An Optimal Distributed Control for Age-dependent Population Diffusion System

The optimal distributed control problem for age-dependent population diffusion system governed by integral partial differential equations is investigated in this paper. As new results, the existence and uniqueness of the optimal distributed control are proposed and proved, a necessary and sufficient conditions for the control to be optimal are obtained, and the optimality system consisting of integro-partial differential equations and variational inequalities are constructed in which the optimal controls can be determined. The applications of penalty shifting method for infinite dimensional systems to approximate solutions of control problems for the population system are researched. An approximation program is structured, and the convergence of the approximating sequences in appropriate Hilbert spaces is derived. The results in this paper may significantly provide theoretical reference for the practical research of the control problem in population systems.