标题： 耦合线性多变量偏微分方程的状态空间表示及基于SDP的稳定性分析 显示英文标题

标题： A State-Space Representation of Coupled Linear Multivariate PDEs and Stability Analysis using SDP

摘要： 在时间和空间中演化的物理过程通常使用偏微分方程（PDE）进行建模。最近，已经展示了如何通过等效的偏积分方程（PIE）表示来更方便地进行单个空间变量耦合PDE的稳定性分析和控制。这种PIE表示的构建基于空间微分算子$\partial_s^{d}$在由边界条件定义的域上的逆的解析表达式。在本文中，我们展示如何通过将域表示为提升的单变量域的交集，将这种单变量表示递归地扩展到多个空间变量。具体来说，我们证明如果每个单变量域是适定的，那么存在一个易于验证的一致性条件，该条件是多变量空间微分算子$D^\alpha=\partial_{s_1}^{\alpha_1}\cdots\partial_{s_N}^{\alpha_N}$在PDE域上存在逆的必要且充分条件。此外，我们证明这个逆是通过多项式半可分离核定义的偏积分（PI）算子的$*$-代数中的一个元素。基于这个算子代数，我们证明任何适当适定的线性多变量PDE的演化都可以由一个PIE描述，该PIE由PI代数的元素参数化。然后，提出了一种用于PDE稳定性的凸计算测试，使用正PI算子的正矩阵参数化，并提供了软件（PIETOOLS），该软件自动化了此类PDE的表示和稳定性分析过程。该软件用于分析二维热方程、波动方程和板方程的稳定性，得到了衰减率的准确界限。 显示英文摘要

摘要： Physical processes evolving in both time and space are often modeled using Partial Differential Equations (PDEs). Recently, it has been shown how stability analysis and control of coupled PDEs in a single spatial variable can be more conveniently performed using an equivalent Partial Integral Equation (PIE) representation. The construction of this PIE representation is based on an analytic expression for the inverse of the spatial differential operator, $\partial_s^{d}$, on the domain defined by boundary conditions. In this paper, we show how this univariate representation may be extended inductively to multiple spatial variables by representing the domain as the intersection of lifted univariate domains. Specifically, we show that if each univariate domain is well-posed, then there exists a readily verified consistency condition which is necessary and sufficient for existence of an inverse to the multivariate spatial differential operator, $D^\alpha=\partial_{s_1}^{\alpha_1}\cdots\partial_{s_N}^{\alpha_N}$, on the PDE domain. Furthermore, we show that this inverse is an element of a $*$-algebra of Partial Integral (PI) operators defined by polynomial semi-separable kernels. Based on this operator algebra, we show that the evolution of any suitably well-posed linear multivariate PDE may be described by a PIE, parameterized by elements of the PI algebra. A convex computational test for PDE stability is then proposed using a positive matrix parameterization of positive PI operators, and software (PIETOOLS) is provided which automates the process of representation and stability analysis of such PDEs. This software is used to analyze stability of 2D heat, wave, and plate equations, obtaining accurate bounds on the rate of decay.