Controllability of an Elliptic equation and its Finite Difference Approximation by the Shape of the Domain

Abstract

In this article we study a controllability problem for an elliptic partial differential equation in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution, with a given right hand side source term, into an open subdomain. The mapping that associates this trace to the shape of the domain is nonlinear. We first consider the linearized problem and show an approximate controllability property. We then address the same questions in the context of a finite difference discretization of the elliptic problem. We prove a local controllability result applying the Inverse Function Theorem together with a "unique continuation" property of the underlying adjoint discrete system. Subject Classification (1991): 35J05, 93B03, 65M06 Mathematics

Approximate controllability of linearized shape-dependent operators for flow problems

“…Our goal is to show approximate controllability for two different linearized shape operators [4]: implies y = 0, then F is approximately controllable.…”

Analytical and numerical methods in shape optimization

“…[26]), the compactness of the shape Hessian at the minimizing domain refers directly to the ill-posedness of the underlying identification problem.…”

Fréchet differentiability of the elasto‐acoustic scattered field with respect to Lipschitz domains

“…This results from the existence and uniqueness of the solution of BVP [50]. Consequently, the Banach space Z given by (10)- (12) is contained in F 0 v,q .0, v, q/Y, where Y is given by (7)- (9). Therefore F 0 v,q .0, v, q/ is a one-to-one and onto operator, and thus F 0 v,q .0, v, q/ defines an isomorphism from the Banach space Y into the Banach space Z.…”

“…The question of the dependence of the solutions of partial differential equations, with respect to their domains, is an important mathematical problem that has been extensively studied in an abstract framework [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], as well as for many specific application problems such as acoustic problems [17][18][19][20][21][22][23], elastic problems [24,25], conductivity problems [26], and various electromagnetic problems [27][28][29][30][31][32][33][34][35][36]. However, to the best of our knowledge, the case of elasto-acoustic scattering problems has not been studied yet in spite of its applied nature and its prevalence in engineering and numerical literature (see, e.g., [37] and the references therein).…”