线性厄米矩阵值动力系统的最优控制与耗散性
Optimal Control and Dissipativity of Linear Hermitian Matrix-Valued Dynamical Systems
AI总结：
针对状态为厄米矩阵的线性脉冲系统，提出统一框架解决最优控制、最优转向、耗散分析和零和微分博弈，基于关键恒等式、扩展Schur补引理和Schur内积分解，给出时变条件下的充分必要条件、LMI表征和显式最优策略。
摘要：
我们为线性脉冲系统开发了一个统一框架，用于线性成本最优控制、有限时间最优转向、耗散性分析和零和微分博弈，其状态是$\mathbb{H}^{n+m}_{\succeq0}$中演化的厄米矩阵，该类包含连续时间和离散时间线性系统以及切换系统作为退化情况，并包括线性（随机）混合系统的二阶矩动力学。整个理论基于三个工具：一个将成本、轨迹和对偶变量联系起来的单一\emph{关键恒等式}、一个扩展Schur补引理和一个Schur内积分解，它们同样应用于流积分和每次跳跃。这些为每个问题类在时变假设下（无需时间不变性或周期性）在有限和无限时域上提供了结构统一的充分必要条件、对偶线性矩阵不等式（LMI）表征和显式最优策略，以及对于允许因果驻留时间策略的问题的因果驻留时间策略。
英文摘要：
We develop a unified framework for linear-cost optimal control, finite-time optimal steering, dissipativity analysis, and zero-sum differential games for linear impulsive systems whose state is a Hermitian matrix evolving in $\mathbb{H}^{n+m}_{\succeq0}$, a class that encompasses continuous- and discrete-time linear systems and switched systems as degenerate cases, and includes the second-order moment dynamics of linear (stochastic) hybrid systems. The entire theory rests on three tools: a single \emph{key identity} relating cost, trajectory, and a dual variable, an Extended Schur complement lemma, and a Schur inner-product decomposition, applied identically to the flow integral and to each jump. These yield structurally uniform sufficient and necessary conditions, dual linear matrix inequality (LMI) characterizations, and explicit optimal policies for every problem class, on both finite and infinite horizons under time-varying assumptions (without time invariance or periodicity), together with causal dwell-time policies for the problems that admit them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
