From pure state and input constraints to mixed constraints in nonlinear systems

Abstract

We survey the results on the problem of pure/mixed state and input constrained control, with multidimensional constraints, for finite dimensional nonlinear differential systems with focus on the so-called admissible set and its boundary. The admissible set is the set of initial conditions for which there exist a control and an integral curve satisfying the constraints for all time. Its boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a generalized minimum-like principle with nonsmooth terminal conditions. Comparisons between pure state constraints and mixed ones are presented on a series of simple academic examples.