Transition paths connecting metastable states are significant in science and engineering, such as in biochemical reactions. In this talk, I will present a stochastic optimal control formulation for transition path problems over an infinite time horizon, modeled by Markov jump processes on Polish spaces. An unbounded terminal cost at a stopping time and a running cost in entropic form for the control velocity regulate the transitions between metastable states. One important feature is that, under optimal control, the original transition bridge (referring to the distribution for paths conditioned on reaching the target states at stopping time) is preserved. The unbounded terminal cost leads to a singular optimal control and poses difficulties in the Girsanov transformation. Through Gamma-convergence techniques and by taking limits in the corresponding Martingale problem, we manage to obtain a singularly optimally controlled transition rate. We demonstrate that the committor function, which solves a backward equation with specific boundary conditions, provides an explicit formula for the optimal path measure. The optimally controlled process realizes the transition paths almost surely but without altering the bridges of the original process.