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 discussed. The existence and uniqueness of the optimal distributed control is proved. The necessary and sufficient condition for a control to be optimal is obtained; and then the optimality system consisting of integral partial differential equations and variational inequalities is deduced. This optimality system can determine optimal controls. The application of penalty shifting method that Di Pillo statemented for infinite dimensional systems to the approximate solution of control problems for the population system is researched. The approximation program is structured, and the convergence of the approximating sequence on appropriate Hilbert spaces is proved.