Algorithms for optimal hybrid control

The authors previously (1994) proposed a general, unified framework for hybrid control problems that encompasses several types of hybrid phenomena and several models of hybrid systems. An existence result was obtained for optimal controls. The value function associated with this problem satisfies a set of "generalized quasi-variational inequalities" (GQVIs). We give a classification of the types of hybrid systems models covered by our framework and algorithms. We review our general framework and results. Then, we outline three explicit approaches for computing the solutions to the GQVIs that arise in optimal hybrid control. The approaches are generalizations to hybrid systems of shooting methods for boundary value problems, impulse control for piecewise-deterministic processes (PDPs), and value and policy iteration for piecewise-continuous dynamical systems. In the central case, we make clear the strong connection between impulse control for PDPs and optimal hybrid control. This allows us to give exact and approximate ("epsilon-optimal") algorithms for computing the value function associated with such problems and give some theoretical results. Also following previous work, we find that we can compute optimal solutions via linear programming (LP). The resulting LP problems are in general large, but sparse. In each case, the underlying feedback controls can be subsequently computed. Illustrative examples of each algorithm are solved in our framework.