# The Differential Stability Problem. Part I: Geometric Contraction for Nonlinear Time-Varying Systems

Mario Spirito

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Mario Spirito. The Differential Stability Problem. Part I: Geometric Contraction for Nonlinear Time-Varying Systems. 2026. hal-05528162

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# The Differential Stability Problem. Part I:
Geometric Contraction for Nonlinear
Time-Varying Systems

Mario Spirito

**Abstract**—This paper develops a unified framework for the analysis of stability, convergence, and con- tractivity of nonlinear time-varying dynamical systems from a geometric perspective. After introducing the re- quired mathematical background (including metric notions, parametrized curves on differentiable manifolds, and Rie- mannian length), we formalize system dynamics on man- ifolds with time-varying vector fields and revisit classical variational and prolonged dynamics in this general setting.

Motivated by the limitations of standard variational equa-
tions in time-varying contexts, we generalize the notion of
variational state by accounting for perturbations in both
the initial conditions and the time argument. The varia-
tional state concept is further extended through a double-
parameterized family of curves connecting system trajecto-
ries, which enables a rigorous characterization of contrac-
tion via time-varying, state-dependent metrics.

Within this framework, we establish regional contraction
conditions for nonlinear time-varying systems and re interpret Lyapunov functions as geometric objects induced by
the underlying metric space associated with the system.

Overall, the proposed approach provides a coherent geo-
metric foundation for trajectory-based stability analysis for
a broad class of nonlinear time-varying systems.

**Index Terms**—Contraction theory, Lyapunov methods, Nonlinear systems, Riemannian Metric, Stability analysis, Time-varying systems, Variational dynamics.

# I. INTRODUCTION

IN stability theory for nonlinear time-varying systems, con-
traction theory has emerged as a powerful and unifying
framework for analyzing stability, convergence, synchroniza-
tion, and robustness properties of nonlinear dynamical sys-
tems. In contrast to classical Lyapunov theory—which studies
convergence toward a distinguished equilibrium—contraction
theory provides a differential, trajectory-based viewpoint in
which stability is characterized through the incremental behav-
ior of solutions. Early precursors of the contraction paradigm
can be traced to the works of Demidovich and Willems [1], [2],
which established sufficient conditions for global convergence
using constant quadratic metrics. These ideas were later revis-
ited and reformulated in the context of nonlinear systems and
observer design in [3]–[5], giving rise to the modern concepts
of convergent systems and incremental stability [6].

A major milestone in the formal development of contraction
theory was the extension of differential stability analysis to
general Riemannian metrics in the seminal contribution [7].
This geometric viewpoint was later expanded into a differ-
ential Lyapunov theory in [8], allowing stability, incremental
stability, and invariance properties to be characterized through
the infinitesimal behavior of virtual displacements. Further
geometric generalizations include transverse contraction for
periodic orbits and hybrid dynamics [9], [10], contraction-
based synchronization and output stability [11], [12], and
extensions to non-Euclidean norms (such as $L_1$ and $L_\infty$ metric
structures) and Finsler-type metrics [8], [13], [14]. In parallel,
in [15] contraction theory was enriched by connections with
differential Hamiltonian structures [16], inequality-constrained
dynamics [17], higher-order differential analysis [18], and
nonlinear Hamiltonian systems [19].

At its core, contraction theory studies the variational dy-
namics of a nonlinear system, i.e., the evolution of infinites-
imal displacements along the flow. By exploiting differen-
tial Lyapunov functions—also called Lyapunov-Finsler met-
rics [8]—one can establish incremental exponential stability,
characterize uniqueness of attractors, analyze robustness, or
infer convergence to invariant manifolds via partial or semi-
contraction [7]. The deep interplay between contraction, dis-
sipativity, and energy-based modeling has been highlighted in
[20], [21], where connections with port-Hamiltonian structures
were formalized.

The contraction paradigm has had a broad impact across
many fields. In control and robotics, contraction metrics
have enabled the design of stabilizing nonlinear controllers,
observers, and distributed feedback laws [12], [22]–[26].
Applications include trajectory tracking [27], underactuated
robotics [28], hybrid locomotion [10], and noncausal iterative
learning control [29]. While applications to robotic motion
planning and observer design are demonstrated in [30], [31].
Contraction-based observers and metric observers were first
introduced in [32]. In neural networks and machine learning,
contraction analysis has been employed to study the stabil-
ity of recurrent networks and the convergence of training
algorithms [33], [34]. In biological and chemical systems,
contraction principles have been employed to analyze stability
and adaptation [7], [35], [36]. Contraction theory has also
been applied to optimization and estimation, including gradient
flows, observer design, and networked systems under commu-
nication constraints [12], [37], as well as in traffic models [38], [39]. Additionally, contraction principles have been leveraged to study input-to-state stability (ISS) and incremental ISS in nonlinear polynomial dynamics and networked systems [39]–[44].

Collectively, these advances demonstrate that contraction theory not only unifies classical and modern notions of stability, incremental stability, and robustness, but also furnishes practically relevant tools for the design of stable nonlinear systems.

## A. Paper contribution

The present paper advances the analysis of contraction theory in two directions. On the one hand, we generalize the notion of variational state by incorporating perturbations not only in the initial condition but also in the temporal domain. On the other hand, the notion of variational state is “orthogonally” extended via double-parameter families of curves connecting trajectories of the system, thereby enabling us to formally characterize contraction for time-varying nonlinear systems through a time-varying metric. Within this framework, we revisit the role of Lyapunov functions in stability analysis and reinterpret them in light of the extended variational dynamics.

