Stabilization of Integral Difference Equations by solving a Corona problem

TL;DR

This paper introduces a novel stabilizing control law for vector systems with integral difference equations, using solutions to a Corona problem to ensure stability under complex delay conditions.

Contribution

It extends IDE stabilization methods to systems with multiple delays and no commensurability constraints by solving a convolution equation via a Corona problem approach.

Findings

Controller kernels satisfy a convolution equation from the Corona problem.

Existence of solutions is guaranteed under spectral stabilizability.

Numerical solutions are obtained through a least-square procedure.

Abstract

This paper proposes a stabilizing state-feedback control law for vector-valued state systems with a scalar control input, governed by a general class of integral difference equations that incorporate both pointwise and distributed input delays. The proposed controller is expressed through integral operators acting on the state and input histories over a finite time horizon. Closed-loop stability is established by characterizing the controller kernels as solutions to a convolution equation arising from a Corona problem. The existence of such solutions is ensured under a suitable spectral stabilizability condition, and a least-square procedure is implemented to find them numerically. The approach extends existing IDE stabilization results to more general settings, allowing for arbitrary numbers of pointwise delays affecting both the state and input, without requiring commensurability assumptions.