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Verifying Well-Posedness of Linear PDEs using Convex Optimization

Declan S. Jagt, Matthew M. Peet

Abstract

Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of $L_{2}$ -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

Verifying Well-Posedness of Linear PDEs using Convex Optimization

Abstract

Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on . We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

Paper Structure

This paper contains 30 sections, 7 theorems, 58 equations, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notation
  4. Strongly Continuous Semigroups
  5. Well-Posedness in the PIE Representation
  6. Bijection between PDE Domain and $L_2$
  7. The PIE Representation of Linear 1D PDEs
  8. A Lumer--Phillips Theorem for PIEs
  9. Dissipativity in the PIE Representation
  10. Surjectivity in the PIE Representation
  11. A PIE-Based Generation Theorem for Quasicontraction Semigroups
  12. A PIE-Based Well-Posedness Test for PDEs
  13. Generation of $C_{0}$-Semigroups on $L_{2}$
  14. An LPI for Well-Posedness of PDEs
  15. Illustrations of the Well-Posedness Test
  16. ...and 15 more sections

Key Result

Corollary 3

If $D\subseteq W_{2}^{d,n}[a,b]$ is PIE compatible, then $\partial_{s}^{d}:D\to L_{2}^{n}[a,b]$ is invertible, and $\mathcal{T}:=(\partial_{s}^{d})^{-1}:L_{2}^{n}[a,b]\to D$ takes the form $(\mathcal{T}\mathbf{ v})(s):=\int_{a}^{b}\boldsymbol{G}(s,\theta)\mathbf{ v}(\theta)\, d\theta,$ where for $\boldsymbol{Q}(s)$, $K$, and $C$ as in Defn. defn:PDE_dom, and $\blacktriangleleft$$\blacktrianglele

Theorems & Definitions (18)

  • Definition 1: Infinitesimal Generator of $C_{0}$-Semigroup
  • Definition 2
  • Corollary 3
  • Definition 4: Well-Posed PDE
  • Definition 5: Well-Posed PIE
  • Definition 6: Contraction Semigroup
  • Theorem 7: Lumer--Phillips Theorem
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 8 more