Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems

Abstract

This chapter is concerned with optimal control problems of dynamical systems described by partial differential equations (PDEs). Firstly, using the Dubovitskii-Milyutin approach, we obtain the necessary condition of optimality, i.e., the Pontryagin maximum principle for optimal control problem of an age-structured population dynamics for spread of universally fatal diseases. Secondly, for an optimal birth control problem of a McKendrick type age-structured population dynamics, we establish the optimal feedback control laws by the dynamic programming viscosity solution (DPVS) approach. Finally, for a well-adapted upwind finite-difference numerical scheme for the HJB equation arising in optimal control, we prove its convergence and show that the solution from this finite-difference scheme converges to the value function of the associated optimal control problem.