On the control of stochastic systems

A general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the “iterative” method of the authors or its more recent adaptations for stochastic operator equations. It is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations; the method, therefore, is applicable to the control problem. Application is made to the state space equation $\text{x}\text{?}\text{= Ax + Bu + C,}\text{where}\text{A, B, C}$ are stochastic matrices corresponding to stochastic operators, i.e., involving randomly time varying elements, e.g., $\text{a}{}_{\mathrm{ij}}\text{(t, ω) ? A, t ? T, ω ? (Ω, F,μ)}$, a p.s. It is emphasized that the processes are arbitrary stochatic processes with known statistics. No assumption is made of Wiener or Markov behavior or of smallness of fluctuations and no closure approximations are necessary. The method differs in interesting aspects from Volterra series expansions used by Wiener and others and has advantages over the other methods. Because of recent progress in the solution of the nonlinear case, it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages.