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A differential Lyapunov framework for contraction analysis

Fulvio Forni, Rodolphe Sepulchre

arXiv:1208.2943v4eess.SYmath.DGmath.DS

TL;DR

The paper addresses how to extend Lyapunov stability analysis from convergence toward a specific solution to contraction between arbitrary solution pairs. It lifts Lyapunov functions to the tangent bundle, equips the state space with a Finsler structure, and integrates infinitesimal contraction to obtain global incremental-stability conclusions. The framework also covers horizontal contraction for systems with symmetries and quotient structures.

  • Problem

    Incremental stability replaces convergence to one target solution with contraction between arbitrary solution pairs, motivating a Lyapunov-style analysis of this stronger property.

  • Method

    The paper lifts Lyapunov functions to the tangent bundle as Finsler-Lyapunov functions and uses their integrated Finsler metrics to connect infinitesimal contraction with distances between solutions.

  • Results

    The differential Lyapunov theorem provides sufficient pointwise conditions for incremental stability, incremental asymptotic stability, and incremental exponential stability, while horizontal contraction handles selected tangent subspaces.

  • Takeaways & Limitations

    The framework unifies differential-geometric contraction approaches and extends contraction analysis to systems with symmetries through horizontal contraction and quotient systems.

Abstract

from arXiv · show

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.

I. INTRODUCTION

The paper seeks an incremental counterpart to Lyapunov stability, replacing convergence to one target with contraction between solution pairs. It bridges Lyapunov and contraction theory by lifting analysis to tangent spaces and integrating local contraction into global distances, including systems with symmetries.

  • I. INTRODUCTION: Incremental stability requires convergence or contraction between any pair of solutions, so solutions forget their initial conditions.The paper motivates this stronger property through applications including tracking, regulation, observer design, coordination, and synchronization.
  • I. INTRODUCTION: The framework lifts a Lyapunov function from the manifold state-space to its tangent bundle and calls the result a Finsler-Lyapunov function.The lifted function endows the manifold with a Finsler structure suitable for constructing distances by integration.
  • I. INTRODUCTION: Pointwise decay of the Finsler-Lyapunov function in tangent space yields decay of an integrated distance between solutions, proving incremental stability.This substitutes a local construction for the often intractable explicit construction of a global distance.
  • I. INTRODUCTION: The differential Lyapunov framework aims to make the broader body of Lyapunov theory available to contraction analysis.The paper illustrates this program with an extension based on LaSalle’s invariance principle.
  • I. INTRODUCTION: Horizontal contraction verifies decay only on a horizontal tangent subspace, allowing symmetry directions without expected contraction to be disregarded.The paper relates this weaker notion to applications such as tracking, observer design, and synchronization.

IV. FINSLER-LYAPUNOV FUNCTIONS

Finsler-Lyapunov functions measure tangent-vector length through a Finsler structure whose integrated curve lengths define distances. The section specifies the required geometric properties and shows why arbitrary tangent-bundle functions may fail to induce valid distances.

  • IV. FINSLER-LYAPUNOV FUNCTIONS: A candidate Finsler-Lyapunov function is a C1 tangent-bundle function associated with a Finsler structure satisfying positivity, homogeneity, and strict convexity conditions.The function combines Lyapunov-function methodology with an asymmetric norm on each tangent space.
  • IV. FINSLER-LYAPUNOV FUNCTIONS: The Finsler structure associated with V induces a distance by integrating F along piecewise C1 curves and taking the infimum over connecting curves.This relation is essential because it converts local tangent-space information into a well-defined global distance.
  • Examples: For a Riemannian structure, V is the quadratic tangent-vector measure and the induced distance is the length of the geodesic connecting the two points.With P(x)=I, V reduces to |δx|^2 and the straight-line curve gives the Euclidean distance squared.
  • Examples: A constant V(x,δx)=1 cannot produce a valid distance through direct integration because reparameterization can make the infimum zero for distinct points.The resulting function satisfies non-negativity and the triangle inequality but fails definiteness.
  • Examples: The form V(x,δx)=|δx|^p1+|δx|^p2 is admissible only when p1=p2; otherwise no compatible Finsler structure satisfies the required homogeneity.Integrating V does not provide a distance unless the exponents coincide.
  • IV. FINSLER-LYAPUNOV FUNCTIONS: The induced Finsler distance can be asymmetric, with d(x,y)≠d(y,x); absolute homogeneity restores symmetry but excludes some metrics such as Randers metrics.Thus, requiring a symmetric distance reduces the generality of the admissible Finsler-Lyapunov class.

