A Linear Quadratic Approach to Linear Time Invariant Stabilization for a Class of Hybrid Systems

In this paper, the problem of linear time invariant state feedback stabilization for a class of hybrid systems is dealt with. The considered class of systems has received a considerable attention in the last years especially as a benchmark for hybrid output regulation, and in this context it turns out to be quite crucial to have stabilization approaches working under minimal hypotheses meanwhile providing linear time invariant solutions. After showing that static linear time invariant state feedback stabilizers might not exist even in the considered simple setting, a new solution is provided by formulating and solving a linear quadratic optimal control problem, which turns out to be a static time varying linear state feedback. It is then shown how such a feedback can be implemented via a stable dynamic time invariant linear state feedback, by exploiting a dynamic extension implementing the stabilized optimal costate dynamics.