Asymptotically-optimum-in-distribution control actions for a linear stochastic system with a quadratic functional (Q1284309)

The authors consider a controlled random process \(x(t)\in \mathbb{R}^n\) of the form \[ dx(t)= (Ax(t)+ Bu(t)) dt+ dw(t), \] where \(u= u(t)\) is an adapted random control action (strategy), \(A\) and \(B\) are some matrices. The strategy price is defined on \([0,T]\) as the functional \[ J_{x_{0,T}}(u)= \int^T_0(x_t' Cx_t+ u_t' Du_t) dt, \] \(0\) and \(D\) are positive definite matrices. The aim of the paper is to study the properties of the optimal strategies connected with the asymptotic distribution of the strategy price. For this purpose the stationary strategy \[ \overline u= \overline u(t)= -D^{-1} B'\Lambda x(t), \] is considered, where \(\Lambda\) is the solution to an appropriate matrix differential equation. The main result is: the stationary strategy \(\overline u\), optimal in the mean-value sense, is asymptotically optimal in distribution in the class of all possible (adaptive) strategies. It means that such a strategy provides the value of a goal functional, which will be, after normalization, in a stochastic sense, no more than the corresponding value for any other strategy.