| In this thesis we consider the problem of minimizing a quadratic functional for a discrete-time, linear stochastic model with multiplicative noise, on a standard probability space, in an infinite time horizon. We show that the necessary and sufficient conditions for the existence of the optimal control for the stochastic model can be formulated as matrix inequalities in frequency-domain. Furthermore, we show that if the optimal control exists, then a certain Lyapunov equation must have a solution. The optimal control is obtained by solving a deterministic discrete-time, linear-quadratic optimal control problem whose objective functional depends on the solution to the Lyapunov equation. Moreover, we show that under certain conditions, solvability of the Lyapunov equation is guaranteed. We also show that, if the frequency-domain inequalities are strict, then the solution is unique up to equivalence.;Keywords: Stochastic Models, Optimal Control Theory, Frequency-domain Inequality, Frequency Lemma, Z-transform, Linear Quadratic Problem, Kalman-Yakubovich Lemma, Kalman-Szego Lemma, Lyapunov equations, Sylvester equations, Mean-square stability. |
