Planning methods for the optimal control and performance certification of general nonlinear switched systems

We consider two problems for discrete-time switched systems with autonomous, general nonlinear modes. The first is optimal control of the switches so as to minimize the discounted infinite-horizon sum of the costs. The second problem occurs when switches are a disturbance, and the worst-case cost under any sequence of switches is sought. We use an optimistic planning (OP) algorithm that can solve general optimal control with discrete inputs such as switches. We extend the analysis of OP to provide sequences of switches with certification (upper and lower) bounds on the optimal and worst-case costs, and to characterize the convergence rate of the gap between these bounds. Since a minimum dwell time between switches must often be ensured, we introduce a new optimistic planning variant that can handle this case, and analyze its convergence rate. Simulations for linear and nonlinear modes illustrate that the approach works in practice.