Numerical solution of a parabolic optimal control problem in economics

In this paper, we construct and study finite difference approximations of optimal control problems arising in the mathematical modeling of certain management and financial problems. For the definiteness we consider two-dimensional in space variables problems, although all the theoretical results remain valid for any dimension. The numerical calculation results are presented for 1D and 2D problems. To approximate the state equations we use implicit (backward Euler) and fractional steps (operator splitting) methods. The solutions of the constructed mesh state equations are strictly positive and keep an analogue of the mass balance condition. We provide the existence results and first order optimality conditions for the corresponding mesh optimal control problems. We use several iterative methods to implement the constructed nonlinear optimization problems and compare their effectiveness. (paper)