[Paper Review] A PIE Representation of Coupled 2D PDEs and Stability Analysis using LPIs

Declan S. Jagt, Matthew M. Peet|arXiv (Cornell University)|Sep 14, 2021

Numerical methods for differential equations15 references4 citations

TL;DR

This paper introduces a Partial Integral Equation (PIE) representation for coupled 2D linear PDEs, extending the PIE framework from 1D systems to two spatial dimensions. By constructing a calculus of Partial Integral (PI) operators on L₂[x,y], the authors establish a bijective mapping between PIE solutions and original PDE solutions, enabling convex optimization-based stability analysis via Linear PI Inequalities (LPIs), with demonstrated low conservatism on 2D heat and wave equations.

ABSTRACT

We introduce a Partial Integration Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -- making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L_2[x,y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R^n x L_2[x] x L_2[y] x L_2[x,y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -- providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -- allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism.

Motivation & Objective

• To extend the PIE framework—previously used for 1D PDEs and time-delay systems—to two spatial dimensions.
• To develop a comprehensive algebra of Partial Integral (PI) operators on L₂[x,y] for composition, adjoint, and inversion.
• To establish a bijective correspondence between solutions of 2D PDEs and their corresponding PIE representations.
• To parameterize the convex cone of 2D PI operators using positive matrices for optimization.
• To derive and validate LPI-based stability conditions for 2D linear PDEs with minimal conservatism.

Proposed method

• Construct an algebra of PI operators on the L₂[x,y] function space, defining operations for composition, adjoint, and inversion.
• Extend the PI operator algebra to Rⁿ × L₂[x] × L₂[y] × L₂[x,y] to accommodate coupled 2D PDEs.
• Demonstrate that every suitable coupled linear 2D PDE has an equivalent PIE representation with explicit solution conversion formulae.
• Parameterize the convex cone of 2D PI operators using positive matrices to enable convex optimization.
• Formulate Linear PI Inequality (LPI) feasibility problems to derive stability conditions for 2D PDEs.
• Apply the LPI framework to analyze stability of 2D heat and wave equations using convex optimization techniques.

Experimental results

Research questions

• RQ1 Can the PIE framework be extended from 1D systems to 2D PDEs with multiple spatial variables?
• RQ2 What algebraic structure governs the composition and inversion of PI operators in two spatial dimensions?
• RQ3 Is there a bijective mapping between solutions of a coupled 2D PDE and its corresponding PIE representation?
• RQ4 How can the convex cone of 2D PI operators be parameterized for optimization-based analysis?
• RQ5 What is the conservatism of the resulting LPI-based stability conditions for 2D linear PDEs?

Key findings

• The paper successfully constructs a complete algebra of PI operators on L₂[x,y], enabling systematic manipulation of 2D PDEs in PIE form.
• A bijective solution mapping is established between original 2D PDEs and their PIE representations, ensuring equivalence.
• The convex cone of 2D PI operators is parameterized using positive matrices, enabling convex optimization for stability analysis.
• The LPI-based stability conditions are shown to have little to no conservatism when tested on 2D heat and wave equations.
• The framework enables stability analysis without requiring continuity constraints or explicit boundary conditions, enhancing applicability.