A Distributed SOS Program For Local Stability Analysis of Polynomial PDEs in the PIE Representation

Abstract: It has recently been shown that the evolution of a state, described by a Partial Differential Equation (PDE), can be more conveniently represented as the evolution of the state's highest spatial derivative (the ``fundamental state''), which lies in $L_2$ and has no boundary conditions (BCs) or continuity constraints. For linear PDEs, this yields a Partial Integral Equation (PIE) parametrized by Partial Integral (PI) operators mapping the fundamental state to the PDE state. In this paper, we show that for polynomial PDEs, the dynamics of the fundamental state can instead be compactly expressed as a distributed polynomial in the fundamental state, parametrized by a new tensor algebra of PI operators acting on the tensor product of the fundamental state. We further define a SOS parametrization of the distributed polynomial and use this to construct a distributed SOS program, for testing local stability of polynomial PDEs.

• The paper presents a novel distributed SOS programming approach that leverages the PIE representation to certify local stability for polynomial PDEs.
• It introduces operator-valued distributed polynomials and Tensor-PI operators that recast PDEs into tractable semidefinite programming problems.
• The proposed method, validated via the Fisher equation, provides tight estimates on exponential decay rates and supports scalable stability certification.

Distributed Sum-of-Squares Programming for Local Stability of Polynomial PDEs via the PIE Framework

Introduction and Motivation

The paper "A Distributed SOS Program For Local Stability Analysis of Polynomial PDEs in the PIE Representation" (2604.01115) systematically addresses structural challenges in the analysis and certification of local stability for nonlinear, polynomial partial differential equations (PDEs). Traditional methodologies in infinite-dimensional systems are often tightly coupled to specific PDE forms and boundary conditions, leading to classical Lyapunov-based approaches that are not only difficult to generalize but also highly sensitive to changes in the PDE’s formulation or boundary specification.

The Partial Integral Equation (PIE) framework was previously introduced to overcome such challenges, providing a representation where the state is described solely by its highest spatial derivative (the "fundamental state"), free from boundary and continuity constraints present in the conventional PDE state. While earlier work extended PIEs to linear and quadratic systems, this paper explicitly generalizes the PIE formalism to arbitrary polynomial PDEs. It further introduces a distributed Sum-of-Squares (SOS) programming approach for constructing Lyapunov functionals (LFs) and systematically devises a tractable method to test local stability in these infinite-dimensional settings.

PIE Representation of Polynomial PDEs

Central to the methodology is the abstract decoupling produced by the PIE representation. For a general one-dimensional, $n$-th order polynomial PDE with linear boundary conditions, the approach defines:

• The PDE state $u(t) \in H_2^n[\Omega]$, subject to boundary constraints,
• The fundamental state $v(t) := \partial^n_s u(t) \in L_2[\Omega]$, which is unencumbered by boundary considerations.

The classical PDE is mapped onto a PIE of the form

$T \dot{v}(t,s) = C Z_d(v), \quad s \in \Omega,$

where $T$ is a 2-PI operator (the inverse map from fundamental state to PDE state), $C$ is defined via a Tensor-PI (T-PI) operator structure, and $Z_d(v)$ is the distributed monomial basis representing polynomial expressions over the state. This abstraction enables the representation of polynomial PDEs as distributed polynomials in the fundamental state, thus decoupling the analysis from boundary conditions and Sobolev regularity constraints. The explicit algebraic structure and operator-theoretic mapping are rigorously constructed, including the necessary existence and invertibility conditions on the PI operators.

Distributed Polynomial and Operator Algebra

A key technical contribution is in defining the class of distributed polynomials and associated operator algebras:

• Tensor-PI Operators: Capture tensor products of standard PI operators, modeling the necessary structure for representing polynomial nonlinearities (e.g., products of $u$, $u_s$, $\cdots$).
• Functional-PI Operators: Facilitate the scalarization of operator expressions via integration with polynomial kernels over the domain, supporting the construction of Lyapunov inequalities and positivity constraints required for SOS certificates.

The authors prove closure properties of these distributed operator classes under addition, multiplication, and integration, ensuring a ring structure essential for encoding and manipulating positivity conditions via SOS program constraints.

Distributed SOS Polynomials and Optimization Program

Building on the insights from sum-of-squares techniques for ordinary and finite-dimensional polynomial systems, the authors introduce distributed SOS polynomials, defined via quadratic forms involving the distributed monomial basis and operator-valued Gram matrices. They derive vectorization theorems converting quadratic forms of PI operators into the linear distributed polynomial basis, which is crucial for converting operator inequalities into standard SOS constraints that are tractable for semidefinite programming (SDP) solvers.

The main SOS program is formulated as follows:

• Lyapunov Functional Candidate: Search for a distributed SOS polynomial $u(t) \in H_2^n[\Omega]$0.
• Stability Conditions: Express decay of $u(t) \in H_2^n[\Omega]$1 along PIE trajectories using the Lie derivative in the PIE setting.
• Feasibility Program: The existence of $u(t) \in H_2^n[\Omega]$2 (and auxiliary multipliers) satisfying the operator-valued SOS constraints certifies exponential stability on an $u(t) \in H_2^n[\Omega]$3-ball around the origin.

All constraints in the program are posed directly as operator equations on the distributed polynomial basis, enabling SDP-based verification for general polynomial PDEs with linear BCs.

Numerical Example: Fisher Equation

A comprehensive computational study is presented using the Fisher equation with Dirichlet boundary conditions:

$u(t) \in H_2^n[\Omega]$4

By constructing the PIE and distributed polynomial representation for this system, the authors numerically solve the distributed SOS program. The computed results yield tight lower bounds on the exponential decay rate $u(t) \in H_2^n[\Omega]$5 ensuring local stability for various sizes of the $u(t) \in H_2^n[\Omega]$6-ball. Notably, the method recovers sharp thresholds for the largest basin of attraction, as confirmed by simulations indicating instability for initial data just above the computed threshold. The approach scales seamlessly to compute decay rates over different local regions (radii $u(t) \in H_2^n[\Omega]$7) in the function space.

Implications and Future Directions

This work rigorously elevates the computational Lyapunov method for PDEs by abstracting away the confounding effects of boundary conditions and regularity restrictions. By encoding PDEs as distributed polynomials in the PIE form, the process of stability certification is reduced to solving general SDP problems in the space of operator-valued polynomials, mirroring finite-dimensional SOS approaches. Several implications arise:

• Automated Stability Certification: The presented framework can be systematically applied to a broad class of 1D polynomial PDEs with linear BCs, with software (PIETOOLS) available for practical implementation.
• Generalizability: Though the paper restricts discussion to scalar, 1D polynomial PDEs, the abstract operator-theoretic framework is poised for extension to coupled systems, multi-component PDEs, and higher spatial dimensions given the necessary generalizations of the PI algebra.
• Integration with Robust and Control Theory: The SOS certification methodology is congruent with modern control and robustness analysis tools, particularly those seeking scalable numerical solutions for infinite-dimensional stability and performance verification.

Conclusion

The paper establishes a principled, operator-theoretic, and computational approach to local stability analysis of nonlinear PDEs by leveraging the PIE representation and distributed SOS polynomials. The rigorous vectorization and closure results ensure that positivity and decay conditions over infinite-dimensional spaces can be recast as feasible semidefinite programs, as demonstrated on canonical nonlinear PDEs. This work lays a strong theoretical and computational foundation for broader applications in nonlinear PDE analysis, robust control, and system-theoretic certification in infinite-dimensional systems.