Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data

Abstract

In this paper we are concerned with the uniqueness of two inverse source problems with lateral overdetermination. We establish uniqueness proofs in a unified manner based on relations with the approximate controllability of the 'adjoint problems'. To solve the inverse problems in a stable manner, one has to apply regularization techniques. Since in general the convergence rate of the regularized solutions to the exact solutions may be arbitrarily slow, it is of practical interest to ask for assumptions that guarantee a reasonable convergence rate of the regularized solutions. For the inverse problems under consideration, these assumptions can be characterized via exact controllability.