About least-squares type approach to address direct and controllability problems

- We discuss the approximation of distributed null controls for partial differential equations. The main purpose is to determine an approximation of controls that drives the solution from a prescribed initial state at the initial time to the zero target at a prescribed final time. As a non trivial example, we mainly focus on the Stokes system for which the existence of square-integrable controls have been obtained in [Fursikov \& Imanuvilov, Controllability of Evolution Equations, 1996]) via Carleman type estimates. We introduce a least-squares formulation of the controllability problem, and we show that it allows the construction of strong convergent sequences of functions toward null controls for the Stokes system. The approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time-in particular the null controllability condition at time T-, and then finding among this class one element satisfying the system. This second step is done by minimizing a quadratic functional, among the admissible corrector functions of the Stokes system. We also discuss briefly the direct problem for the steady Navier-Stokes system. The method does not make use of any duality arguments and therefore avoid the ill-posedness of dual methods, when parabolic type equation are considered

Numerical null controllability of a semi-linear heat equation via a least squares method

This note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fern´andez-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani’s theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points.Contrôlabilité exacte à zéro d’une equation de la chaleur semi-linéaire par une méthode des moindres carrés. Cette note concerne la détermination effective de contrôles à zéro pour une équation de la chaleur semi-linéaire, dans le cas lègérement surlinéaire. Sous des conditions de croissances optimales, l’existence de contrôles a été obtenue dans [Fernández-Cara & Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] par un argument de point fixe ; voir aussi [Barbu, Exact controllability of the superlinear heat equation, 2000]. Précisément, des inégalités de Carleman et le théorème de Kakutani impliquent l’existence de points fixes pour un opérateur de contrôle linéarisé associé. En pratique, la difficulté est d’extraire des itérés de Picard une sous-suite convergente. Cette note propose et analyse une reformulation du problème par une approche de type moindres carrés : on montre que celle-ci garantit une construction explicite de points fixes

Inverse problems for linear parabolic equations using mixed formulations -Part 1 : Theoretical analysis

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

On the exact boundary controllability of semilinear wave equations

We address the exact boundary controllability of the semilinear wave equation $\partial_{tt}y-\Delta y + f(y)=0$ posed over a bounded domain $\Omega$ of $\mathbb{R}^d$. Assuming that $f$ is continuous and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f(r)\vert /(\vert r\vert \ln^p\vert r\vert)\leq \beta$ for some $\beta$ small enough and some $p\in [0,3/2)$, we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in $L^2(\Omega)\times H^{-1}(\Omega)$. Then, assuming that $f$ is in $\mathcal{C}^1(\mathbb{R})$ and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f^\prime(r)\vert/\ln^p\vert r\vert\leq \beta$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation

A mixed formulation for the direct approximation of L2-weighted controls for the linear heat equation

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.Coordenação de aperfeiçoamento de pessoal de nivel superiorMinisterio de Ciencia e Innovació

Stabilisation interne optimale de l'équation des ondes par une méthode de lignes de niveau

International audienceOn considère une équation des ondes linéaire définie sur un domaine Ω régulier du plan, et amortie sur un sous-domaine interne ω⊂Ω. On considère le problème de la position et de la forme optimale de ω minimisant l'énergie du système à un instant T>0. La méthode de dérivation de forme conduit à la variation de l'énergie vis-à-vis de ω exprimée comme une intégrale curviligne le long de ∂ω. La méthode des lignes de niveau ramène alors le problème à la résolution d'une équation non linéaire d'Hamilton-Jacobi dont le terme d'advection est l'intégrant de la dérivée de forme. L'efficacité de la méthode est numériquement confirmée