Manuel De la Sen

SUFFICIENCY-TYPE CONDITIONS FOR A TYPE OF STRICTLY DECREASING SOLUTIONS OF LINEAR CONTINUOUS-TIME DIFFERENTIAL SYSTEMS WITH BOUNDED POINT TIME-VARYING DELAYS

Introduction

Sufficiency-type conditions for a type of strictly decreasing solutions of linear continuous-time differential systems with bounded point time-varying delays. Sufficiency conditions for strictly decreasing solutions in linear differential systems with time-varying, piecewise bounded delays. No prior delay function knowledge needed.

Abstract

This paper investigates sufficiency-type conditions for strictly decreasing solutions of linear time-delay differential systems subject to a finite number of time-varying bounded point delays. The delay functions are not required to be time-differentiable nor even continuous but simply piecewise bounded continuous. It is not also required for the delay functions at any time instant to be upper-bounded. It is not necessary to have the knowledge of either the delay functions or their lower and upper bounds. It is proved that the supremum of any vector norm of the solution trajectory on consecutive time intervals of finite lengths is strictly decreasing under either stability conditions on the matrix which describes the delay-free dynamics, or on the one which describes the zero-delay auxiliary system, provided in both cases the contribution of the delayed dynamics is sufficiently small related to the convergence abscissas of the above matrices. Received: March 6, 2024Accepted: April 22, 2024

Review

This paper presents a rigorous investigation into sufficiency-type conditions for ensuring strictly decreasing solutions in linear continuous-time differential systems affected by a finite number of time-varying point delays. The problem of characterizing solution decay is fundamental in the analysis of time-delay systems, with broad implications for stability, control, and performance in various scientific and engineering applications. The study effectively targets a crucial aspect of system behavior under delay, seeking to establish general conditions that guarantee a specific, desirable form of solution trajectory. A significant strength of this work lies in its remarkably relaxed assumptions regarding the nature of the delay functions. The authors advance the field by demonstrating that their results hold even when delay functions are not time-differentiable or continuously differentiable, requiring only piecewise bounded continuity. Furthermore, the analysis impressively bypasses the need for delay functions to be upper-bounded at any instant, and crucially, does not necessitate prior knowledge of the delay functions themselves or their specific bounds. The core contribution is the derivation of conditions under which the supremum of any vector norm of the solution trajectory on consecutive finite time intervals is strictly decreasing, contingent upon stability conditions of either the delay-free dynamics matrix or an auxiliary zero-delay system, provided the contribution of delayed dynamics is adequately small. The presented research offers valuable theoretical advancements, extending the applicability of strictly decreasing solution analysis to a much broader class of time-delay systems where detailed knowledge or restrictive regularity conditions on delays are impractical or unavailable. While the proviso that the delayed dynamics' contribution remains "sufficiently small" is a common constraint in such analyses, it appropriately delineates the domain of applicability. This paper contributes meaningfully to the robust stability theory of delay differential equations, providing a robust framework for assessing and ensuring desirable transient behaviors under highly generalized delay characteristics, making it a valuable addition to the literature.

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