Smooth Fit for Some Bellman Equations

Two linear-convex deterministic singular control problems in dimensions one and two are considered in this paper. They are solved using the dynamic programming method. The interest here is the explicitness of the results and the relation between the geometry of the drift along the free boundary of these problems and the principle of smooth t. The method of dynamic programming reduces the study of an optimal control problem to the study of a nonlinear partial diierential equation, the Hamilton-Jacobi-Bellman equation (see 6]). The value function for the optimal control problem is a solution of this equation. Singular optimal control problems generally give rise to a free boundary problem for said p.d.e. A basic step in solving the p.d.e. then is that of nding the free boundary. In the problems we consider here, convexity leads to the value function being C 1;1 (Lipschitz rst partial derivatives) across the free bounday. Then a central issue is to determine whether or not the value function is C 2 across the free boundary. The C 2 case is referred to as satisfying the smooth t principle. In stochastic control problems, smooth t occurs in the case of the linear-quadratic-Gaussian problem and it is also known to occur in other examples with nondegenerate diiusion (see 8] and its references). In fact, the property of smooth t was instrumental in solving the celebrated monotone follower problem 1]. On the other hand 8] gives an example where smooth t does not occur. In this paper we present problems in one and two dimensions having no diiusion at all; i.e., deterministic problems. These problems have linear dynamics and a nonnegative control. Other work dealing with problems with nonnegative control appear in 2]-5],,7],,9],10]. Because of the singular nature of our variational problems , we expect the optimal control to be extreme or to be singular. Since our controls are nonnegative this implies that we expect optimal controls to equal zero, innnity, or to be singular. The free boundary separates the null region (where the