Flow functions, control flow functions, and the reach control problem

The paper studies the reach control problem (RCP) to make trajectories of 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 a flow function, which provides the analog of a Lyapunov function for the equilibrium stability problem. A flow function comprises a scalar function that decreases along closed-loop trajectories, and its existence is a necessary and sufficient condition for closed-loop trajectories to exit the polytope. It provides an analysis tool for determining if a specific instance of RCP is solved, without the need for calculating the state trajectories of the closed-loop system. Results include a variant of the LaSalle Principle tailored to RCP. An open problem is to identify suitable classes of flow functions. 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. Next we introduce the notion of a control flow function. It is shown that the Artstein-Sontag theorem of control Lyapunov functions has direct analogies to RCP via control flow functions.