On the set-valued approach to optimal control of sliding mode processes

This paper is devoted to a theoretic framework for a general optimal control problem (OCP) associated with the classic sliding mode process. The sliding dynamic behavior is interpreted here as a special kind of additional constraints related to the main optimization problem. We are specially interested in the development of some adequate constructive approximations of the original OCPs. The mathematical approach based on the set-valued analysis allows to study the discontinuity of sliding mode dynamics in the abstract setting. Moreover, we also establish some sensitivity properties of the optimal solutions. The obtained results provide an universal analytical tool for the corresponding conceptual approximation schemes related to the original OCPs. The constructive approximations proposed in this paper are numerically stable and can be applied to various classes of optimal control processes governed by the affine control systems.