Hybrid LQ-optimization using dynamic programming

In this paper we study the linear quadratic optimal control problem for linear hybrid systems in which transitions between different discrete locations occur autonomously when the continuous state intersects given switching surfaces. In particular, we make an explicit connection between the newly developed, Pontryagin-type Hybrid Maximum Principle and the Bellman Dynamic Programming approach. As a consequence, we extend the classic Riccati-formalism, derive the associated Riccati-type equations, and prove the discontinuity of the full "hybrid" Riccati matrix. Finally, we discuss some computational aspects of the obtained theoretical results and propose a numerical algorithm in the framework of an optimal feedback control law.