Smooth and Nonsmooth Optimal Lipschitz Control — a Model Problem

Abstract

For a semilinear second order ordinary differential equation a control problem with pointwise state constraints is considered. The control of the system is realized via the outer function of the nonlinear term in the state equation. We prove a simple theorem stating the existence of optimal controls within the class of Lipschitz continuous functions and derive two necessary optimality conditions depending on their smoothness properties. For a smooth optimal control function the given optimality condition looks somehow like Pontryagin’s maximum principle. In the general case the optimality condition involves Clarke’s generalized directional derivative. Furthermore a modified problem is considered, for which by means of Ekeland’s variational principle a suboptimality condition can be proved.