V. A FINSLER-LYAPUNOV THEOREM

The theorem lifts Lyapunov analysis to the tangent bundle: contraction of a Finsler-Lyapunov function along variational dynamics integrates into shrinking distances between solutions. It establishes incremental stability variants and extends the framework to horizontal contraction and nonconstant geometric structures.

  • Theorem: Theorem 1 yields incremental stability when α is zero, incremental asymptotic stability when α is a class K function, and incremental exponential stability when α(s) = λs > 0.These properties follow from the corresponding decay conditions imposed on the lifted Lyapunov function.
  • Theorem: Contraction of infinitesimal neighborhoods integrates along solution curves to make the distance between any pair of solutions shrink to zero.The induced distance is obtained by integrating the Finsler structure along curves connecting states.
  • Extensions: Horizontal contraction weakens the pointwise decay requirement by restricting it to a horizontal subspace, allowing noncontracting symmetry directions.The framework is motivated as a unification of Riemannian and matrix-measure approaches to contraction.
  • Example: A nonconstant Riemannian structure captures a maximal contracting region on S1 \ {π}, whereas the constant structure provides only incremental stability on a boundary-inclusive compact set.For compact sets excluding π, the state-dependent choice V2 yields incremental exponential stability; including points where cos(ϑ) = 0 weakens the conclusion.
  • Proof: The proof uses smooth curves between initial conditions and their parameterized solutions, avoiding any assumption that geodesics exist.Virtual displacements are tangent vectors to the evolving family of curves, and their contraction yields contraction of the induced distance.
  • Theorem: A Finsler-Lyapunov function on TM decreases along the variational system, providing a differential analogue of Lyapunov’s second theorem.The function induces a Finsler structure on the manifold, while the variational dynamics describe infinitesimal displacements.

V (Z)(H(Y )). Thus,

The coordinate condition is invariant under changes of coordinates, so the contraction criterion is geometric rather than tied to one chart.

  • Equation (26) proves that the contraction condition is independent of the chosen coordinates.

A. Riemannian contraction, matrix measure contraction, and incremental stability

The differential Finsler-Lyapunov framework contains matrix-measure, Riemannian, and incremental Lyapunov conditions as related special cases. It uses local contraction on the tangent bundle to infer global distance contraction by integration.

  • Comparison: The proof strategy generalizes earlier Euclidean and Riemannian contraction arguments to general manifolds and Finsler structures.
  • Matrix-measure contraction: Matrix-measure contraction implies incremental exponential stability and fits the framework with V(x, δx) = |δx| and α(s) = cs.For convex forward-invariant sets, the induced distance is the underlying norm; mildly regular nonconvex sets yield an equivalent bounded distance.
  • Riemannian contraction: Quadratic Riemannian structures and matrix inequalities are particular cases of the differential framework for suitable positive-definite matrices.The paper relates conditions involving P, Q, and the Jacobian to incremental exponential stability under a constant Riemannian structure.
  • Riemannian contraction: The framework also covers contraction conditions based on state-dependent quadratic structures M(x), connecting them directly to the lifted condition on V.The relation follows by choosing V(x, δx) = δxT M(x)δx.
  • Incremental stability: Incremental Lyapunov functions study the distance between two states directly, whereas the differential approach contracts infinitesimal neighborhoods and recovers global distance through integration.Both approaches fall within the proposed differential Finsler-Lyapunov framework, although their direct constructions differ.

B. Contractive systems forget initial conditions

Contractive systems lose dependence on initial conditions: bounded solutions converge toward one another and therefore toward a unique or input-induced steady-state solution under the stated assumptions. Virtual systems extend this consequence to tracking and state-estimation settings.

