⚡ electrical engineering

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

This paper introduces a distributed Sum-of-Squares (SOS) program for testing the local stability of polynomial Partial Differential Equations (PDEs) by reformulating their dynamics as distributed polynomials in the fundamental state using a novel tensor algebra of Partial Integral (PI) operators.

Original authors: Carl R Richardson, Declan S Jagt, Matthew M Peet, Antonis Papachristodoulou

Published 2026-04-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Carl R Richardson, Declan S Jagt, Matthew M Peet, Antonis Papachristodoulou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a ripple moves across a pond, or how heat spreads through a metal rod. In the world of physics and engineering, these are described by Partial Differential Equations (PDEs). They are the "rules of the road" for how things change over space and time.

However, analyzing these equations is notoriously difficult. It's like trying to navigate a maze where the walls keep moving, and you have to worry about the exact rules at the very edges of the maze (the "boundary conditions"). If you make a tiny mistake at the edge, your whole prediction can collapse.

This paper introduces a clever new way to look at these problems, turning a tangled mess into a neat, solvable puzzle. Here is the breakdown using simple analogies:

1. The Problem: The "Edge" Trap

Think of a PDE as a complex machine. Usually, to understand how the machine works, you have to track every single part, including the ones stuck to the walls (the boundaries).

The Old Way: Imagine trying to describe a dance by tracking every dancer's feet, their hands, and the fact that they must stay within the stage ropes. It's messy. If you change the ropes (the boundary conditions), you have to rewrite the whole dance description.
The Result: This makes it very hard to prove that the system is "stable" (i.e., that the dance won't turn into a chaotic pile-up).

2. The Solution: The "Fundamental State" (The Core)

The authors propose a radical shift in perspective. Instead of tracking the whole dancer, they suggest tracking only the highest derivative—the most "extreme" part of the movement.

The Analogy: Imagine you are watching a wave. Instead of tracking the water level at every point (which has to be zero at the shore), you only track the steepness of the wave.
Why it helps: The "steepness" (the fundamental state) doesn't care about the ropes at the edge. It floats freely in a mathematical space called $L_2$ . By focusing only on this core, the "boundary conditions" (the ropes) are automatically baked into the math, rather than being annoying constraints you have to check constantly.

3. The New Language: "Distributed Polynomials"

Once they switched to this "core" view, the equations looked different. They weren't just simple lines anymore; they were complex, multi-dimensional shapes.

The Analogy: Think of a standard polynomial (like $x^2 + 2x + 1$ ) as a flat drawing on a piece of paper. The new method uses "Distributed Polynomials." Imagine that instead of a flat drawing, you have a 3D sculpture where every point in space has its own little polynomial attached to it.
The Magic: The authors invented a new "grammar" (using something called Tensor-PI operators) to write these 3D sculptures down on paper. This grammar allows them to treat these complex, space-filling shapes just like simple algebraic equations.

4. The Stability Test: The "Sum of Squares" (SOS)

Now that they have this new language, they need to answer the big question: "Is this system stable?"

The Metaphor: Imagine you want to prove a ball rolling in a bowl will eventually stop at the bottom. You need a "Lyapunov Function" (a fancy term for an energy meter). If the energy meter always goes down, the ball is safe.
The SOS Trick: In math, proving something is always positive (like energy) is hard. But if you can prove something is a Sum of Squares (like $x^2 + y^2$ ), you know it's automatically positive because squares are never negative.
The Innovation: The authors created a "Distributed SOS Program." This is a computer algorithm that automatically searches for that perfect "energy meter" (Lyapunov function) for these complex 3D polynomial sculptures. It checks if the energy always goes down, proving the system won't explode.

5. The Real-World Test: The Fisher Equation

To prove their method works, they tested it on the Fisher Equation, which models how a population (like bacteria or a species) spreads and grows.

The Result: They used their new "Distributed SOS" tool to calculate exactly how big a population could get before it became unstable.
The Analogy: It's like finding the exact size of a balloon you can blow up before it pops. Their method calculated this "pop limit" with high precision, showing that for small populations, the system is stable, but if you blow it too big, it becomes chaotic.

Summary: Why This Matters

Before: Analyzing these equations was like trying to solve a Rubik's cube while blindfolded, with the rules changing every time you touched a corner.
Now: The authors gave us a pair of glasses (the Fundamental State) and a new set of instructions (Distributed Polynomials).
The Benefit: This allows engineers and scientists to use computers to automatically verify that complex physical systems (like fluid flow, heat transfer, or population dynamics) are safe and stable, without getting bogged down in the messy details of the edges.

In short, they turned a messy, boundary-constrained nightmare into a clean, algebraic puzzle that a computer can solve instantly.

1. Problem Statement

The paper addresses the challenge of analyzing the local stability of polynomial Partial Differential Equations (PDEs) subject to linear boundary conditions (BCs).

Current Limitations: Standard PDE representations involve states in Sobolev spaces with complex boundary and continuity constraints. This makes stability analysis difficult, particularly for nonlinear systems, because:
- Lyapunov functions (LFs) must account for BCs explicitly.
- The derivative of the LF along system trajectories often depends on higher-order derivatives and BCs in ways that are hard to verify algebraically.
- Existing methods often rely on specific PDE structures (e.g., specific nonlinearities or specific BC types like Dirichlet), lacking a general framework for arbitrary polynomial PDEs.
Goal: To develop a unified, computational framework for testing local stability of 1D polynomial PDEs on an $L_2$ -ball, handling arbitrary linear BCs (Dirichlet, Neumann, Robin) without explicitly constraining the state space during the analysis.

