Successive approximations to an optimal control of quasilinear stochastic difference equations (Q2735903)

Many processes in automatic regulation, physics, mechanics, etc. can be simulated by stochastic difference equations. One of the problems of the theory of optimal control lies in the following: It is necessary to find an admissible control such that the performance functional is at its minimum. This control is called the optimal control. If the optimal control does not exist or if it is difficult to find then it arises the problem of constructing successive approximations to the optimal control. In this paper the problem of the optimal control of the system NEWLINE\[NEWLINE\begin{multlined} x(i+1)= \eta(i+l)+ \sum_{j=0}^i K(i,j)x(j)+ \sum_{j=0}^i a(i,j)u(j)+ \varepsilon \Biggl[\Phi (i+1, x_{i+1})+ \sum_{j=0}^i b(i,j,x_j)\xi(j)\Biggr],\\ i=0,\dots,N-1,\;x_0= \varphi_0 \end{multlined}NEWLINE\]NEWLINE with the quadratic performance functional NEWLINE\[NEWLINEJ(u)={\mathbf E} \Biggl[x'(N) Fx(N)+ \sum_{j=0}^{N-1} u'(j) G(j) u(j)\Biggr]NEWLINE\]NEWLINE is considered. An algorithm for constructing successive approximations to the optimal control is described.