Linear-Quadratic Optimal Control for Hybrid Systems with State-driven Jumps

In this paper, we lay preliminary foundations towards the comprehensive formulation, together with some constructive results, of the Linear-Quadratic (LQ) optimal control problem for hybrid systems in the presence of state-driven jumps. This objective is achieved by introducing the notion of N-jump optimal control law in terms of a policy that minimizes a certain (quadratic) cost functional and, at the same time, is capable of inducing N (state-driven) jumps of the resulting closed-loop hybrid system. Herein, we then focus on the constructive solution to the 1-jump optimal control problem, by stating necessary and sufficient conditions, together with the rather unexpected implications of the comparison with the, somewhat trivial, 0-jump optimal solution. The paper is concluded by an illustrative example that highlights some of the interesting features of the LQ optimal control problem for such a class of hybrid systems, which derive from the fact that linearity of the resulting hybrid arc is not preserved.