Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach

The problem of determining the pair w := {F(x, t); T-0(t)}of source terms in the parabolic equation ut = (k(x)u(x))(x) + F(x, t) and Robin boundary condition -k(l)u(x)(l, t) = v[u(l, t) - T-0(t)] from the measured final data mu(T)(x) = u(x, T) is formulated. It is proved that both components of the Frechet gradient of the cost functional J(w) = parallel to mu(T)(x) - u(x, T : w)parallel to(2)(0) can be found via the same solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove existence of a quasi-solution of the considered inverse problem, as well as to construct a monotone iteration scheme based on a gradient method. (c) 2006 Elsevier Inc. All rights reserved.