Neighboring feasible trajectories and value functions for state constrained control problems

The inward pointing condition (IPC) for a control system whose state is constrained in a smooth set C requires that at each point of the boundary of C the intersection between the dynamics and the interior of the tangent cone of C be nonempty. Thanks to IPC, for every system trajectory y on an interval [0, T], possibly violating the constraint by an amount d, one can construct a new system trajectory x that both satisfies the constraint and whose distance from y is bounded by a quantity proportional to d. When IPC does not hold, the construction of such a trajectory is not possible in general. This paper is devoted to a state constrained dynamics (possibly containing an uncontrolled drift) that does not satisfy IPC. We formulate an inward pointing condition involving Lie brackets of the dynamics' vector fields and, by implementing a suitable ``rotating'' control strategy, we construct a constrained trajectory whose distance from the reference one is bounded by a quantity proportional to the square root of d. As an application, we establish the continuity up to the boundary of the value function of related optimal control problems with finite and infinite horizon.