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Asymptotic stability equals exponential stability -- while you twist your eyes

Wouter Jongeneel

TL;DR

The paper proves that global asymptotic stability (GAS) of equilibria under two vector fields can be connected by a GAS-preserving homotopy of semiflows, effectively transforming GAS into exponential stability via orientation-preserving homeomorphisms. The main result extends Grüne–Sontag–Wirth-type coordinate changes to a continuous, GAS-preserving path and generalizes to discontinuous vector fields and manifolds, with a parallel treatment for input-to-state stability (ISS). An additional viewpoint ties the construction to optimal transport, using density paths between Lyapunov potentials to illuminate the GAS-to-exponential transition. These contributions offer a partial answer to Conley’s converse question and suggest practical implications for stability-preserving design in control and learning frameworks.

Abstract

Suppose that two vector fields on a smooth manifold render some equilibrium point globally asymptotically stable (GAS). We show that there exists a homotopy between the corresponding semiflows such that this point remains GAS along this homotopy.

Asymptotic stability equals exponential stability -- while you twist your eyes

TL;DR

The paper proves that global asymptotic stability (GAS) of equilibria under two vector fields can be connected by a GAS-preserving homotopy of semiflows, effectively transforming GAS into exponential stability via orientation-preserving homeomorphisms. The main result extends Grüne–Sontag–Wirth-type coordinate changes to a continuous, GAS-preserving path and generalizes to discontinuous vector fields and manifolds, with a parallel treatment for input-to-state stability (ISS). An additional viewpoint ties the construction to optimal transport, using density paths between Lyapunov potentials to illuminate the GAS-to-exponential transition. These contributions offer a partial answer to Conley’s converse question and suggest practical implications for stability-preserving design in control and learning frameworks.

Abstract

Suppose that two vector fields on a smooth manifold render some equilibrium point globally asymptotically stable (GAS). We show that there exists a homotopy between the corresponding semiflows such that this point remains GAS along this homotopy.

Paper Structure

This paper contains 11 sections, 3 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Dynamical systems
  4. Stability
  5. Further comments on related work
  6. Stability preserving homotopies
  7. Input-to-State Stability
  8. A view from optimal transport
  9. Conclusion and future work
  10. Continuation and the Conley index
  11. Added in proof

Key Result

Proposition 3.2

Let $0\in \mathbb{R}^n$, for $n \neq 5$, be strongly GAS, in the sense of Filippov, under $\dot{x}\in \mathcal{F}[X](x)$, with $X$ satisfying Assumption ass:reg:X. Then,

Figures (2)

  • Figure 3.1: Example \ref{['ex:invextoconvex']}: on the left, the graph of $v_i$ around $x=0$; and on the right, the graph of $v_q$ around $x=0$. In between, steps of the homotopy that connects $v_i$ to $v_q$, while preserving that $x=0$ is the global minimizer.
  • Figure 3.2: Example \ref{['ex:StereoS2']}: on the left, some flow lines under $(\Pi_N^{-1})_* X$; and on the right, some flow lines under $(\Pi_N^{-1})_* Y$. In between, two steps of the homotopy that connects $(\Pi_N^{-1})_* X$ to $(\Pi_N^{-1})_* Y$, through vector fields that render $S$ GAS on $\mathbb{S}^2\setminus \{N\}$.

Theorems & Definitions (20)

  • Example 1.1: Trivial convex combinations can fail
  • Remark 1.2: On global stability
  • Definition 1.3: Strong Lyapunov pairs ref:clarke1998asymptotic
  • Example 1.4: A semiflow corresponding to a vector field with bounded discontinuities
  • Example 1.5: An irregular gradient flow
  • Example 2.1: Stability-preserving homotopies for $n=1$
  • Example 2.2: Stability preserving homotopies for linear ODEs
  • Proposition 3.2: Strong global asymptotic stability and homotopic semiflows on $\mathbb{R}^n$
  • proof
  • Example 3.3: Homotopic vector fields, preserving stability
  • ...and 10 more