Bilateral Solution Bounds and Successive Estimation of Boundedness and Stability Regions for Vector Delay Nonlinear Time-Varying Systems

TL;DR

This paper introduces a novel method for estimating the boundedness and stability regions of vector nonlinear systems with variable delays, improving accuracy and reducing conservatism in stability analysis.

Contribution

It develops a successive approximation scheme and residual norm estimates to construct bilateral bounds and stability criteria for complex delay systems.

Findings

Bounds rapidly approach reference boundaries with iterations

Bilateral bounds converge as initial conditions remain within regions

Proposed method effectively estimates stability regions

Abstract

Stability and boundedness analysis for vector nonlinear systems with variable delays and coefficients remains challenging due to the conservatism of existing methods. Moreover, estimates of the transient behavior of solution norms remain insufficiently developed. This paper presents an approach to estimate the temporal evolution of solution norms and applies it to the analysis of boundedness and stability of vector nonlinear systems with variable delays and coefficients. The method is based on a novel scheme for successive approximations of the original solutions, complemented by the estimates of the corresponding residual norms. This leads to the construction of a scalar nonlinear delay equation whose solutions provide upper bounds for the evolution of residual norms. As a result, bilateral bounds on the original solution norms are obtained, yielding effective boundedness and stability criteria and enabling estimation of the associated regions. Simulations demonstrate that the proposed approximations rapidly approach the reference boundaries of the regions of interest as the iteration count increases. Moreover, the bilateral bounds progressively approach each other and the norm of the reference solution when the initial function remains within the considered regions.