Planning for optimal control and performance certification in nonlinear systems with controlled or uncontrolled switches

We consider three problems for discrete-time switched systems with autonomous, general nonlinear modes. The first is optimal control of the switching rule so as to optimize the infinite-horizon discounted cost. The second and third problems occur when the switching rule is uncontrolled, and we seek either the worst-case cost when the rule is unknown, or respectively the expected cost when the rule is stochastic. We use optimistic planning (OP) algorithms that can solve general optimal control with discrete inputs such as switches. We extend the analysis of OP to provide certification (upper and lower) bounds on the optimal, worst-case, or expected costs, as well as to design switching sequences that achieve these bounds in the deterministic case. In this case, since a minimum dwell time between switching instants is often required, we introduce a new OP variant to handle this constraint, and analyze its convergence rate. We provide consistency and closed-loop performance guarantees for the sequences designed, and illustrate that the approach works well in simulations.