Hybrid systems and optimal control

The aim of this paper is to treat some optimal control problems for a class of hybrid systems. We start providing a definition of hybrid system inspired by the concept introduced by Artstein (1995), who defines hybrid control in relation to stabilization problems for a classical control system. The same definition proved to be successful to tackle other stabilization problems. In this paper, we consider a class of systems with hybrid features and optimization problems for this class of systems. The word hybrid is motivated by the fact that these systems are characterized by the presence of both a continuous time evolution and a discrete time evolution. A trajectory for these systems evolves following some dynamical constraint and at some fixed or variable times (called location switching times) it jumps following the rules of a discrete time evolution. The definition of hybrid system we give is quite general and covers many interesting applications.