发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出最优耗散二次化（ODQ）方法，联合优化稳定器增益和表示规范，为多项式系统生成谱范数最优的二次Lyapunov证书，显著扩大认证吸引域。
AI 中文摘要
多项式系统的吸引域（ROA）证书随着状态维数和阶数的增长而变得昂贵：直接平方和（SOS）公式需要组合增长的单项式基。二次化在二次系统的不变流形上精确表示多项式向量场，允许二次Lyapunov函数来认证ROA。对于固定的提升，稳定器增益塑造了流形外扩展和横向动力学，而表示规范改变矩阵表示但不改变向量场。两者都影响谱范数证书，然而先前的工作在优化规范之前通过可行性启发式固定增益。我们提出了最优耗散二次化（ODQ），针对固定的单项式提升、参考扩展、稳定器分解和Lyapunov权重$Q=I$，联合设计增益和规范。增益位于一个预定的紧致Hurwitz盒子内。在每个增益处，一个精确的半定规划在规范上全局最小化谱范数界。残差感知界产生一个经过认证的闭合Lyapunov子水平集，考虑了浮点Lyapunov残差。在我们陈述的假设下，理想化外部搜索的每个聚点都是盒子-Clarke平稳的。有限运行返回最佳独立验证的候选；不声称增益搜索的全局最优性。在一个平面五次系统上，优化增益将认证面积比匹配的零增益规范提高了$2.238$倍。在16个具有稳定器自由度的异构多项式系统上，ODQ优于固定增益提升基线。中继基准上的所有36次ODQ运行均完成，九个完全配对案例中的全部27次重复级别比较都倾向于ODQ，相对于具有固定二次Lyapunov函数的SOS基线，在固定方向代理和构建时间上均如此。与直接SOS方法的更广泛比较仍然好坏参半。
英文摘要
Region-of-attraction (ROA) certificates for polynomial systems become expensive as state dimension and degree grow: direct sum-of-squares (SOS) formulations require combinatorially growing monomial bases. Quadratization represents a polynomial vector field exactly on an invariant manifold of a quadratic system, allowing a quadratic Lyapunov function to certify the ROA. For a fixed lift, stabilizer gains shape the off-manifold extension and transverse dynamics, while representation gauges change the matrix representation but not the vector field. Both affect the spectral-norm certificate, yet prior work fixes the gain by a feasibility heuristic before optimizing the gauge. We formulate optimal dissipative quadratization (ODQ), jointly designing gains and gauges for a fixed monomial lift, reference extension, stabilizer factorization, and Lyapunov weight $Q=I$. Gains lie in a prescribed compact Hurwitz box. At each gain, an exact semidefinite program globally minimizes the spectral-norm bound over the gauge. Residual-aware bounds yield a certified closed Lyapunov sublevel set, accounting for the floating-point Lyapunov residual. Under our stated assumptions, every accumulation point of the idealized outer search is box-Clarke stationary. A finite run returns the best independently verified candidate; global optimality of the gain search is not claimed. On a planar quintic, optimizing the gain increases the certified area by a factor of $2.238$ over a matched zero-gain gauge. Across 16 heterogeneous polynomial systems with stabilizer freedom, ODQ improves on both fixed-gain lifted baselines. All 36 ODQ runs on the relay benchmark complete, and all 27 repeat-level comparisons across nine fully paired cases favor ODQ over an SOS baseline with a fixed quadratic Lyapunov function in both the fixed-direction proxy and construction time. Broader comparisons with direct SOS methods remain mixed.
Comments43 pages, 4 figures, 7 tables; includes appendices
