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Incremental stability in $p=1$ and $p=\infty$: classification and synthesis

Simon Kuang, Xinfan Lin

Abstract

All Lipschitz dynamics with the weak infinitesimal contraction (WIC) property can be expressed as a Lipschitz nonlinear system in proportional negative feedback -- this statement, a ``structure theorem,'' is true in the $p=1$ and $p=\infty$ norms. Equivalently, a Lipschitz vector field is WIC if and only if it can be written as a scalar decay plus a Lipschitz-bounded residual. We put this theorem to use using neural networks to approximate Lipschitz functions. This results in a map from unconstrained parameters to the set of WIC vector fields, enabling standard gradient-based training with no projections or penalty terms. Because the induced $1$- and $\infty$-norms of a matrix reduce to row or column sums, Lipschitz certification costs only $O(d^2)$ operations -- the same order as a forward pass and appreciably cheaper than eigenvalue or semidefinite methods for the $2$-norm. Numerical experiments on a planar flow-fitting task and a four-node opinion network demonstrate that the parameterization (re-)constructs contracting dynamics from trajectory data. In a discussion of the expressiveness of non-Euclidean contraction, we prove that the set of $2\times 2$ systems that contract in a weighted $1$- or $\infty$-norm is characterized by an eigenvalue cone, a strict subset of the Hurwitz region that quantifies the cost of moving away from the Euclidean norm.

Incremental stability in $p=1$ and $p=\infty$: classification and synthesis

Abstract

All Lipschitz dynamics with the weak infinitesimal contraction (WIC) property can be expressed as a Lipschitz nonlinear system in proportional negative feedback -- this statement, a ``structure theorem,'' is true in the and norms. Equivalently, a Lipschitz vector field is WIC if and only if it can be written as a scalar decay plus a Lipschitz-bounded residual. We put this theorem to use using neural networks to approximate Lipschitz functions. This results in a map from unconstrained parameters to the set of WIC vector fields, enabling standard gradient-based training with no projections or penalty terms. Because the induced - and -norms of a matrix reduce to row or column sums, Lipschitz certification costs only operations -- the same order as a forward pass and appreciably cheaper than eigenvalue or semidefinite methods for the -norm. Numerical experiments on a planar flow-fitting task and a four-node opinion network demonstrate that the parameterization (re-)constructs contracting dynamics from trajectory data. In a discussion of the expressiveness of non-Euclidean contraction, we prove that the set of systems that contract in a weighted - or -norm is characterized by an eigenvalue cone, a strict subset of the Hurwitz region that quantifies the cost of moving away from the Euclidean norm.

Paper Structure

This paper contains 12 sections, 7 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Enforcing Contraction
  4. Extension to weighted norms
  5. Lipschitz by construction: neural networks
  6. Numerical Examples
  7. Toy example ($1$-norm contraction)
  8. Opinion network ($\infty$-norm contraction)
  9. Discussion: where are all the p in 1, infinity-contractions?
  10. Diversity
  11. Physicality
  12. Conclusion

Key Result

Theorem 1

Suppose that a dynamic system $\dot x = f(x)$ is WIC with respect to the $p$-norm. Then for any trajectories $x_1(t), x_2(t)$ of the ODE $\dot x = f(x)$, for all $t \geq 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 3: Phase portrait comparison for the toy example. Left: origin-destination pairs. Right: learned contractive neural ODE.
  • Figure 4: Opinion network example. Top left: the 4-node weighted digraph defining $A$. Top middle: training and test loss. Top right: ground truth (solid) and learned predictions (dashed) for one initial condition in the test dataset. Bottom row: pairwise projections of the ground-truth (blue) and learned (orange) vector fields onto the $(x_0,x_1)$, $(x_0,x_2)$, and $(x_1,x_3)$ planes.
  • Figure 5: The $(\tau,\delta)$-plane for diagonalizable $2\times 2$ matrices with $\tau = \operatorname{tr} A$ and $\delta = \det A$. The shaded marginally stable region ($\tau < 0$, $\delta > 0$) characterizes $p = 2$ WIC. The hatched subregion below the parabola $\delta = \tau^2/2$ characterizes $p \in \{1,\infty\}$ WIC. The dashed curve $\delta = \tau^2/4$ is the zero set of the discriminant $\tau^2 - 4\delta$: eigenvalues are real below it and complex above it.

Theorems & Definitions (15)

  • Definition 1: Matrix Measure
  • Definition 2: bullo_contraction_2026
  • Theorem 1: bullo_contraction_2026
  • Lemma 1: Matrix Measure for Common Norms
  • Lemma 2: Matrix Norms for Common Norms
  • Proposition 1: Properties of Matrix Measure
  • Definition 3: Lipschitz constant
  • Theorem 2: Structure Theorem for WIC
  • proof
  • Remark 1
  • ...and 5 more