Application of a Novel Model Reduction Technique to the Assessment of Boundedness/Stability of Some Delay Time-Varying Vector Nonlinear Systems

Abstract

Assessing the boundedness and stability of vector nonlinear systems with variable delays and coefficients remains a challenging problem with broad applications in science and engineering. Existing methods tend to produce overly conservative criteria that offer limited practical value and often fail to explicitly characterize the temporal evolution of solution norms. This paper presents a novel framework for evaluating the evolution of solution norms in such systems. This approach constructs scalar counterparts for the original vector equations. We prove that the solutions to these scalar nonlinear equations, which also include delays and variable coefficients, provide upper bounds for the norms of the original solutions, if the history functions for both equations are properly matched. This reduction enables the evaluation of the boundedness and stability of vector systems through the analysis of the dynamics of their scalar counterparts, which can be performed via straightforward simulations or simplified analytical reasoning. Consequently, we introduce new criteria for boundedness and stability and estimate the radii of the balls containing history functions that yield bounded or stable solutions for the original vector systems. Finally, we validated our inferences through representative simulations that also assessed the accuracy of the proposed approach.