Numerical solution of a parabolic optimal control problem arising in economics and management

Abstract

We investigate a parabolic optimal control problem that serves as a mathematical model for a class of problems of economics and management. The problem is to minimize the objective functional that is not convex and not coercive, and the state equation is a two-dimensional linear parabolic diffusion-advection equation controlled by the coefficients of the advective part. The main property of a solution of state equation is its non-negativity for a non-negative initial data. We prove that implicit (backward Euler) and fractional steps (operator splitting) approximations of the state equation have strictly positive solutions for positive time and use this fact to prove the existence of a solution for a discrete optimal control problem without imposing any additional constraints on the control function and mesh parameters. We derive first order optimality conditions and apply gradient-type iterative methods for the constructed discrete optimal control problem. Numerical tests confirm theoretically valid results and demonstrate the effectiveness of the proposed solution methods.