In recent years, we have witnessed the appearance in the control literature of claims regarding the behavior of the trajectories of closed-loop systems, which are valid only for a specific set of initial conditions (ICs). Being these “trajectory-dependent” claims, a natural question that arises is whether these claims are robust, in some well-defined sense. Regarding Lyapunov stability claims, it is well-known that this property is equivalent to a form of continuity of solutions with respect to the ICs. For linear time-invariant (LTI) systems characterized by transfer matrices, the property of internal stability is widely adopted as a necessary condition to ensure robustness. However, to the best of our knowledge, this question has not been addressed for claims—different from stability—concerning the behavior of trajectories of nonlinear time-varying (NLTV) systems, which is the scenario in the aforementioned claims. The main objective of this note is to propose a framework for the characterization of robustness or fragility (with respect to ICs) of claims of this nature for NLTV systems.

中文翻译：

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对最近关于适用于特定初始条件的控制系统轨迹的说法的评论

近年来，我们在控制文献中目睹了关于闭环系统轨迹行为的声明的出现，这些声明仅对一组特定的初始条件 (IC) 有效。作为这些“依赖于轨迹的”声明，一个自然而然的问题是这些声明在某种定义明确的意义上是否可靠。关于Lyapunov 稳定性声明，众所周知，该属性等价于 IC 的解的连续性形式。对于以传递矩阵为特征的线性时不变（LTI）系统，内部稳定性的性质被广泛用作确保稳健性的必要条件。然而，据我们所知，对于非线性时变 (NLTV) 系统的轨迹行为的声明（与稳定性不同），这个问题尚未得到解决，这是上述声明中的场景。本说明的主要目的是提出一个框架，用于表征 NLTV 系统的这种性质的声明的稳健性或脆弱性（相对于 IC）。