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A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Declan S. Jagt, Matthew M. Peet

Abstract

We introduce a Partial Integral 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.

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Abstract

We introduce a Partial Integral 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.

Paper Structure

This paper contains 38 sections, 31 theorems, 289 equations, 5 figures, 3 tables.

Table of Contents

  1. INTRODUCTION
  2. Notation
  3. Algebras of PI Operators
  4. An Algebra of 2D to 2D PI Operators
  5. An Algebra of 011-PI Operators
  6. An Algebra of 0112-PI Operators
  7. Useful Properties of PI Operators
  8. Inverse of 011-PI Operators
  9. Differentiation and 2D-PI Operators
  10. Adjoint of 2D-PI Operators
  11. A Standardized PDE Format in 2D
  12. Definition of Solution
  13. The Fundamental State on 2D
  14. Map From Fundamental State to PDE State
  15. PDE to PIE conversion
  16. ...and 23 more sections

Key Result

Lemma 1

For any $N \in \mathcal{N}_{2D}^{n\times p}$ and $M \in \mathcal{N}_{2D}^{p\times m}$, there exists a unique $Q \in \mathcal{N}_{2D}^{n\times m}$ such that $\mathcal{P}[N]\mathcal{P}[M]=\mathcal{P}[Q]$. Specifically, we may choose where $\mathcal{L}_{2D}:\mathcal{N}_{2D}^{n\times p} \times \mathcal{N}_{2D}^{p\times m} \rightarrow \mathcal{N}_{2D}^{n\times m}$ is defined in Eqn. eq_composition_2Dt

Figures (5)

  • Figure 1: Parameters $K_1$, $K_2$, $H_1$ and $H_2$ defining the mappings in Lemma \ref{['lem_uhat_to_u']} and in Corollary \ref{['cor_uhat_to_BC']}
  • Figure 2: Parameters $T$ describing PI operator $\mathcal{T}=\mathcal{P}[T]$ mapping the fundamental state back to the PDE state in Theorem \ref{['thm_Tmap']}
  • Figure 3: Parameters $\hat{Q}$ describing the inverse PI operator $\mathcal{P}[\hat{Q}]=\mathcal{P}[Q]^{-1}$ in Lemma \ref{['lem_inv_appendix']}
  • Figure 4: Parameters $T$ describing PI operator $\mathcal{T}=\mathcal{P}[T]$ mapping the fundamental state back to the PDE state in Theorem \ref{['thm_Tmap_appendix']}
  • Figure 5: Parameters $N$ describing the positive PI operator $\mathcal{P}[N]=\mathcal{Z}^* P\mathcal{Z}$ in Proposition \ref{['prop_pos_PI_appendix']}

Theorems & Definitions (57)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 47 more