SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2616-2638, January 2020.
The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds on the JSR. All results of this paper are presented for the general case of constrained switched systems, that is, systems for which the switching signal is constrained by an automaton.

中文翻译：

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使用平方和编程来验证切换系统的不稳定

SIAM控制与优化杂志，第58卷，第4期，第2616-2638页，2020年1月。
一组矩阵的联合光谱半径（JSR）表示该组矩阵的无限乘积的最大渐近增长率。这个数量出现在许多应用中，包括交换式和混合式系统的稳定性。用于这些系统稳定性分析的一种流行方法是使用凸优化工具搜索Lyapunov函数。我们研究此方法的双重表述，并利用这些双重程序为JSR开发新的分析工具。我们表明，这个凸问题的对偶搜索具有高渐近增长率的轨迹的职业度量。我们都展示了如何生成保证高渐近增长率的序列，以及如何检测可以在JSR上提供较低界限的情况。