A constructive method for parabolic equations with opposite orientations arising in optimal control

An optimal control model governed by parabolic equations is usually analyzed by carrying out the formulation of the so called “optimality system”, that consists of equations with opposite orientations. PDE-constrained optimization is employed in different fields, such as in economics and in ecology for allocating resources and managing ecosystems. Under the assumption that any solution exists, we provide an original proof of its uniqueness. This result is original and can be applied to a wide range of problems. Moreover, the same proof can be exploited to carry out a constructive method for approximating the solution, which is not available in closed form in the most cases. The method is based on successive approximations converging to a fixed-point that is the required solution. Due to the structure of the problem, this kind of approximation performs a forward-backward integration, giving raise to different iterative schemes. We investigate their convergence in the continuous setting under an original approach, by adapting the same proof provided for the exact solution uniqueness. Another innovative issue is related to the numerical implementation which involves exponential integration in time: up to our knowledge, the use of exponential integrators is new and original in the setting of PDE-constrained optimization. The effectiveness of the proposed approach is shown by providing some numerical results.