Sufficient approximate optimality condition for the inverse one-phase Stefan problem

We investigate the identification problem for the onephase Stefan problem. As the inverse Stefan problem is not well posed, an optimal control problem is considered instead. In the paper we develop a dual dynamic programming approach to derive sufficient approximate optimality conditions for that optimal control problem. As a next step we formulate and prove a verification theorem for approximate solution. The verification Theorem 4.1 is the basis for the development of a numerical algorithm. Having the verification theorem we do not need the convergence of our algorithm.