发表机构
School of Electrical and Computer Engineering, Tel Aviv University（特拉维夫大学电气与计算机工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究线性系统时间最优控制中切换次数随时间的渐近增长，针对任意谱给出下界，并在飞机模型上验证。
AI 中文摘要
我们研究了在时间范围[0,T]上可控线性系统的时间最优控制，重点关注大T时的渐近切换密度。当系统矩阵仅具有实特征值时，众所周知，切换次数在T上一致有上界；当系统矩阵具有复特征值时，不存在这种一致有界性，切换次数通常随T增长。我们针对具有任意谱的系统矩阵刻画了这一增长，同时允许实特征值、复特征值以及非平凡的Jordan结构。如果主导模态是复的，则切换次数至少随T线性增长，其显式下界通过平均运动问题和Bohl-Weyl-Wintner公式表达。我们在线性化的飞机俯仰/高度模型上说明了该理论，显示出预测的渐近切换速率与数值计算的时间最优控制之间具有良好的一致性。
英文摘要
We study the time-optimal control of a controllable linear system on a time horizon [0,T], focusing on the asymptotic switching density for large T. When the system matrix has only real eigenvalues, it is well-known that the number of switches is upper bounded uniformly in T; when it has complex eigenvalues, no such uniform bound exists, and the switching count instead typically grows with T. We characterize this growth for a system matrix with an arbitrary spectrum, allowing simultaneously for real eigenvalues, complex eigenvalues, and a non-trivial Jordan structure. If the dominant mode is complex, the number of switches grows at least linearly in T, with an explicit lower bound expressed via the mean motion problem and the Bohl-Weyl-Wintner formula. We illustrate the theory on a linearized aircraft pitch/altitude model, showing close agreement between the predicted asymptotic switching rate and numerically computed time-optimal controls.
