论文标题
$ L_2 $ - 使用线性PI不平等的线性2D PDE的分析
$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities
论文作者
论文摘要
在本文中，我们提出了一种新方法，用于估算由两个空间变量（使用半芬派编程）在两个空间变量中由第二阶线性偏微分方程（PDE）估算的系统的$ L_2 $结构。以前已经证明，对于任何此类PDE，都可以得出等效的部分积分方程（PIE）。这些馅饼是根据部分积分（PI）操作员在$ l_2 [ω] $中映射状态的表示，并且不含PDE中出现的边界和连续性约束。在本文中，我们将2D PIE表示形式扩展到在$ \ mathbb {r}^n $中包括输入和输出信号，从而在PDE和PIE的解决方案之间得出了一条bijective映射，以及必要的公式在两个表示之间转换。接下来，使用PI运算符的代数属性，我们证明，可以通过测试线性PI不平等的可行性（LPI）来验证$ L_2 $ - 派的上限，这是由PI操作员对PI操作员绘制$ \ Mathbb {r}^n \ tile l_2 [$ l_2 $ fim的阳性约束所定义的。最后，我们使用阳性矩阵将$ \ Mathbb {r}^n \ times l_2 [ω] $的正PI运算符参数化，从而使$ l_2 $ gain-gain lpi的可行性可以使用SemideFinite编程进行测试。我们在MATLAB工具箱Pietools中实现了此测试，并证明了这种方法允许在$ L_2 $ gain的PDE上的上限估算几乎没有保守主义。
In this paper, we present a new method for estimating the $L_2$-gain of systems governed by 2nd order linear Partial Differential Equations (PDEs) in two spatial variables, using semidefinite programming. It has previously been shown that, for any such PDE, an equivalent Partial Integral Equation (PIE) can be derived. These PIEs are expressed in terms of Partial Integral (PI) operators mapping states in $L_2[Ω]$, and are free of the boundary and continuity constraints appearing in PDEs. In this paper, we extend the 2D PIE representation to include input and output signals in $\mathbb{R}^n$, deriving a bijective map between solutions of the PDE and the PIE, along with the necessary formulae to convert between the two representations. Next, using the algebraic properties of PI operators, we prove that an upper bound on the $L_2$-gain of PIEs can be verified by testing feasibility of a Linear PI Inequality (LPI), defined by a positivity constraint on a PI operator mapping $\mathbb{R}^n\times L_2[Ω]$. Finally, we use positive matrices to parameterize a cone of positive PI operators on $\mathbb{R}^n\times L_2[Ω]$, allowing feasibility of the $L_2$-gain LPI to be tested using semidefinite programming. We implement this test in the MATLAB toolbox PIETOOLS, and demonstrate that this approach allows an upper bound on the $L_2$-gain of PDEs to be estimated with little conservatism.
