SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 400-414, February 2024.
Abstract. We prove that, for every discrete-time linear switching system in two complex variables and with finitely many switching states, either the system is Lyapunov stable or there exists a trajectory which escapes to infinity with at least linear speed. We also give a checkable algebraic criterion to distinguish these two cases. This dichotomy was previously known to hold for systems in two real variables but is known to be false in higher dimensions and for systems with infinitely many switching states.

中文翻译：

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二维离散时间线性开关系统的稳定性二分法

SIAM 控制与优化杂志，第 62 卷，第 1 期，第 400-414 页，2024 年 2 月。
摘要。我们证明，对于每个具有两个复变量且具有有限多个切换状态的离散时间线性切换系统，该系统要么是李雅普诺夫稳定，要么存在一条至少以线速度逃逸到无穷大的轨迹。我们还给出了一个可检查的代数准则来区分这两种情况。以前已知这种二分法适用于两个实变量的系统，但已知在更高维度和具有无限多个切换状态的系统中是错误的。