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Strongly Regular Differential Variational Systems

机译：强正则微分变分系统

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A differential variational system is defined by an ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution of a finite-dimensional variational inequality containing the state variable of the system. This paper addresses two system-theoretic topics for such a nontraditional nonsmooth dynamical system; namely, (non-)Zenoness and local observability of a given state satisfying a blanket strong regularity condition. For the former topic, which is of contemporary interest in the study of hybrid systems, we extend the results in our previous paper, where we have studied Zeno states and switching times in a linear complementarity system (LCS). As a special case of the differential variational inequality (DVI), the LCS consists of a linear, time-invariant ODE and a linear complementarity problem. The extension to a nonlinear complementarity system (NCS) with analytic inputs turns out to be non-trivial as we need to use the Lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state. Further extension to a differential variational inequality is obtained via its equivalent Karush-Kuhn-Tucker formulation. For the second topic, which is classical in system theory, we use the non-Zenoness result and the recent results in a previous paper pertaining to the B-differentiability of the solution operator of a nonsmooth ODE to obtain a sufficient condition for the short-time local observability of a given strongly regular state of an NCS. Refined sufficient conditions and necessary conditions for local observability of the LCS satisfying the P-property are obtained
机译：微分变分系统由一个普通的微分方程（ODE）定义，该方程由代数变量参数化，该代数变量必须是包含系统状态变量的有限维变分不等式的解。本文讨论了这种非传统非光滑动力系统的两个系统理论主题。即，满足总体强规则性条件的给定状态的（非）Zenoness和局部可观察性。对于在混合系统研究中具有当代意义的前一个主题，我们将其结果扩展到之前的论文中，在该论文中我们研究了线性互补系统（LCS）中的芝诺状态和切换时间。作为差分变分不等式（DVI）的特例，LCS由线性，时不变的ODE和线性互补问题组成。带有分析输入的非线性互补系统（NCS）的扩展被证明是不平凡的，因为我们需要使用解析函数的Lie导数来获得给定状态附近的解轨迹的扩展。通过其等效的Karush-Kuhn-Tucker公式，可以进一步扩展到差分变分不等式。对于第二个主题，这是系统理论中的经典问题，我们使用非Zenoness结果以及先前论文中有关非光滑ODE解算子的B可微性的最新结果来获得充分的条件， NCS的给定强规则状态的时间局部可观察性。获得了满足P属性的LCS的局部观测的精炼充分条件和必要条件

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