Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data

Abstract

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation u(t)(x,t) = (k(x)u(x)(x,t))(x) with mixed boundary conditions k(0)u(x)(0,t) = psi(0), u(1, t) = psi(1). The aim of this paper is to investigate the distinguishability of the input-output mappings Phi[.]: kappa -> H-1,H-2[0,T], Psi[.] : kappa -> H-1,H-2[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Phi[.] and Psi[.] have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability property of these mappings. Moreover, in the light of measured output data (boundary observations) f(t) := u(0,t) or/and h(t) := k(1)u(x)(1, t), the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x = 0 and x = 1, respectively, can be determined explicitly. In addition to these, the values k'(0) and k'(1) of the unknown coefficient k(x) at x = 0 and x = 1, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f (t) and h(t) can be determined analytically by an integral representation. Hence the input-output mappings Phi[.] : kappa -> H-1,H-2[0,T], Psi[.] : kappa -> H-1,H-2[0,T] are given explicitly in terms of the semigroup.