A guaranteed control problem for a linear stochastic differential equation

Valeriy L Rozenberg

doi:10.15826/umj.2015.1.007

Abstract

A problem of guaranteed closed-loop control under incomplete information is considered for a linear stochastic differential equation (SDE) from the viewpoint of the method of open-loop control packages worked out earlier for the guidance of a linear control system of ordinary differential equations (ODEs) to a convex target set. The problem consists in designing a deterministic open-loop control providing (irrespective of a realized initial state from a given finite set) prescribed properties of the solution (being a random process) at a terminal point in time. It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problem for the SDE is reduced to an equivalent problem for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. Solvability conditions for the problems in question are written.

Highlights

The problem of constructing optimal strategies of guaranteed feedback control under conditions of uncertainty is one of the most important in mathematical control theory and its applications
Krasovskii’s school [1,2,3], the approach based on the so-called method of open-loop control packages originating from the technique of nonanticipating strategies from the theory of differential games [4] is applied to solving the guidance problem for a linear stochastic differential equation (SDE)
The method tested on the guidance problems under incomplete information for linear controlled systems of ordinary differential equations (ODEs) consists in reducing the problems of guaranteed control formulated in the class of closed-loop strategies to equivalent problems in the class of open-loop control packages

Summary

Introduction

The problem of constructing optimal strategies of guaranteed feedback control under conditions of uncertainty is one of the most important in mathematical control theory and its applications. The method tested on the guidance problems under incomplete information for linear controlled systems of ODEs consists in reducing the problems of guaranteed control formulated in the class of closed-loop strategies to equivalent problems in the class of open-loop control packages. The latter class contains the families of open-loop controls parameterized by admissible initial states and possessing the property of nonanticipation with respect to the dynamics of observations [5,6,7]. Note that the procedure reducing a problem for a linear SDE to the corresponding problem for the ODE, which is conceptually close to the one proposed in the paper, was used, in particular, in [10] for solving the problem of dynamic reconstruction of an unknown disturbance characterizing the level of random noise in a linear SDE on the base of measuring some realizations of the SDE’s phase vector

Problem statement

Reduction of the original problem

The method of open-loop control packages: a brief review of results for ODE

Properties of the statistical estimates

Findings

Illustrative example