  • Contractive systems forget initial conditions: Under standard completeness assumptions, all bounded solutions of a contractive system converge to a unique steady-state solution.This property supports control applications such as tracking and observer design.
  • Contractive systems forget initial conditions: For an exogenous signal w, incremental asymptotic stability makes every bounded solution converge toward the steady-state solution induced by w.The induced steady state satisfies ˙x∗(t) = f(x∗(t), w(t)).
  • Virtual systems: A virtual system ẋ-style dynamics with ˆf(t, x, x) = f(t, x) can represent tracking control or observer dynamics through state feedback or output injection.The reference trajectory is embedded as a solution of the virtual system.
  • Virtual systems: If the virtual system satisfies the theorem uniformly in the reference state, each of its solutions converges asymptotically to the corresponding reference solution.The proposition applies when the relevant sets are connected and forward invariant and the Finsler-Lyapunov condition holds.
  • Virtual systems: The virtual-system decomposition is useful for tracking, state estimation, and analysis because incremental convergence transfers to convergence toward the reference trajectory.

VII. LASALLE-LIKE RELAXATIONS

The paper extends LaSalle’s invariance principle to the differential Lyapunov framework, using nonincreasing Finsler-Lyapunov functions and invariant sets to establish incremental asymptotic stability. The extension is developed for time-invariant differential equations and illustrated with a linear system and a boost-converter model.

  • VII. LASALLE-LIKE RELAXATIONS: LaSalle’s relaxation extends to Finsler-Lyapunov functions by replacing strict decay with invariance of the largest set where the decay rate vanishes.The analysis is developed for time-invariant systems ˙x = f(x).
  • VII. LASALLE-LIKE RELAXATIONS: A bounded variational-system solution has a nonempty compact invariant positive limit set because incremental stability bounds trajectories and tangent displacements.The proof uses forward invariance of C, boundedness, and monotonicity of V.
  • VII. LASALLE-LIKE RELAXATIONS: If the largest invariant subset of Π is C × {0}, every tangent displacement converges to zero, yielding incremental asymptotic stability on C.Here Π is the set where α vanishes, and the limiting argument forces V to converge to zero.
  • VII. LASALLE-LIKE RELAXATIONS: The authors state that an invariance principle had not appeared previously in contraction theory, highlighting the potential of a Lyapunov framework for contraction analysis.The result is presented as an extension of the classical LaSalle theorem.
  • VII. LASALLE-LIKE RELAXATIONS: The linear example applies Theorem 2 to an incremental-energy Finsler-Lyapunov function and obtains Πτ = R2 × {0} for any τ > 0.The resulting linear system is incrementally asymptotically stable, and in fact exponentially stable.

A. Contraction and symmetries

The paper generalizes contraction to selected horizontal directions by defining horizontal Finsler-Lyapunov functions on a tangent-bundle decomposition. The resulting pseudo-distance supports stability conclusions ranging from nonexpansion to exponential contraction, including systems with symmetry directions.

  • A. Contraction and symmetries: Horizontal contraction restricts infinitesimal decay checks to a horizontal subspace, allowing noncontracting symmetry directions to be excluded.This makes the framework relevant to applications such as synchronization.
  • A. Contraction and symmetries: Horizontal Finsler-Lyapunov functions are positive only on selected horizontal subspaces Hx ⊆ TxM, which determine the directions represented by the induced Finsler structure.The tangent space is decomposed into complementary vertical and horizontal distributions.
  • A. Contraction and symmetries: The induced metric is generally a pseudo-distance because vertical directions have zero length and distinct states can therefore have zero distance.The pseudo-distance measures only horizontal components of connecting curves.
  • A. Contraction and symmetries: If α(s) = 0, the dynamics do not expand the pseudo-distance; if α is a K function, they asymptotically contract it; and if α(s) = λs > 0, they exponentially contract it.The exponential case satisfies d(ψt0(t, x1), ψt0(t, x2)) ≤ Ke−λ(t−t0)d(x1, x2).
  • A. Contraction and symmetries: Invariance of the horizontal distribution lets the horizontal projection evolve consistently along the dynamics, so Theorem 3’s conclusions follow from the differential contraction condition.The proof decomposes tangent displacements into horizontal and vertical components and uses the projection identity.
  • A. Contraction and symmetries: The framework permits piecewise continuously differentiable and locally Lipschitz Finsler-Lyapunov functions when the required inequality holds almost everywhere.This is given as a relaxation of the regularity assumption in Definition 4.

