A SINGULAR STOCHASTIC CONTROL PROBLEM ARISING FROM A DETERMINISTIC PROBLEM WITH NON-LIPSCHITZIAN MINIMIZERS (CALCULUS, VARIATIONS, HAMILTON-JACOBI EQUATIONS, OPTIMAL

A stochastic control problem is obtained as a "small noise" approximation to a deterministic optimal control problem. Two classes of admissible controls are considered for each of these problems and the optimal control policies are explicitly determined for each admissible class.;For the deterministic problem, the larger admissible class consists of controls with L('1) trajectories while the smaller class consists of controls with L('2) trajectories. Even though one admissible class is dense in the other, the optimal control in the larger class (the absolute minimizer) provides a cost strictly smaller than the minimum cost achievable while working in the smaller admissible class.;For the stochastic problem, the larger admissible class contains controls referred to as singular stochastic controls. For this class, the cumulative control has bounded variation trajectories. The smaller admissible class contains the standard stochastic controls, whose cumulative effect has absolutely continuous trajectories. The optimal singular control provides a cost strictly smaller than the minimum cost achievable when only standard stochastic controls are admissible. In particular, this shows that it is not always possible to approach the optimal cost for singular control if only the standard stochastic controls are used.