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首页> 外文期刊>International Journal of Partial Differential Equations >Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay
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Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay

机译：具有状态依赖时滞的中立型微分方程解的存在性和近似可控性

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This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.
机译：本文分为两部分。在第一部分中，我们研究了具有状态相关延迟和非瞬时脉冲的二阶中立型偏微分方程。通过非紧致性的Hausdorff测度和Darbo Sadovskii不动点定理，研究了温和溶液存在和唯一的条件。因此，我们无需在关联的运算符族上假设紧凑性假设。针对希尔伯特空间中相应线性系统的近似可控制性，研究了中性二阶系统的近似可控制性条件。一个简单的范围条件用于证明近似可控性。因此，我们不再需要假设作者使用的可控性算子的可逆性（Balachandran和Park，2003年），如果关联的半群是紧凑的，则它不会存在于无限维空间中。我们的方法还消除了检查作者（Dauer和Mahmudov，2002年）中使用的可控性Gramian算子的可逆性和相关极限条件的需要，而这实际上很难验证和应用。提供示例来说明所提出的理论。

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