PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Abstract

This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverseproblem is transformed into a PDE-constrained optimal control problem and these two problems areequivalent under some assumptions. Different from the existing literature which is also based onthe optimal control theory, we consider the state equation in the sense of very weak solution definedby the transposition technique. In this way, it does not need to impose any regularity requirementon the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization termis introduced. The regularized problem is well-posed and its solution converges to the non-regularizedcounterpart as the regularization parameter approaches zero. We establish the finite element approximationto the regularized control problem and the convergence of the discrete problem is also investigated.Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.