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 $L_{2}$ -- 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.

• The paper introduces a novel convex optimization framework that reformulates the well-posedness verification of linear PDEs into tractable operator inequalities.
• It utilizes the Partial Integral Equation (PIE) approach to transform boundary conditions into semi-separable polynomial kernels, enabling numerical certification.
• The method provides rigorous well-posedness certificates and quantitative growth rate bounds through semidefinite programming on canonical PDEs such as heat, transport, and wave equations.

PIE-Based Verification of Well-Posedness for Linear PDEs

Introduction

The paper "Verifying Well-Posedness of Linear PDEs using Convex Optimization" (2604.00573) develops a novel operator-theoretic and optimization-based approach to verifying the well-posedness of linear partial differential equations, leveraging the Partial Integral Equation (PIE) framework. The methodology centers on overcoming traditional obstacles presented by domain-specific analysis for PDE generators and reframing well-posedness certification as a tractable convex optimization problem. The PIE representation serves as the linchpin, enabling the transformation of boundary-constrained PDEs into operator inequalities accessible to numerical solution via PIETOOLS and semidefinite programming. This essay details the paper's technical approach, main results, and broader implications for PDE systems analysis and computational control.

Technical Background

Establishing well-posedness for dynamical models governed by PDEs—ensuring existence, uniqueness, and continuous dependence on the initial state—remains technically challenging, especially due to the need for compatible boundary conditions and intricate domain regularity. The $C_0$-semigroup framework serves as the analytic backbone, transforming the well-posedness question into operator-theoretic properties of the PDE generator $A:D\to H$, where $D$ is a Sobolev subspace of a Hilbert space $H$. For dissipative systems, the Lumer--Phillips theorem provides necessary and sufficient generation conditions via dissipativity and surjectivity constraints, but verifying these in practice is hampered by the complexity of the operator domain $D$ and the lack of systematic parameterizations for the generator.

PIE Representation and Computational Reformulation

The PIE framework addresses these difficulties by constructing a bijection $\mathcal{T}:L_2 \to D$, associating each element in $D$ (the PDE domain) with an $L_2$-integrable state. In the PIE formulation, PDEs of the form $\dot{u}(t) = A u(t)$ with PDE-compatible boundary and regularity constraints are equivalently represented as $\partial_t \mathcal{T}v(t) = \mathcal{A} v(t)$, where both $A:D\to H$0 and $A:D\to H$1 are bounded (partial integral) operators on $A:D\to H$2. This allows boundary conditions, which previously made analysis intractable, to be encoded into $A:D\to H$3 with explicit semi-separable polynomial kernels. The PIE system equivalently describes the original PDE, providing an analytic bridge for generation-theoretic results to be transferred and reformulated.

PIE-Based Lumer--Phillips Theorem

A central contribution is the development of a PIE-analog of the Lumer--Phillips theorem. Dissipativity is reformulated as an operator inequality on $A:D\to H$4:

$A:D\to H$5

for all $A:D\to H$6 and $A:D\to H$7 the desired exponential growth rate. The surjectivity of the resolvent, typically intractable for unbounded differential operators, is converted into a bounded, numerically verifiable condition: for some $A:D\to H$8, the operator $A:D\to H$9 is surjective on $D$0, holding if and only if its adjoint is bounded below (by the open mapping theorem). This allows well-posedness conditions to be checked by numerically feasible operator inequalities.

The authors further extend these results from contraction semigroups ($D$1) to quasicontraction and general $D$2-semigroups, accommodating systems with nonzero exponential growth rate. Well-posedness can thus be established for PDEs that generate $D$3-semigroups with arbitrary gain and growth rate, broadening applicability to dissipative and certain unstable systems.

Convex Optimization Formulation

The equivalence between PIE-based operator inequalities and well-posedness motivates the formulation of Linear Partial Integral Inequalities (LPI), whose feasibility certifies the generation property. The main result gives that, for a fixed $D$4, existence of a coercive PI operator $D$5 and a scalar $D$6 such that:

• Dissipativity: $D$7
• Surjectivity: $D$8

ensures well-posedness (i.e., $D$9 is the infinitesimal generator of a $H$0-semigroup) with explicit exponential bound on the solution. These operator inequalities are reduced to semidefinite programs via the polynomial parameterization of PI operators, and solved using the PIETOOLS software suite.

Empirical Results

The efficacy of the approach is demonstrated on several canonical PDEs:

• Transport Equation: Both well-posed and ill-posed boundary conditions are considered. For admissible settings, the method certifies well-posedness with $H$1 and verifies contraction for arbitrary boundary data. In the ill-posed case (boundary compatibility violated), the LPI is infeasible, aligning with analytic theory.
• Heat Equation: Tight numerical upper bounds on growth rate $H$2 are obtained, closely matching analytic results $H$3 for a reaction parameter $H$4. Reverse-time (ill-posed) variants are correctly detected by infeasibility.
• Wave Equation: The method distinguishes between well-posed state representations and incompatible ill-posed formulations, verifying semigroup generation only in the former scenario.
• Speculative Coupled PDEs: The framework extends to complex, multi-component, and non-standard PDEs, establishing well-posedness without requiring bespoke analytic techniques.

The approach provides not only a yes/no certificate of well-posedness but also a computable least upper bound on the solution growth rate.

Implications and Future Directions

The PIE-based convex optimization framework unifies well-posedness verification across a broad spectrum of linear PDEs, systematically removing the necessity for ad hoc analytic proofs for each new system. The numerical LPI test provides both rigor (as a certificate of semigroup generation) and quantitative insight (growth rate estimation) using efficient SDP solvers. This facilitates subsequent controller, observer, and stability synthesis directly within the partial integral operator algebra, with immediate applicability to large-scale, coupled, and distributed-parameter systems.

Theoretical implications include the possibility of transferring additional semigroup-theoretic results into the PIE setting and investigating the extension to certain classes of nonlinear or time-varying PDEs, potentially using parameter-dependent or adaptive operators within the PIE framework. Practically, the methodology enables rapid, rigorous prototyping and design in control and modeling of distributed-parameter systems, with error certification and stability guarantees.

Conclusion

By recasting the abstract generation conditions of the Lumer--Phillips theorem into PIE-based operator inequalities and translating the verification task into convex optimization, this work provides a principled, computationally viable methodology for certifying the well-posedness of a large class of linear PDEs. The PIE approach, together with algorithmic implementations, opens new avenues for robust analysis and design in infinite-dimensional systems theory and computational PDE control.