$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

TL;DR

This work addresses bounding the $L_2$-gain from disturbances to outputs for linear 2D PDEs by recasting the PDEs as Partial Integral Equations (PIEs) and formulating a Linear PI Inequality (LPI) that certifies an upper bound $\gamma$ on the gain. The key idea is to use a bijection between PDEs and PIEs via operators $\mathcal{T}_0,\mathcal{T}_1$ and to parameterize positive PI operators $\mathcal{P}$ with positive matrices, enabling the LPI to be solved as an SDP (LMIs). This yields a practical, low-conservatism method for $L_2$-gain estimation implemented in the MATLAB toolbox PIETOOLS, demonstrated on 2D parabolic PDEs (e.g., KISS model) with comparisons to discretization-based bounds. The contributions include a full PDE-to-PIE conversion framework for inputs/outputs, a PSD-PI operator parameterization strategy, and an SDP-based LMI that verifies $\|z\|_{L_2} \leq \gamma \|w\|_{L_2}$ with provable guarantees. The approach offers scalable, provably valid bounds for a broad class of 2D PDEs, reducing conservatism and avoiding large finite-dimensional truncations.

Abstract

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.

Figures (8)

• Figure 1: Parameters $K_1$, $K_2$, $H_1$ and $H_2$ defining the mappings in Lemma \ref{['lem:vhat_to_v']} and in Corollary \ref{['cor:vhat_to_BC']}
• Figure 2: Parameters $T_0$ and $T_1$ defining the PI operators $\mathcal{T}_0=\mathcal{P}[T_0]$ and $\mathcal{T}_1=\text{M}[T_1]$ mapping the fundamental state back to the PDE state in Theorem \ref{['thm:Tmap']}
• Figure 3: Parameters $\hat{G}$ defining the adjoint $\mathcal{P}[\hat{G}]=\mathcal{P}^*[G]$ of the PI operator $\mathcal{P}[G]$ in Lem. \ref{['lem:adjoint']}
• Figure 4: Bounds on the $L_2$-gain of System \ref{['eq:Heat_eq']} computed using the LPI methodology, parameterizing $\mathcal{P}\in\Xi_d$ in Thm. \ref{['thm:KYP_PDE']} using monomials of degree at most $d=1$. Estimates of the gain computed through discretization are also shown, using a grid of $12\times 12$ uniformly distributed points.
• Figure 5: Parameters $\hat{G}$ defining the adjoint $\mathcal{P}[\hat{G}]=\mathcal{P}^*[G]$ of the PI operator $\mathcal{P}[G]$ in Lem. \ref{['lem:appx_adjoint']}

Theorems & Definitions (32)

• Definition 1: 011-PI Operators, $\Pi_{011}$
• Definition 2: 2D-PI Operators, $\Pi_{2D}$
• Definition 3: 0112-PI Operators, $\Pi_{0112}$
• Proposition 4
• Proposition 5
• Proposition 6
• Example 7
• Lemma 8
• proof
• Definition 9: Solution to the PDE