A generalized global Hartman-Grobman theorem for asymptotically stable semiflows

1. Problem Statement

The paper addresses a fundamental limitation in the classical Hartman-Grobman theorem, which states that a nonlinear dynamical system is locally topologically conjugate to its linearization near a hyperbolic equilibrium point.

• Limitations of Classical Theory: The classical theorem is strictly local and relies on hyperbolicity (eigenvalues with non-zero real parts). It does not apply globally, nor does it handle non-hyperbolic equilibria or systems with finite-time stability.
• The Gap: Recent work by Kvalheim and Sontag extended this to a global setting for continuous vector fields by leveraging Lyapunov functions to bypass the hyperbolicity requirement. However, their result assumes continuity at the equilibrium.
• The Specific Challenge: Many modern control systems (e.g., those with discontinuous feedback, sliding mode control, or finite-time stabilizers) generate semiflows that are discontinuous at the equilibrium or exhibit finite-time stability. In these cases, the standard flow definition fails (solutions cannot be extended backward in time indefinitely), and the classical linearization intuition breaks down. The paper asks: To what extent can these discontinuous, finite-time stable semiflows be "linearized" globally?

2. Methodology

The author employs a topological approach, extending the framework of Kvalheim and Sontag to handle discontinuous vector fields and semiflows.

• System Class: The study focuses on vector fields $X$ on a smooth manifold $M$ that satisfy Assumption II.1:
  • Locally essentially bounded.
  • Possibly set-valued (discontinuous) at the equilibrium point $x^*$.
  • Locally Lipschitz on $M \setminus \{x^*\}$.
  • Solutions are interpreted in the sense of Filippov (differential inclusions), ensuring existence and uniqueness of solutions for the attractor.
• Topological Tools:
  • Lyapunov Functions: The existence of a smooth converse Lyapunov function $V$ is guaranteed for asymptotically stable systems under these conditions.
  • Level Sets: The method utilizes level sets of $V$ (specifically $L_\epsilon = V^{-1}(\epsilon)$) which are homeomorphic to spheres ($S^{n-1}$).
  • Time-to-Reach Maps: Instead of relying on global backward time (which doesn't exist for finite-time stable systems), the proof constructs a map based on the time required to reach a specific level set $L_\epsilon$ from a point $x$.
• Construction of the Homeomorphism:
  • The author constructs a homeomorphism $h: B(x^*) \to \mathbb{R}^n$ that maps the basin of attraction to Euclidean space.
  • The map is defined piecewise:
    1. Outside a small neighborhood ($U_\epsilon$): The map uses the time $\tau(x)$ to reach the level set $L_\epsilon$ and a projection $P$ onto the sphere.
    2. Inside the neighborhood: The map is rescaled using the Lyapunov function value to ensure continuity at the equilibrium ($h(x^*) = 0$).
  • Crucially, the construction avoids time reparametrization, preserving the original time scale of the dynamics.

3. Key Contributions

1. Generalization to Discontinuous Vector Fields: The paper extends the global Hartman-Grobman theorem to systems where the vector field is discontinuous at the equilibrium, specifically covering Filippov solutions.
2. Handling Semiflows and Finite-Time Stability: It provides a rigorous framework for semiflows (where backward time is not defined for all $t$). It demonstrates that even systems with finite-time stability (where trajectories reach the equilibrium in finite time) can be topologically conjugated to a linear system, provided the conjugacy is restricted to the region outside an arbitrarily small neighborhood of the equilibrium.
3. "Practical" Linearization: The main result (Theorem II.2) establishes that any asymptotically stable semiflow is practically indistinguishable from the canonical linear system $\dot{y} = -y$ (or $e^{-t}y$) outside any chosen neighborhood of the equilibrium.
4. Explicit Construction: The proof is semi-constructive, providing explicit formulas for the homeomorphism and its inverse, particularly for the case of normalized vector fields on $\mathbb{R}^n$.

4. Main Results

Theorem II.2 (Global Hartman-Grobman for Semiflows):
For an asymptotically stable equilibrium $x^*$ with basin of attraction $B(x^*)$ generated by a vector field satisfying the regularity assumptions:

• Case I (Global Domain): If the system allows backward extension indefinitely outside $x^*$, there exists a homeomorphism $h_r$ such that the dynamics are conjugate to $\dot{y} = -y$ for all $y$ outside a ball $B(0, r)$. The conjugacy holds for a time duration determined by a class-$\mathcal{K}_\infty$ function $\gamma_r$.
• Case II (Bounded Domain): If the domain is bounded or backward extension is limited, a similar conjugacy exists on an annular region $cl(B(0, R)) \setminus B(0, r)$.

Key Implications of the Result:

• Topological Equivalence: All asymptotically stable semiflows (including finite-time stable ones) are topologically equivalent to the canonical linear decay $\dot{y} = -y$ outside an arbitrarily small neighborhood of the origin.
• Preservation of Stability Type: The homeomorphism preserves the nature of stability. If the original system is finite-time stable, the conjugated system will exhibit finite-time stability inside the small neighborhood $B(0, r)$, while appearing as asymptotic (exponential) decay outside.
• No Time Reparametrization: The linearization is achieved via a spatial coordinate change (homeomorphism) only; the time variable $t$ is not rescaled.

5. Significance and Applications

• Bridging Theory and Practice: The result validates the use of linear analysis tools for a broader class of nonlinear systems, particularly those arising in robust control and sliding mode control where discontinuities are intentional.
• Koopman Operator Theory: The paper addresses a central open challenge in Koopman theory: finding global coordinate systems where dynamics appear linear. It suggests that while global linearization of all dynamics is impossible (due to finite-time vs. asymptotic differences), "practical" global linearization is achievable.
• Data-Driven Learning: The finding that stable systems are "practically indistinguishable" implies that data-driven models trained on stable systems might converge to the same topological structure, regardless of whether the underlying system is finite-time or asymptotically stable, as long as the data does not resolve the immediate neighborhood of the equilibrium.
• Topological Classification: It refines the understanding of how to distinguish dynamical systems, showing that the distinction between finite-time and asymptotic stability is a local property near the equilibrium, while the global structure is topologically identical.

6. Illustrative Example

The paper uses the normalized vector field $\dot{x} = -x/\|x\|^2$ (which reaches 0 in finite time) as a running example.

• Transformation: The constructed homeomorphism $h_r$ maps the finite-time decay of the original system to a trajectory that decays exponentially ($e^{-t}$) outside a ball $B(0, r)$ and transitions to a finite-time decay behavior inside $B(0, r)$.
• Visualization: The transformed system looks like a standard linear decay until it hits the "inner" region, where the finite-time nature is recovered, demonstrating the "practical" nature of the linearization.

In summary, Jongeneel provides a powerful topological tool that generalizes the Hartman-Grobman theorem to discontinuous and finite-time stable systems, proving that their global behavior is topologically equivalent to linear decay, with the specific stability characteristics (finite-time vs. asymptotic) confined to an arbitrarily small neighborhood of the equilibrium.