发表机构
Institute for Problems in Mechanical Engineering, Russian Academy of Sciences（俄罗斯科学院机械问题研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于对数仿射密度族的凸优化方法，通过最大化区域收缩裕度来设计密度函数，从而为非线性系统提供可计算的区域相体积收缩证书，并解决普通散度无法检测的收缩问题。
AI 中文摘要
本文研究了非线性系统的区域加权相体积收缩问题，采用从有限维对数仿射族中选取的正密度函数。密度散度对设计参数的仿射依赖性将密度选择转化为一个半无限凸优化问题，该问题在紧致区域上最大化一个经过认证的均匀收缩裕度。这为测试预先选定的密度提供了一种建设性替代方案，并确保在紧致参数约束下设计问题的适定性。有限活跃点和极小极大特征刻画了决定最优解的最坏情况状态，而严格的采样界允许从有限个不等式恢复连续统收缩保证。对于标量幂密度，可行性由精确的区间条件表征。相同的结构在离散时间中得以保留，并允许距离加权和共形几何解释。总体而言，该框架提供了可计算的证书，用于检测可能对普通散度隐藏的区域收缩。一个双参数极限环示例展示了当普通相体积和自然单基密度均失效时的这种效应。
英文摘要
Regional weighted phase-volume contraction for nonlinear systems is studied using positive densities selected from a finite-dimensional log-affine family. The affine dependence of the density divergence on the design parameters turns density selection into a semi-infinite convex optimization problem that maximizes a certified uniform contraction margin on a compact region. This provides a constructive alternative to testing a preselected density and ensures a well-posed design problem under compact parameter constraints. Finite active-point and minimax characterizations identify the worst-case states governing the optimum, while rigorous sampling bounds allow continuum contraction guarantees to be recovered from finitely many inequalities. For scalar power densities, feasibility is characterized by an exact interval condition. The same structure is preserved in discrete time and admits distance-weighted and conformal-geometric interpretations. Overall, the framework provides computable certificates for detecting regional contraction that may remain hidden from ordinary divergence. A two-parameter limit-cycle example demonstrates this effect when both ordinary phase volume and a natural single-basis density fail.
