Optimal Control of Functional Differential Systems

This paper presents a unified approach to diverse optimal control problems for hereditary differential systems (HDS). An abstract local maximum principle is established via the Dubovitskii–Milyutin method. It yields necessary conditions for the optimal control of HDS towards surfaces in $R^n $ and towards target sets in function spaces. Nondegeneracy criteria are included. It is shown that the necessary conditions are sufficient in the case of linear HDS with convex cost functionals. Analogous results are obtained for systems described by Fredholm equations with general control action. For Fredholm systems with targets in function spaces, the attainability space $\mathcal{A}$ is investigated, criteria for $\mathcal{A}$ to be closed are established and full attainability is characterized.