The continuity and uniqueness of the value function of the hybrid optimal control problem with reach time to a target set

TL;DR

This paper proves the continuity and uniqueness of the value function in a hybrid optimal control problem involving both continuous and discrete dynamics, using viscosity solutions and dynamic programming.

Contribution

It establishes the continuity and uniqueness of the value function for hybrid systems with reach time, characterizing it as a unique viscosity solution of a quasi-variational inequality.

Findings

Proved the value function's continuity under certain boundary conditions.

Characterized the value function as a unique viscosity solution.

Applied dynamic programming to hybrid systems with reach time.

Abstract

The hybrid optimal control problem with reach time to a target set is addressed and the continuity and uniqueness of the associated value function is proved. Hybrid systems involves interaction of different types of dynamics: continuous and discrete dynamics. The state ofa continuous system is evolved by an ordinary differential equation until the trajectory hits the predefined jump sets: an autonomous jump set and a controlled jump set . At each jump the trajectory is moved discontinuously to another Euclidean space by a discrete system. We study the hybrid optimal control problem with reach time to a target set, prove the continuity of the associated value function with respect to the initial point under the assumption that is lower semicontinuous on the boundary of a target set, and also characterize it as an unique solution of a quasi-variational inequality in a viscosity sense using the dynamic programming principle.