B. Contraction on quotient manifolds

Horizontal contraction provides a way to analyze systems on quotient manifolds by ignoring tangent directions along symmetry fibers. When the induced pseudo-distance separates quotient classes, contraction in the total space becomes incremental stability of the quotient system.

  • B. Contraction on quotient manifolds: A quotient-system representation preserves equivalence classes under the dynamics, with vertical directions tangent to fibers and horizontal directions representing quotient displacements.Consensus dynamics provide an example where the vertical space is Span({1}) and the horizontal space is its orthogonal complement.
  • B. Contraction on quotient manifolds: If the pseudo-distance is nonzero between distinct equivalence classes, it becomes a distance on the quotient manifold.The quotient distance is defined through equivalence classes and inherits separation from condition (48).
  • B. Contraction on quotient manifolds: Under this separation condition, nonexpansion, asymptotic contraction, and exponential contraction of the lifted system imply the corresponding incremental stability properties of the quotient system.The three cases correspond respectively to α(s) = 0, α a K function, and α(s) = λs > 0.
  • B. Contraction on quotient manifolds: A fiber-invariant Finsler structure supplies a sufficient condition for the pseudo-distance on the total space to define a distance on the quotient.The condition requires invariance under fiber functions and supports applications including tracking, coordination, and synchronization.
  • B. Contraction on quotient manifolds: For consensus systems, V is non-increasing along tangent dynamics and decreases exponentially under uniform connectivity conditions.Integration then gives incremental exponential stability of the quotient system and exponential convergence of solutions to [0].
  • B. Contraction on quotient manifolds: For phase synchronization, a constant horizontal quadratic function contracts in the forward-invariant region where pairwise phase differences are less than Π/2.A nonconstant horizontal metric can establish almost-global contraction, and the resulting quotient analysis excludes balanced phase and saddle points.

C. Forward contraction

Forward contraction applies contraction analysis to selected horizontal directions, including the system’s flow direction, and supports attractor analysis under boundedness assumptions.

  • Horizontal contraction is not restricted to quotient systems or systems with first integrals.
  • Forward contraction defines convergence between each trajectory and its time-shifted version as time tends to infinity.
  • The horizontal distribution Hx = Span({f(x)}) is invariant along the dynamics, enabling attractor analysis based on forward contraction.
  • Under Theorem 4, forward contraction rules out periodic orbits within a forward invariant set.
  • If solutions are bounded, the absence of periodic orbits supports asymptotic attraction to a forward invariant attractor A.
  • The proof uses lower bounds on the horizontal Finsler-Lyapunov function and its decay estimate to derive contradictions on omega-limit sets.

IX. CONCLUSIONS

The paper develops a differential Lyapunov framework that turns local tangent-bundle contraction into global distances between solutions. It presents Finsler geometry and horizontal contraction as extensions supporting broader contraction analysis and future Lyapunov-based developments.

  • The framework extends the classical Lyapunov theorem from stability to incremental stability by lifting the Lyapunov function to the tangent bundle.
  • Integrating the lifted local function along curves constructs a decreasing global distance between solutions.
  • Finsler geometry is presented as a natural framework unifying Riemannian and matrix-measure approaches to contraction.
  • Formulating results on differentiable manifolds addresses incremental-stability questions arising in nonlinear spaces.
  • The paper illustrates extending Lyapunov tools to contraction analysis through LaSalle’s invariance principle and anticipates further generalizations.
  • Horizontal contraction provides a differential-geometric framework for systems with symmetries by disregarding symmetry directions without expected contraction.
  • The authors anticipate further developments beyond the basic theory presented.
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