A priori estimates for operational differential inclusions and necessary conditions for optimality.(English)Zbl 0715.49010

Analysis and optimization of systems, Proc. 9th Int. Conf., Antibes/Fr. 1990, Lect. Notes Control Inf. Sci. 144, 519-528 (1990).
Summary: [For the entire collection see Zbl 0699.00041.]
Derivation of necessary conditions for optimality in optimal control theory relies on variational calculus of control systems. We provide here such calculus for the semilinear differential inclusion $(1)\quad x'\in Ax+F(t,x),\quad x(0)=x_ 0$ where A is the infinitesimal generator of a $$C_ 0$$-semigroup on a separable Banach space X and F: [0,T]$$\times X\rightsquigarrow X$$ is a set-valued map. Such inclusion encompass a number of control systems, including those with state dependent controls. We prove a relaxation theorem and investigate infinitesimal generators of reachable sets and variational inclusions. Results are applied to derive necessary conditions for optimality for some semilinear optimal control problems.

MSC:

 49J27 Existence theories for problems in abstract spaces 49K27 Optimality conditions for problems in abstract spaces 34A60 Ordinary differential inclusions 49J15 Existence theories for optimal control problems involving ordinary differential equations 49J20 Existence theories for optimal control problems involving partial differential equations

Zbl 0699.00041