论文标题
轻度溶液，常数公式的变化以及延迟微分方程的线性稳定性
Mild solutions, variation of constants formula, and linearized stability for delay differential equations
论文作者
论文摘要
普通微分方程（ODES）常数变化的方法和公式是分析平衡附近颂歌的动力学的基本工具。很自然地期望这种公式适用于延迟微分方程（DDES），但是，众所周知，DDE的公式存在概念上的困难。在这里，我们通过引入\ textit {温和解决方案}的概念来讨论DDES常数公式的变化，该概念是具有不连续历史函数的初始条件下的解决方案。然后将\ textIt {主基本矩阵解}定义为矩阵值的轻度解，我们通过此功能获得了常数公式的变化。这也是在Volterra卷积积分方程的框架中获得的，但是这里的处理本身就可以理解。我们还应用该公式来显示线性化稳定性的原理和DDES的Poincaré-Lyapunov定理，我们不需要假设溶液的唯一性。
The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a \textit{mild solution}, which is a solution under an initial condition having a discontinuous history function. Then the \textit{principal fundamental matrix solution} is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré-Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.
