AI 中文总结
研究无限维脉冲系统在巴拿赫和希尔伯特空间上的一致指数稳定性分析，结合混合系统与无限维系统思想得出基于算子的稳定性条件，给出充要条件，应用于线性切换系统，通过理论和数值示例说明，尤其用于时滞系统采样数据控制。
AI 中文摘要
我们考虑在巴拿赫或希尔伯特空间上定义的无限维脉冲系统的一致指数稳定性分析，其流由固定的\(C_0\)-半群生成器控制，跳跃发生在规定的时间序列。虽然流和跳跃映射本身是时不变的，但时间触发的脉冲使传播子成为真正的时变演化族，这是分析困难的根源。我们结合混合系统理论和无限维系统的思想，得出基于算子的稳定性条件，可通过凸规划进行解析或数值检验。在固定脉冲时间序列以及任意、恒定、最小和范围驻留时间的情况下，使用非强制和强制李雅普诺夫泛函获得了巴拿赫空间上脉冲系统一致指数稳定性的充要条件。然后将其中一些结果应用于希尔伯特空间上的系统和二次李雅普诺夫泛函。作为应用，线性切换系统被证明是一个精确的特殊情况：重新表述为具有单位范数选择器跳跃的脉冲系统，它们在巴拿赫和希尔伯特空间上继承了非强制和时钟相关的驻留时间稳定性条件。给出了理论和数值示例进行说明，特别是在时滞系统的采样数据控制方面。
英文摘要
We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are themselves time-invariant, the time-triggered impulses render the propagator a genuinely time-varying evolution family, which is the source of the analysis difficulty addressed here. We combine ideas from hybrid systems theory and infinite-dimensional systems to produce operator-based stability conditions, which can be analytically or numerically checked via convex programming. Necessary and sufficient conditions for the uniform exponential stability of impulsive systems on Banach spaces are obtained in the context of a fixed impulse-times sequence but also of arbitrary, constant, minimum, and range dwell-times using both non-coercive and coercive Lyapunov functionals. Some of those results are then adapted to systems on a Hilbert space and quadratic Lyapunov functionals. As an application, linear switched systems are shown to be an exact special case: reformulated as impulsive systems with unit-norm selector jumps, they inherit non-coercive and clock-dependent dwell-time stability conditions on both Banach and Hilbert spaces. Theoretical and numerical examples are given for illustration, notably on the sampled-data control of time-delay systems.
Comments80 pages
