State Constraints, Higher Order Inward Pointing Conditions, and Neighboring Feasible Trajectories

The classical inward pointing condition (IPC) for a control system whose state x is constrained in the closure C := \= \Omega of an open set \Omega prescribes that at each point of the boundary x \in \partial \Omega the intersection between the dynamics and the interior of the tangent cone of \= \Omega at x is nonempty. Under this hypothesis, for every system trajectory x(.) on a time interval [0, T], possibly violating the constraint, one can construct a new system trajectory \^x(.) that satisfies the constraint and whose distance from x(.) is bounded by a quantity proportional to the maximal deviation d := dist(\Omega , x([0, T])). When IPC is violated, the construction of such a constrained trajectory is not possible in general. However, in this paper we prove that a ``higher order"" inward pointing condition involving Lie brackets of the dynamics' vector fields (together with a nonpositiveness curvature-like assumption) and the implementation of a suitable ``rotating"" control strategy allows for a novel construction of a constrained trajectory \^x(.) whose distance from the reference trajectory x(.) is bounded by a quantity proportional to \surd d. As an application, we establish the continuity up to the boundary of the value function V of a connected optimal control problem, a continuity that allows one to regard V as the unique constrained viscosity solution of the corresponding Bellman equation.