Inverse heat conduction problems with boundary and final time measured output data

Abstract

This article presents a systematic study of inverse problems of identifying the unknown source term F(x, t) of the heat conduction (or linear parabolic) equation ut = (k(x)ux)x + F(x, t) from measured output data in the form of Dirichlet h(t) ≔ u(0, t), Neumann f(t) ≔ −k(0)ux(0, t) types of boundary conditions, also in the form the final time overdetermination uT(x) ≔ u(x, T). In the first part of this article the adjoint problem approach is used to derive formulas for the Fréchet gradient of cost functionals via solutions of the corresponding adjoint problems. It is proved that all these gradients are Lipschitz continuous. A necessary conditions for unicity and hence distinguishablity of solutions of all the three types of inverse source problems are derived. In the second part of this article the semigroup theory is used to obtain a general representation of a solution of the inverse source problem for the abstract evolution equation ut = Au + F with final data overdetermination. This representation shows non-uniqueness structure of the inverse problem solution, and also permits one to derive a sufficient condition for unicity.