Nearly Optimal Singular Controls for Wideband Noise Driven Systems

摘要

Singular control problems with diffusion or Weiner process systems have been occuring with increasing frequency as models of a wide variety of applications; e.g., storage, inventory, finite fuel, consumption and investment, limits of impulsive control problems, etc. Here, the increment of the control force is not of the usual form u(t)dt, but is the differential of a non-decreasing and suitably adapted process. The models used (Wiener or diffusion processes) are only approximations in some sense to some 'physical' process - perhaps a 'wideband' noise driven system or a suitably scaled discrete parameter process. The optimal controls for these 'physical' processes are usually nearly impossible to obtain. Thus, it is of considerable interest to know whether the optimal (or delta-optimal control for the diffusion model is 'nearly' optimum when applied to the physical problem, when compared to the optimal or delta optimal control for the latter problem. This is true, under broad conditions. The discounted and average cost per unit time problems are treated. The main methods are those of weak convergence theory.