Linear-Convex Singular Control In Two Dimensions

. A two-dimensional linear-convex deterministic singular control problem is posed and solved. The interest here is the explicitness of the results and the relation between the geometry of the drift along the free boundary of the problem and the principle of smooth fit. 0. Introduction We start with the two dimensional linear system in canonical controllable form x = y; x(0) = x (0:1) y = u; y(0) = y (0:2) where the control u is a nonnegative measurable function of time. We define v u (x; y) = Z 1 0 e \Gammat [u(t) + f(x(t); y(t))]dt; (0:3) where f is a convex C 2 function of polynomial growth. We set v(x; y) = inffv u (x; y): u(\Delta) 0g: (0:4) The problem we address in this paper is the determination of the so-called value function v of the variational problem (0.1)-(0.4). This problem is a prototype of a class of multi-dimensional singular control problems involving linear dynamics and linear cost; as such, this class of problems should be applicable in a wide setti..