Optimal stability of linear stochastic systems in the presence of information constraints

Abstract

This paper is devoted to the problem of optimal stability of a semi-linear stochastic system with controlled parameters. Systems of this type are described by linear random differential equations with multiplicative perturbations, the matrices of which are generally nonlinear and control-dependent. The quality standard is a modification of the classical quadratic management quality standard. The task is to reduce the norm on the set of acceptable control operations. This formulation of the problem is interesting because it allows us to study a wide range of optimization issues of linear systems with multiplicative perturbations, including: optimization of system design parameters, problems of optimal stabilization in the presence of constraints on the coefficients of the linear regulator in the form of inequalities, problems of optimal stabilization of linear stochastic systems in the presence of information constraints. The main result of the work is the necessary conditions for optimizing the parameter vector in the problem of optimal stability of a semi-linear random system with controlled parameters. A gradienttype procedure is proposed for the synthesis of the optimal parameter stabilization vector. In addition, based on the optimal conditions obtained, an algorithm is created for the synthesis of suboptimal program control in the problem under consideration. The result of the proposed algorithm is a continuous multi-definition control, which gives the value of the criterion guaranteed to be no worse than the optimal stabilization vector. The resulting algorithm is simple and allows calculation in real time. The obtained results are applied to the problem of optimal stability with information constraints, where the necessary optimal conditions are also obtained and a procedure for the synthesis of Gradient-Type control is proposed. The use of the results obtained is illustrated by a model example.