Nearly Optimal State Feedback Controls for Stochastic Systems with Wideband Noise Disturbances

Abstract

Much of optimal stochastic control theory is concerned with diffusion models. Such models are often only idealizations (or limits in an appropriate sense) of the actual physical process, which might be driven by a wide band-width (not white) process or be a discrete parameter system with correlated driving noises. Optimal or nearly optimal controls, derived for the diffusion models, would not normally be useful, or even of much interest, if they were not also “nearly optimal” for the physical system that the diffusion approximates. It turns out that, under quite broad conditions, the “nearly optimal” controls for the diffusions do have this desired robustness property and are “nearly optimal” for the physical (say wideband noise driven) process, even when compared to controls that can depend on all the (past) driving noise. We treat the problem over a finite time interval, as well as the average cost per unit time problem. Extensions to discrete parameter systems, and to systems stopped on first exit from a bounded domain, are also discussed. Weak convergence methods provide the appropriate analytical tools.

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