General LQG Homing Problems in One Dimension

Optimal control problems for one-dimensional diffusion processes in the interval ( ) are considered. The aim is either to maximize or to minimize the time spent by the controlled processes in ( ). Exact solutions are obtained when the processes are symmetrical with respect to . Approximate solutions are derived in the asymmetrical case. The one-barrier cases are also treated. Examples are presented. 1. Introduction Let be a one-dimensional controlled diffusion process defined by the stochastic differential equation where is the control variable, and are Borel measurable functions, and are constants, and is a standard Brownian motion. The set of admissible controls consists of Borel measurable functions. Remark 1.1. We assume that the solution of (1.1) exists for all and is weakly unique. We define the first-passage time where . We want to find the control that minimizes the expected value of the cost function where and are constants. Notice that if is negative, then the optimizer wants to maximize the survival time of the controlled process in the interval , taking the quadratic control costs into account. In general, there is a maximal value that the parameter can take. Otherwise, the expected reward becomes infinite. When the relation holds for some positive constant , using a theorem in Whittle [1, p. 289], we can express the value function in terms of a mathematical expectation for the uncontrolled process obtained by setting in (1.1). Actually, for the result to hold, ultimate entry of the uncontrolled process into the stopping set must be certain, which is not a restrictive condition in the case of one-dimensional diffusion processes considered in finite intervals. In practice, the theorem in Whittle [1] gives a transformation that enables us to linearize the differential equation satisfied by the function . In Lefebvre [2], using symmetry, the author was able to obtain an explicit and exact expression for the optimal control when is a one-dimensional controlled standard Brownian motion process (so that and ), and . Notice that the relation in (1.4) does not hold in that case. The author assumed that the parameter in the cost function is negative, and he found the maximal value that this parameter can take. Previously, Lefebvre [3] had computed the value of when , but with the cost function rather than the function defined above. We cannot appeal to the theorem in Whittle [1] in that case either. However, the author expressed the function in terms of a mathematical expectation for an uncontrolled geometric Brownian motion. In Section 2, we will