Open loop optimal control problems with linear hybrid ( discrete/ continuous) systems embedded are often approximated as dynamic optimization problems. We propose a deterministic global optimization algorithm for linear hybrid systems with varying transition times. First, the control parametrization enhancing transform is used to transform the problem from a linear hybrid system with scaled discontinuities and varying transition times into a nonlinear one with stationary discontinuities and fixed transition times. Next, a theory is developed for constructing convex relaxations of arbitrary Bolza-type functionals subject to the transformed hybrid system. Finally, the convex relaxations are utilized in a branch-and-bound framework to obtain the solution to e global optimality within a finite number of iterations.