  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.
