Representation of linear PDEs with spatial integral terms as Partial Integral Equations
Representation of linear PDEs with spatial integral terms as Partial Integral Equations
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DOI：
10.23919/acc55779.2023.10156465
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发表时间：
2022-12
期刊：
影响因子：
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通讯作者：
Sachin Shivakumar;Amritam Das;M. Peet
中科院分区：
文献类型：
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作者：
Sachin Shivakumar;Amritam Das;M. Peet
在本文中，我们提出了一维线性偏微分方程（PDE）的偏积分方程（PIE）表示，其中 PDE 具有出现在动力学和边界条件中的空间积分项。 PIE 表示是通过执行变量更改来获得的，其中使用微积分基本定理将每个 PDE 状态替换为其最高的、明确定义的导数，以获得新的方程 (PIE)。我们证明了从 PDE 表示到 PIE 表示的转换可以用从 PDE 参数到 PIE 参数的显式映射来编写。最后，我们通过凸优化方法对偏微分方程进行稳定性分析，给出数值示例来演示 PIE 表示的应用。
In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.