利用半定规划获得全局吸引子的收敛外逼近
Converging outer approximations to global attractors using semidefinite programming
AI总结：
本文提出一种基于半定规划层级的方法，通过Borel测度空间的无限维LP及其对偶的平方和收紧，生成多项式动力系统全局吸引子的收敛外逼近，方法简便且有收敛保证，经数值算例验证。
摘要：
本文提出了一种获取连续时间和离散时间非线性动力系统全局吸引子的有保证外逼近的方法。该方法基于规模递增的半定规划问题层级，可保证收敛到全局吸引子。所采用的方法遵循既定的推理思路：我们首先通过Borel测度空间中的无限维线性规划问题（LP）来刻画全局吸引子。该LP的对偶问题定义在连续函数空间上，其可行解能给出全局吸引子的有保证外逼近。对于具有多项式动力学特性的系统，对偶LP的有限维平方和收紧层级可生成一系列全局吸引子的外逼近，且在体积差异趋于零的意义下具有收敛保证。该方法使用非常简便，完全基于凸优化。文中通过数值算例（代码可在线获取）对该方法进行了验证。
英文摘要：
This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.
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