Generalized flow conditions for reach control on polytopes

The paper studies the reach control problem (RCP) to make an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope. We introduce the notion of generalized flow conditions, which give a necessary and sufficient condition for closed-loop trajectories to exit the polytope. In analogy with Lyapunov stability theory, the generalized flow condition comprises a functional that decreases along closed-loop trajectories. We provide a set of results to analyze whether an instance of RCP is solved, without resorting to exhaustive simulation of the closed-loop system. This includes a variant of the LaSalle principle tailored to RCP. An open problem is to identify suitable classes of functionals that give rise to a generalized flow condition. We explore functions of the form V (x) = max{Vi(x)}, and we give evidence that these functions arise naturally when RCP is solved using continuous piecewise affine feedbacks.