Optimal control under a dynamic fuel constraint

We present a new approach to solve optimal control problems of the monotone follower type. The key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint. Instead of dynamic programming, we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the Snell envelope of the objective functional's gradient at the optimum. The optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in Bank and El Karoui ( 2004), Ann. Probab., 32, pp. 1030 - 1067. As an illustration, we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics.