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For a general optimal control problem for dynamical systems with hybriddynamics, we study the dependency of the optimal cost and of the value functionon the initial conditions, parameters, and perturbations. We show that upperand 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 andupper/lower semicontinuity of the optimal cost. In particular, by exploitingproperties of finite horizon reachable sets for hybrid systems, we show thatthe 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 ofsolutions. Consequently, when the system is both (nominally) inner and outerwell-posed and the aforementioned assumption holds, the optimal cost variescontinuously, and optimal solutions vary upper semicontinuously. We furthershow that even in the absence of this solution-based assumption, the optimalcost can be continuously approximated. The results are demonstrated in finitehorizon optimization problems with hybrid dynamics, both theoretically andnumerically.

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