Identification of the unknown diffusion coefficient in a linear parabolic equation via semigroup approach

Abstract

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u(t)(x,t) = u(xx) + qu(x)(x, t) + p(t) u(x,t), with mixed boundary conditions u(x)(0, t) = psi(0), k(1) u(1, t) = psi(1). The main purpose of this paper is to investigate the distinguishability of the input-output mapping Phi[.] : P -> H-1,H-2[0, T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only the zero function, then the input-output mapping Phi[.] has the distinguishability property. It is also shown that both types of boundary conditions and also the region in which the problem is defined play an important role in the distinguishability property of the input-output mapping. Moreover, the input data can be used to determine the unknown parameter p(t) at (x, t) = (0, 0) and also the unknown coefficient q. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Phi[.] : P -> H-1,H-2[0, T] is given explicitly in terms of the semigroup.