The Maximum Principle

The classical maximum principle was introduced in Chap. 1 . This fundamental theorem is a set of necessary conditions of optimality for dynamic optimization problems with endpoint cost and endpoint constraints, in which the dynamic constraint takes the form of a controlled differential equation. It unifies earlier optimality conditions in the calculus of variations and extends them to take account of the dynamic constraint. This chapter provides a generalization of these conditions, known as the Clarke nonsmooth maximum principle, that covers problems in which the endpoint constraint and cost are expressed in terms of nonsmooth functions and general closed sets, and the right side of the controlled differential equation is nonsmooth w.r.t. the state variable. Special cases of interest and simple extensions are discussed in detail. The derivation of Clarke’s nonsmooth principle appearing in this chapter is based on his perturbation technique. The necessary conditions in their full generality are arrived at in stages, starting with necessary conditions for a simple problem with a smooth endpoint cost function and no endpoint constraints. Subsequent stages introduce refinements (additional constraints, nonsmoothness, removal of temporary simplifying hypotheses). The key idea is that, at each stage, when we seek necessary conditions for a newly refined version of the problem, we construct a minimizer to a perturbed problem, which is simpler and for which necessary conditions are available from the previous stage. Necessary conditions for the new problem are then obtained in the limit, from the necessary conditions for the perturbed problems. Construction of the perturbed problems is accomplished by various techniques, which include application of Ekeland’s theorem and the use of quadratic inf convolutions. Limit taking of the necessary conditions is carried out with the help of subdifferential calculus.