A New State-Space Representation of Lyapunov Stability for Coupled PDEs and Scalable Stability Analysis in the SOS Framework - with Lemma Proofs
A New State-Space Representation of Lyapunov Stability for Coupled PDEs and Scalable Stability Analysis in the SOS Framework - with Lemma Proofs
复制标题
耦合偏微分方程 Lyapunov 稳定性的新状态空间表示和 SOS 框架中的可扩展稳定性分析 - 带引理证明
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发表时间：
2018
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通讯作者：
M. Peet
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作者：
M. Peet
在本文中，我们提出了一个框架的耦合线性偏微分方程系统的稳定性分析。本文所考虑的偏微分方程组包括抛物型、椭圆型和双曲型方程组，它们具有Dirichelet、Neuman和混合边界条件。本文中的结果适用于系统与一个单一的空间变量，并假设存在和连续的解决方案，除非在这种情况下，存在和连续性可以推断从存在的李雅普诺夫函数。我们的方法是基于一个新的概念，PDE系统的状态，使我们能够直接表示为线性算子不等式的李雅普诺夫函数的衍生物，并允许任何类型的适当适定的边界条件。这种方法避免了对分部积分、间隔函数或类似数学分支的需要。由此产生的算法在Matlab中实现，在几个激励性的例子中进行测试，代码已经发布在网上。数值试验表明，该方法有很少或没有保守性的一大类系统。
In this paper, we present a framework for Stability Analysis of Systems of Coupled Linear Partial-Differential Equations. The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as a Linear Operator Inequality directly on $L_2$ and allows for any type of suitably well-posed boundary conditions. This approach obviates the need for integration by parts, spacing functions or similar mathematical encumbrances. The resulting algorithms are implemented in Matlab, tested on several motivating examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems.