Regularity of Optimal Solutions and the Optimal Cost for Hybrid Dynamical Systems via Reachability Analysis

TL;DR

This paper investigates how the optimal cost and solutions of hybrid dynamical systems depend on initial conditions and parameters, establishing conditions for continuity and approximation of optimal costs through reachability analysis.

Contribution

It provides new theoretical insights into the regularity of optimal solutions and costs in hybrid systems using reachability, including conditions for continuity and approximation.

Findings

Optimal cost varies upper semicontinuously under outer well-posedness.

Optimal cost varies lower semicontinuously under inner well-posedness with additional assumptions.

Optimal solutions vary upper semicontinuously when system is both inner and outer well-posed.

Abstract

For a general optimal control problem for dynamical systems with hybrid dynamics, we study the dependency of the optimal cost and of the value function on the initial conditions, parameters, and perturbations. We show that upper and lower semicontinuous dependence of solutions on initial conditions -- properties that are captured by outer and inner well-posedness, respectively -- lead to the existence of a solution to the hybrid optimal control problem and upper/lower semicontinuity of the optimal cost. In particular, by exploiting properties of finite horizon reachable sets for hybrid systems, we show that the optimal cost varies upper semicontinuously when the hybrid system is (nominally) outer well-posed, and lower semicontinuously when it is (nominally) inner well-posed and an additional assumption requiring partial knowledge of solutions. Consequently, when the system is both (nominally) inner and outer well-posed and the aforementioned assumption holds, the optimal cost varies continuously, and optimal solutions vary upper semicontinuously. We further show that even in the absence of this solution-based assumption, the optimal cost can be continuously approximated. The results are demonstrated in finite horizon optimization problems with hybrid dynamics, both theoretically and numerically.