Verifying Well-Posedness of Linear PDEs using Convex Optimization

TL;DR

This paper presents a convex optimization-based method to verify the well-posedness of linear PDEs by reformulating the Lumer-Phillips conditions through the PIE representation.

Contribution

It introduces a novel approach to test PDE well-posedness using convex optimization on the PIE reformulation, simplifying verification process.

Findings

Successfully verified well-posedness for classical parabolic PDEs.

Successfully verified well-posedness for classical hyperbolic PDEs.

Established a least upper bound on the exponential growth rate of solutions.

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.