Approximate Feedback Minimum Principle for Suboptimal Processes in Non-smooth Optimal Control Problems

Abstract

We consider a non-smooth optimal control problem with Lipschitz dynamics with respect to state variables and a terminal functional, which is defined by a semiconcave function (the difference between smooth and continuous convex functions). For suboptimal processes of this problem, the variational type necessary optimality condition is obtained. This condition, on one hand, is a generalization of the so-called Feedback Minimum Principle obtained by the author in previous publications, and on the other hand, it significantly strengthens \(\varepsilon \)-Maximum Principle for suboptimal processes obtained by I. Ekeland. An important feature of our result is that the obtained condition of suboptimality is formulated using a family of auxiliary problems of dynamic optimization, due to the multiplicity of solutions of the adjoint inclusion and the plurality of subgradients of the terminal function.