First-Order Necessary Optimality Conditions for Nonlinear Optimal Control Problems

The aim of this dissertation is to provide first-order necessary conditions of optimality for local optimal solutions x^* of the problem

Minimize L(x), subject to ƒ(x) ∈ Ax,

where (X,‖ · ‖ _X),(Y,‖ · ‖ _Y) are Banach spaces, L : X → I̅R̅, ƒ : X → Y, and A : X → 2^Y. Here L is locally Lipschitz continuous or of the form g(y) + h(u), where x = (y,u), g is locally Lipschitz continuous, and h is proper convex lower semicontinuous, ƒ is locally Lipschitz continuous or constant, and A is linear or strongly monotone.

The main part of this dissertation deals with the construction of a general approach to the maximum principle followed by direct applications of the maximum principle to optimal control problems governed by abstract state equations.

This dissertation concludes by presenting two specific examples of applications of our necessary optimality conditions to problems governed by nonlinear partial differential equations of elliptic type. The necessary conditions are given in terms of Clarke's generalized gradient and Jacobian.