Abstract
The estimation and control of linear stochastic systems with delays in the state, control, and observations are studied. First, the deterministic optimal control problem with quadratic cost over an infinite time interval is examined. Using an extended notion of stabilizability, the existence and characterization of the optimal control law is obtained. Using the additional assumption of detectability the optimal closed-loop system is shown to be $L^2 $-stable. Next, the stochastic filtering problem is studied. A new version of the duality relations between optimal control and filtering is developed. This combined with a suitable notion of detectability, is exploited to show convergence of the filter gains. Under the additional assumption of stabilizability, the optimal stationary filter is shown to be $L^2 $-stable. Finally, by putting together the optimal control and filtering results, a stable constant stochastic control law is obtained for the linear-quadratic-Gaussian problem.
Published Version
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