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Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control

Yue Sun, Xianping Wu, Xun Li

Abstract

In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvability and the solution of the generalized Riccati equation is given. On the other hand, when the system is merely open-loop solvable, we have found the equivalent existence form of the optimal solution by perturbation method, which is said to be a weak closed-loop solution. Moreover, it obtains that there is an open-loop optimal control with a linear feedback form of the state. The essential technique is to solve the forward and backward difference equations by iteration. An example sheds light on the theoretical results established.

Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control

Abstract

In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvability and the solution of the generalized Riccati equation is given. On the other hand, when the system is merely open-loop solvable, we have found the equivalent existence form of the optimal solution by perturbation method, which is said to be a weak closed-loop solution. Moreover, it obtains that there is an open-loop optimal control with a linear feedback form of the state. The essential technique is to solve the forward and backward difference equations by iteration. An example sheds light on the theoretical results established.

Paper Structure

This paper contains 6 sections, 10 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Open-Loop Solvability by Perturbation Method
  4. Weak Closed-Loop Solvability of Problem (LQ)
  5. Numerical Example
  6. Conclusion

Key Result

Lemma 1

Under Assumption A1-A2, if Problem (LQ) is open-loop solvable, then $u$ subject to (1) satisfies the equilibrium condition where the co-state satisfies the following backward equation with the terminal value $\lambda_{N-1}=Hx_{N}$ and

Theorems & Definitions (27)

  • Example 1
  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 17 more