  In this work we provide explicit conditions on the existence of optimal
feedback controls for stochastic processes with regime-switching. We use the
compactification method which needs less regularity conditions on the
coefficients of the studied stochastic systems. Two different kinds of controls
have been considered: one is the control on the coefficients of the diffusion
processes, another is the control on the transition rate matrices of the
continuous-time Markov chains. Moreover, the dynamic programming principle is
established after showing the continuity of the value function.
