Discontinuous solutions to the Hamilton Jacobi equation under second order interiority hypotheses

This paper concerns the characterization of a lower semicontinuous value function in terms of the unique generalized solution of the Hamilton Jacobi equation (HJE), for optimal control problems with end point and pathwise state constraints. There is an extensive literature on this topic, in the contexts both of finite and infinite horizon problems. A key step in establishing this link is to validate a certain 'interiority' property, specifically to show that an arbitrary state trajectory satisfying the state constraint can be approximated by an interior state trajectory. The step has traditionally been accomplished under the hypothesis that, at each point in the boundary of the state constraint set, there exist admissible velocities strictly pointing away from the boundary of the state constraint set. This is a significant restriction, since this 'strict sense' outward/inward pointing condition is violated by control of systems, typically encountered in mechanical control, where the control action affects the rate of change of the state, not directly, but through the intermediary of a dynamic system. A class of control systems has been recently identified that enjoy this interiority property, but for which the strict sense outward pointing condition is violated. These advances have been used to establish uniqueness of continuous viscosity solutions of the relevant Hamilton Jacobi equation for an infinite horizon formulation of the problem involving such systems. In this paper, we exploit this newly discovered property also to provide the desired characterization of the value function for finite horizon problems involving endpoint and pathwise state constraints, when the value function is possibly discontinuous and the traditional strict outward pointing condition is replaced by a softer, second order condition.