Numerical solution to the optimal birth feedback control of a population dynamics: Viscosity solution approach

This paper is concerned with the optimal birth control of a McKendrick-type age-structured population dynamic system. We use the dynamic programming approach in our investigation. The Hamilton-Jacobi-Bellman equation satisfied by the value function is derived. It is shown that the value function is the viscosity solution of the Hamilton-Jacobi-Bellman equation. The optimal birth feedback control is found explicitly through the value function. A finite difference scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is verified numerically by comparing with different controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989.