State bounding for positive coupled differential - difference equations with bounded disturbances

TL;DR

This paper introduces a new method for computing state bounds in positive coupled differential-difference equations with delays and disturbances, using nonnegative matrices and optimization, extending to systems with bounded disturbances.

Contribution

The paper presents the first approach for state bounding of positive CDDEs with bounded disturbances, combining finite-time bounds, comparison methods, and state transformations.

Findings

Componentwise upper bounds are derived for systems without disturbances.

Ultimate bounds and invariant sets are established for systems with disturbances.

Numerical example demonstrates the effectiveness of the proposed method.

Abstract

In this paper, the problem of finding state bounds is considered, for the first time, for a class of positive time-delay coupled differential-difference equations (CDDEs) with bounded disturbances. First, we present a novel method, which is based on nonnegative matrices and optimization techniques, for computing a like-exponential componentwise upper bound of the state vector of the CDDEs without disturbances. The main idea is to establish bounds of the state vector on finite-time intervals and then, by using the solution comparison method and the linearity of the system, extend to infinite time horizon. Next, by using state transformations, we extend the obtained results to a class of CDDEs with bounded disturbances. As a result, componentwise upper bounds, ultimate bounds and invariant set of the perturbed system are obtained. The feasibility of obtained results are illustrated through a numerical example.