Optimization method for an evolutional type inverse heat conduction problem

摘要

This paper deals with the determination of a pair (q, u) in the heat conduction equation u(t) - u(xx) + q(x, t) u = 0, with initial and boundary conditions u(x, 0) = u(0)(x), u(x|x=0) = u(x|x=1) = 0, from the overspecified data u(x, t) = g(x, t). By the time semi-discrete scheme, the problem is transformed into a sequence of inverse problems in which the unknown coefficients are purely space dependent. Based on the optimal control framework, the existence, uniqueness and stability of the solution (q, u) are proved. A necessary condition which is a couple system of a parabolic equation and parabolic variational inequality is deduced.