# Optimistic planning for the near-optimal control of nonlinear switched discrete-time systems with stability guarantees

## TL;DR

This paper introduces OPmin, an improved optimistic planning algorithm for nonlinear switched systems, addressing previous limitations and providing near-optimal, stable control with broader cost function applicability.

## Contribution

The paper develops OPmin, a modified optimistic planning algorithm that overcomes key limitations of OP, enabling near-optimal and stable control for nonlinear switched systems with general cost functions.

## Key findings

- OPmin achieves near-optimal control guarantees.
- OPmin ensures stability under certain assumptions.
- The algorithm broadens applicability to various cost functions.

## Abstract

Originating in the artificial intelligence literature, optimistic planning (OP) is an algorithm that generates near-optimal control inputs for generic nonlinear discrete-time systems whose input set is finite. This technique is therefore relevant for the near-optimal control of nonlinear switched systems, for which the switching signal is the control. However, OP exhibits several limitations, which prevent its application in a standard control context. First, it requires the stage cost to take values in [0,1], an unnatural prerequisite as it excludes, for instance, quadratic stage costs. Second, it requires the cost function to be discounted. Third, it applies for reward maximization, and not cost minimization. In this paper, we modify OP to overcome these limitations, and we call the new algorithm OPmin. We then make stabilizability and detectability assumptions, under which we derive near-optimality guarantees for OPmin and we show that the obtained bound has major advantages compared to the bound originally given by OP. In addition, we prove that a system whose inputs are generated by OPmin in a receding-horizon fashion exhibits stability properties. As a result, OPmin provides a new tool for the near-optimal, stable control of nonlinear switched discrete-time systems for generic cost functions.