[论文解读] Bilateral Solution Bounds and Successive Estimation of Boundedness and Stability Regions for Vector Delay Nonlinear Time-Varying Systems

Mark A. Pinsky|arXiv (Cornell University)|Jan 9, 2026

Stability and Control of Uncertain Systems被引用 0

一句话总结

本文提出一种逐次逼近方案，用于估计带变量延迟和系数的向量非线性时变系统的解范数，给出双边界且具有可扩展性的稳定性/有界区间。

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.

研究动机与目标

• 为带变量延迟和系数的向量非线性系统的稳定性与有界性分析提供更少保守性的需求动机。
• 通过逐次迭代及残差范数估计，建立近似原始解的方案。
• 构造一个标量非线性延迟方程，其解界定残差范数演化的边界。
• 获得原始解范数的双边界，以推导有效的有界性和稳定性判据。
• 证明该方法能够估计相关区域并展现迭代收敛性。

提出的方法

• 为原始向量延迟非线性时变系统提出逐次逼近框架。
• 估计随每一步近似而来的残差范数。
• 推导一个标量非线性延迟方程，其解界定残差范数演化的边界。
• 利用标量界限获得原始解范数的双边界。
• 将双边界应用于建立有界性和稳定性判据及区域估计。
• 通过仿真验证该方法收敛于参考边界的趋势。

实验结果

研究问题

• RQ1 如何对向量延迟非线性时变系统的残差范数演化进行有效界定？
• RQ2 一个标量非线性延迟方程是否能为残差范数演化提供可靠的上界？
• RQ3 对解范数的双边界是否能给出实际的有界性和稳定区域估计？
• RQ4 迭代近似到参考区域边界需要多快的收敛？

主要发现

• 构建了一个标量非线性延迟方程，其解界定残差范数演化的边界。
• 获得了原始解范数的双边界，从而实现了稳定的有界性判据。
• 该方法给出向量延迟非线性时变系统的有界性与稳定性区域的估计。
• 仿真表明提出的近似在迭代增加时迅速趋近区域边界。
• 当初始函数位于区域内时，双边界逐步收敛彼此以及收敛到参考解的范数。