# On linear-quadratic optimal control of implicit difference equations

## TL;DR

This paper studies linear-quadratic optimal control problems for implicit difference equations, deriving new conditions for feasibility and solution existence using spectral analysis of matrix pencils, extending continuous-time results.

## Contribution

It introduces a discrete-time Lur'e equation and a Kalman-Yakubovich-Popov inequality for implicit difference equations, providing weaker assumptions than traditional algebraic Riccati approaches.

## Key findings

- Feasibility conditions for the optimal control problem.
- Existence and uniqueness criteria for optimal controls.
- Spectral characterization of solvability via palindromic matrix pencils.

## Abstract

In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyze a discrete-time Lur'e equation and a corresponding Kalman-Yakubovich-Popov inequality. We show that solvability of the Kalman-Yakubovich-Popov inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

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Source: https://tomesphere.com/paper/1703.01217