Existence of feasible approximating trajectories for differential inclusions with obstacles as state constraints

In the analysis of optimal control problems with state constraints, a key step (establishing `metric regularity´) is to show that an arbitrary state trajectory can be approximated by a feasible state trajectory, whose distance from the original state trajectory is linearly related to the state constraint violation. It is then possible, for example, to give conditions for the state constraint maximum principle to apply in normal form (the cost multiplier can be taken non-zero), to establish regularity properties of the value function and to characterize it as the unique viscosity solution of the Hamilton Jacobi Bellman equation. While a great deal of attention has been given in the literature to conditions for metric regularity, cases in which the state constraint region is the complement of a collection of boxes with corners has not previous treated. Such cases are important because they arise in the optimal control of AUVs which are required to avoid box regions in the state space. This paper provides conditions for metric regularity that cover such situations.