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

TL;DR

This paper investigates the relationships between open-loop, closed-loop, and weak closed-loop solvability in discrete-time linear-quadratic control, providing new characterizations and solution forms, especially when only open-loop solvability is present.

Contribution

It establishes the equivalence between closed-loop solvability and generalized Riccati equation solutions, and introduces a weak closed-loop solution framework for open-loop solvable systems.

Findings

Closed-loop solvability is equivalent to the solution of a generalized Riccati equation.

Weak closed-loop solutions exist for systems only open-loop solvable.

An example illustrates the theoretical results.

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.