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Exact Controllability of Discrete-Time Stochastic System with Multiplicative Noise

Juanjuan Xu, Huanshui Zhang

Abstract

This paper is concerned with the exact controllability of discrete-time stochastic system which is one of the basic problems of modern control theory. Though the exact controllability of continuous-time system governed by Ito stochastic differential equations has been well studied in S. Peng, Progress in Natural Science, 1994, the counterpart of the discrete-time case is still open due to the adaptiveness constraint of the controllers and the solvability challenging of stochastic difference equation with terminal value. The main contribution in this paper is to present both the Gramian matrix criterion and the Rank criterion for the exact controllability of discrete-time stochastic system. The novelty lies in the transformation of the forward stochastic difference equation into a novel backward one.

Exact Controllability of Discrete-Time Stochastic System with Multiplicative Noise

Abstract

This paper is concerned with the exact controllability of discrete-time stochastic system which is one of the basic problems of modern control theory. Though the exact controllability of continuous-time system governed by Ito stochastic differential equations has been well studied in S. Peng, Progress in Natural Science, 1994, the counterpart of the discrete-time case is still open due to the adaptiveness constraint of the controllers and the solvability challenging of stochastic difference equation with terminal value. The main contribution in this paper is to present both the Gramian matrix criterion and the Rank criterion for the exact controllability of discrete-time stochastic system. The novelty lies in the transformation of the forward stochastic difference equation into a novel backward one.
Paper Structure (9 sections, 12 theorems, 74 equations)

This paper contains 9 sections, 12 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Equivalent BSDE
  4. Exact Controllability
  5. Stochastic Systems with Output Measurements
  6. Stochastic Systems with Input Delay
  7. Stochastic Systems with State Delay
  8. Numerical Examples
  9. Conclusions

Key Result

Lemma 1

Under (a3) and the invertibility of the matrix $A-L\bar{A}$, FSDE (c1) is equivalently reformulated as the following BSDE: where $C\triangleq (A-L\bar{A})^{-1}, \bar{C}\triangleq-(A-L\bar{A})^{-1}L$ and $D\triangleq-(A-L\bar{A})^{-1}F$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 3
  • Theorem 2
  • ...and 15 more