Regularity of optimal solutions and the optimal cost for hybrid dynamical systems via reachability analysis

Abstract

For a general optimal control problem for dynamical systems with hybrid dynamics, we study the dependency of the optimal cost on perturbations to problem constraints and mappings. The former limits the allowable values for the state, input, initial and terminal condition, and the terminal time, while the latter is given by the right-hand sides of the differential/difference inclusions defining the dynamics, as well as the functions defining the cost functional 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 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 outer well-posed, and lower semicontinuously when it is inner well-posed and an additional assumption requiring partial knowledge of solutions holds. Consequently, when the system is both inner and outer well-posed and the aforementioned assumption holds, the optimal cost varies continuously, and optimal solutions vary outer/upper semicontinuously. We further show that even in the absence of this solution-based assumption, the optimal cost (respectively, solutions) can be continuously (respectively, outer/upper semicontinuously) approximated. Results are demonstrated by examples, theoretically and numerically.