Dynamic optimization problems with linear hybrid (discrete/continuous) systems embedded whose transition times vary are inherently nonconvex. For a wide variety of applications, a certificate of global optimality is essential, but this cannot be obtained using conventional numerical methods. We present a deterministic framework for the solution of such problems in the continuous time domain. First, the control parametrization enhancing transform is used to transform the embedded dynamic system from a linear hybrid system with scaled discontinuities and varying transition times into a nonlinear hybrid system with stationary discontinuities and fixed transition times. Next, a recently developed convexity theory is applied to construct a convex relaxation of the original nonconvex problem. This allows the problem to be solved in a branch-and-bound framework that can guarantee the global solution within epsilon optimality in a finite number of iterations.