A fixed-point iteration method for optimal control problems governed by parabolic equations

The analysis of an optimal control model governed by parabolic differential equations is usually developed by carrying out the formulation of the so called “optimality system”, which consists of partial differential equations with opposite orientations. Even though existence and uniqueness of the solution may be investigated, in most cases the solution itself is not available in closed form. For this reason, our aim consists of evaluating an accurate approximation. Thus, we provide a constructive method based on a scheme of successive approximations which converge to a fixed-point representing the required solution. Successive approximations are evaluated by applying Finite Element method for the spatial semi-discretization; then the resulting ODE system is solved by exponential integrators. Some numerical results are provided in order to show the effectiveness of the proposed approach.