2. Methodology

The authors propose a novel framework based on the Partial Integral Equation (PIE) representation and Sum-of-Squares (SOS) optimization.

A. The PIE Representation

Instead of analyzing the PDE state $u(t,s)$ directly, the authors map the system to the evolution of its fundamental state, defined as the highest spatial derivative $v(t,s) = \partial^n u / \partial s^n$ .

Key Advantage: The fundamental state $v$ lies in $L_2$ and is free of boundary conditions and continuity constraints.
Mapping: A linear, invertible Partial Integral (PI) operator $T$ maps the fundamental state $v$ back to the PDE state $u$ (i.e., $u = Tv$). The BCs are embedded into the operator $T$ rather than constraining the state space.
Dynamics: The PDE dynamics are transformed into a PIE where the time derivative of $v$ is expressed as a distributed polynomial in $v$ .

B. Distributed Polynomials and Tensor-PI Operators

To handle the nonlinear terms in the PDE (which become polynomial terms in the PIE), the authors define a new algebraic structure:

Distributed Polynomials: Polynomials where the variables are functions in $L_2$ , and coefficients are operators.
Tensor-PI (T-PI) Operators: These operators act on tensor products of the fundamental state (e.g., $v \otimes v$ ). They allow the representation of complex nonlinear interactions (like $u \cdot u_s$ ) as linear operations on tensor products of $v$ .
Functional-PI (F-PI) Operators: These map function-valued objects to scalars (integrals). They are used to represent Lyapunov functions and semi-algebraic sets (like the $L_2$ -ball).
Vectorization: A crucial mathematical tool introduced to convert integrals of T-PI operators into F-PI operators, enabling the enforcement of equality constraints.

C. Distributed SOS Parametrization

Inspired by SOS programming for ODEs, the authors define Distributed SOS Polynomials:

A distributed polynomial $q(x)$ is SOS if it can be written as a quadratic form $\langle Z_d(x), Q Z_d(x) \rangle$ where $Q$ is a positive semi-definite operator matrix.
This parametrization ensures positivity by construction.
The paper provides a method to convert the quadratic SOS representation into a linear distributed polynomial form, allowing the use of equality constraints to match coefficients.

D. The Distributed SOS Program

The stability test is formulated as a convex optimization problem (Semidefinite Programming - SDP):

Lyapunov Candidate ( $V$ ): A distributed SOS polynomial in the fundamental state $v$ .
Region of Attraction: Defined as an $L_2$ -ball of radius $r$ in the PDE domain, mapped to a semi-algebraic set in the $L_2$ space of $v$ .
Conditions:
- $V(v) \geq \epsilon \|u\|_{L_2}^2$ (Positive definite).
- $V(v) \leq C \|u\|_{L_2}^2$ (Upper bound).
- $\dot{V}(v) \leq -2\lambda V(v)$ (Exponential decay), where $\dot{V}$ is the Lie derivative along the PIE dynamics.
Implementation: The conditions are enforced by searching for SOS multipliers and coefficients that satisfy the polynomial identities derived from the PIE representation.

3. Key Contributions

Generalized PIE for Polynomial PDEs: Extends the PIE framework from linear and quadratic PDEs to arbitrary polynomial PDEs using Tensor-PI operators.
Distributed Polynomial Algebra: Defines a rigorous algebraic structure for distributed polynomials, including Tensor-PI and Functional-PI operators, and proves closure properties (e.g., the product of two distributed polynomials is a distributed polynomial).
Distributed SOS Framework: Introduces the concept of distributed SOS polynomials and provides the necessary vectorization techniques to convert them into a form suitable for SDP solvers.
Unified Stability Test: Constructs a distributed SOS program that tests local exponential stability for any 1D polynomial PDE with linear BCs, eliminating the need for case-specific derivations of Lyapunov functions.
Open-Source Integration: The methodology is designed to be implemented in the existing PIETOOLS software package.

4. Results

The authors validate the method using the Fisher Equation with Dirichlet boundary conditions:
$u_t = u_{ss} + \alpha u - \beta u^2$

Stability Radius: The program successfully verified local stability on an $L_2$ -ball of radius $r \approx 4.048$ . Numerical simulations showed instability for $r > 4.05$ , suggesting the bound is tight.
Decay Rates: The authors computed the largest lower bound on the exponential decay rate $\lambda$ $λ$ for various radii $r$ $r$ .
- As $r \to 0$ , the computed $\lambda$ converges to the linearized system's decay rate ( $\approx 4.869$ ).
- As $r$ increases, $\lambda$ decreases, reflecting the impact of nonlinearity.
Visualization: Simulations confirmed that the derived exponential bounds tightly envelope the actual system trajectories.

5. Significance

Automation of Stability Analysis: This work moves PDE stability analysis from a manual, case-by-case derivation of Lyapunov functions to an automated, computational procedure.
Handling Nonlinearity: It provides a systematic way to handle polynomial nonlinearities and arbitrary linear boundary conditions, which have historically been difficult to treat simultaneously.
Scalability: By working in the $L_2$ space without explicit BC constraints, the method simplifies the search space for Lyapunov functions, making it more scalable for complex systems.
Foundation for Control: The framework lays the groundwork for future extensions to distributed control and estimation of nonlinear PDEs, similar to how SOS methods revolutionized ODE control.

Limitations: Currently, the method is restricted to scalar, 1D PDEs. The authors note that extension to coupled systems and higher dimensions is a potential future direction.