Existence of solutions to an optimal control problem for a hybrid system

In this note we consider an optimal control problem for a hybrid dynamical system. In Russian literature such systems are also called discrete-continuous or mixed logical dynamical systems. Hybrid systems usually appear as mathematical models of various technical processes. For example, they describe the functioning of automobile transmissions, temperature control systems, certain processes with hysteresis, dynamical systems with collisions or Coulomb friction, and many others. Mathematical theory of optimal control for such systems is currently well-developed; in particular, necessary and sufficient optimality conditions are found and numerical algorithms are constructed. On the other hand, the authors are not aware of any results on existence of optimal controls. The aim of the paper is to fill the above mentioned gap. Recall that to prove the existence is enough to show that the initial optimal control problem is equivalent to a nonlinear optimization problem that consists in minimizing a continuous function on a reachable set of the control system. Then, according to the Weierstrass theorem, conditions ensuring compactness of the reachable set also ensure the existence of an optimal control. In this work we show that a similar approach can be applied to the hybrid dynamical system. The auxiliary nonlinear optimization problem is slightly different, so that in order to prove the compactness of the feasible set one must use properties of the integral funnel of a control system rather than those of its reachable sets.