of any discrete approximation, of the uniform coercivity property (typically here some uniform discrete observability inequality for the adjoint solution) of the discrete functional with respect to the approximation parameter.As far as we know, this delicate issue has received answers only for specific and somehow academic situations (uniform Cartesian approximation of Ω, constant coefficients in (1), etc). We refer to [4,30].• Second, in view of the regularization property of the heat kernel, the space of initial data H for which the corresponding solution of (1) belong to L 2 (q T ) is a huge space. Its contains in particular the negative Sobolev space H −s (Ω) for any s > 0 and therefore is very hard to approximate numerically. For this reason, the reconstruction of the initial condition y 0 of (1) from a partial observation in L 2 (q T ) is therefore known to be numerically severally ill-posed and requires, within this framework, a regularization to enforce that the minimizer belongs, for instance, to L 2 (Ω) much easier to approximate (see [13]). The situation is analogous for the so-called backward heat problem, where the observation on q T is replaced by a final time observation. We refer to ([8, 31, 33]) where this ill-posedness is discussed.The main reason of this work is to reformulate problem (LS) and show that the use of variational methods may overcome these two drawbacks.Preliminary, we also mention that the quasi-reversibility method initia...