On the Optimal Control of Heterogeneous Systems

Heterogeneity plays an important role in modeling demographic, epidemic, biological and economical processes. The mathematical formulation of such systems can vary widely: age-structured systems, trait-structured systems, or systems with endogenously changing domains are some of the most common. Controls in such systems may be non-distributed (targeting the whole system), or distributed (targeting one particular part of the system); some have to further satisfy constraints, such as integral constraints. This thesis investigates necessary optimality conditions of Pontryagin’s type involving a Hamiltonian functional. At first, infinite horizon age-structured systems are analyzed. They are governed by partial differential equations with boundary conditions, coupled by non-local integral states. Despite the numerous applications in population dynamics and economics, which are naturally formulated on an infinite horizon, a complete set of optimality conditions is missing, because of the difficult task of defining appropriate transversality conditions. For problems on the infinite horizon, the objective value may become infinite. Therefore, the necessary optimality conditions are derived for controls that are weakly overtaking optimal. To prove the result, a new approach (recently developed for ordinary differential equations by S. Aseev and V. Veliov) has been used for a system affine in the states, but non-linear in the controls and with a non-linear objective function. Furthermore, a problem which arises in demography is studied. Due to a low birth rate, many countries need immigration to sustain their population size. The age of the immigrants has severe implications on the stability of social security systems, therefore, the optimal age-pattern of immigrants is studied. The problem is on the infinite time horizon with a rather specific equality constraint. It is shown that there exists an optimal solution (although the problem is non-concave), and that this optimal control is time-invariant. A numerical case study is carried out for the Austrian population. A second focus lies on heterogeneous systems with a fixed domain of heterogeneity, which are