On Optimal Stochastic Control of Discrete-Time Systems in Hilbert Space

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Abstract
A general system described by a linear difference equation in a Hilbert space is considered. Three types of disturbances, control-dependent noise, state-dependent noise and purely additive noise, are taken into account. The cost function is assumed to be quadratic. The existence of an optimal stationary strategy and the uniqueness of the stationary measure related to this strategy are proved. Special attention is paid to the related Riccati operator difference equation and the asymptotic behavior of the solution of such an equation is investigated. Under certain assumptions, the existence and uniqueness of the solution of the algebraic Riccati equation are proved, too.

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