Optimistic planning with long sequences of identical actions for near-optimal nonlinear control

Optimistic planning for deterministic systems (OPD) is an algorithm able to find near-optimal control for very general, nonlinear systems. OPD iteratively builds near-optimal sequences of actions by always refining the most promising sequence; this is done by adding all possible one-step actions. However, OPD has large computational costs, which might be undesirable in real life applications. This paper proposes an adaptation of OPD for a specific subclass of control problems where control actions do not change often (e.g. bang-bang, time-optimal control). The new algorithm is called Optimistic Planning with K identical actions (OKP), and it refines sequences by adding, in addition to one-step actions, also repetitions of each action up to K times. Our analysis proves that the a posteriori performance guarantees are similar to those of OPD, improving with the length of the explored sequences, though the asymptotic behaviour of OKP cannot be formally predicted a priori. Simulations illustrate that for properly chosen parameter K, in a control problem from the class considered, OKP outperforms OPD.