To support these developments, we introduce the necessary mathematical background, such as the concept of distance, metric and $L^1$ spaces, of double-parameter curves on differentiable manifolds and the associated Riemannian length. A structured review and comparison of classical, incremental, and differential notions of stability is also provided to position the results within the broader stability framework.

## B. Structure of the paper

The remainder of the paper is organized as follows. We begin in Section II with the mathematical foundation required for our developments. This includes metric spaces, and $L^1$ functions space, parametrized curves on differentiable manifolds and the related notion of Riemannian length.

Next, Section III formalizes the system-theoretic framework. We consider dynamics on manifolds under a time-varying vector field, discuss the associated flow map, recall the notion of variational state and variational dynamics, and introduce the prolonged dynamics.

In Section IV we compare and unify different notions of stability with an overview of these concepts for nonlinear systems. Classical stability concepts, including simple, asymptotic, and exponential stability, are reviewed in Section IV-A, while incremental stability and convergent dynamics in Section IV-B. Contractive dynamics and differential stability notions are described and introduced in Section IV-C.

The generalized variational dynamics is introduced in Section V. After motivating the need for this generalization in the time-varying nonlinear setting, we show that variations may arise both from the initial conditions and from temporal perturbations, with the classical variational dynamics and pure temporal variations appearing as special cases.

Section VI develops the extended variational framework. We define two-parameter curves that connect trajectories

initialized at distinct points. The main contraction result is established in this section by proving the vanishing length of connecting curves on a region $\mathcal{X}$ under an appropriately defined time-varying state-dependent metric. We also provide an interpretation of Lyapunov functions in Section VI-A.

The paper concludes in Section VII with a summary of the main contributions and a discussion of future research directions.

## C. Notation

The set of real (complex) numbers is denoted by $\mathbb{R}(\mathbb{C})$, and $\mathbb{N}$ denotes the set of natural numbers. Given $n \in \mathbb{N}$, vectors are assumed to belong to $\mathbb{R}^n$ and matrices to $\mathbb{R}^{n \times n}$. The symbol $\| \cdot \|$ is used as the standard Euclidean norm for vectors and the induced 2-norm for matrices, unless stated otherwise.

A function $\alpha : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is said to be of class $\mathcal{K}$ if it is continuous, strictly increasing, and satisfies $\alpha(0) = 0$. It is of class $\mathcal{K}_{\infty}$ if it is of class $\mathcal{K}$ and $\alpha(r) \xrightarrow{r \to \infty} = \infty$. It is of class $\mathcal{KL}$ if $\alpha(\cdot, t)$ is of class $\mathcal{K}$ for each fixed $t \ge 0$, and $\alpha(r, \cdot)$ is decreasing and converges to zero as $t \to \infty$ for each fixed $r \ge 0$. $C^k(\mathcal{X})$ denotes the class of continuously $k$ times differentiable functions on $\mathcal{X}$.

Given a sufficiently smooth function $f : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$, the Jacobian matrix with respect to the state variable is denoted by $\frac{\partial f(x, t)}{\partial x}\bigg|_{(x,t)}$.

For a matrix $M \in \mathbb{R}^{n \times n}$, $\sigma(M)$ denotes its spectrum, $\lambda_i(M)$ its $i$-th eigenvalue, and $\sigma_i(M)$ its $i$-th singular value, defined as $\sigma_i(M) = \sqrt{\lambda_i(M^\top M)}$. The minimum and maximum singular values are denoted by $\sigma_{\min}(M)$ and $\sigma_{\max}(M)$, respectively, and the condition number of $M$ is $\mu(M) = \frac{\sigma_{\max}(M)}{\sigma_{\min}(M)}$. The 2-norm for matrices comes then directly as $\|M\| = \sigma_{\max}(M)$ and $\|M^{-1}\| = \sigma_{\min}(M)^{-1}$.

For any matrix $M \in \mathbb{R}^{n \times n}$, its skew-symmetric and symmetric parts are defined as skew$(M) = \frac{1}{2}(M - M^\top)$ and sym$(M) = \frac{1}{2}(M + M^\top)$. A matrix $P$ is said to be symmetric if $P = P^\top$. A symmetric matrix $P \in \mathbb{R}^{n \times n}$ is positive definite, denoted by $P > 0$, if $x^\top P x > 0$, for all $x \in \mathbb{R}^n \setminus \{0\}$.

# II. SOME PRELIMINARIES

We begin by recalling a number of fundamental notions from functional analysis, following standard references [45]. These concepts provide the basic mathematical setting used throughout the paper.

**Definition II.-1 (Distance and metric).** Let $\mathcal{X}$ be a set. A map $d: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ satisfying, for all $x, y, z \in \mathcal{X}$, the following properties:

a) **Non-negativity:** $0 \le d(x, y) < \infty$,

b) **Identity of indiscernibles:** $d(x, y) = 0 \iff x = y$,

c) **Symmetry:** $d(x, y) = d(y, x)$,

d) **Triangle inequality:** $d(x, z) \le d(x, y) + d(y, z)$,

is called a *distance* on $\mathcal{X}$ and we say that the space $\mathcal{X}$ is endowed with a *metric*. The pair $(\mathcal{X}, d)$ is then called a *metric space*.
The first application of distance is the definition of ball of radius $r$ and centered in $x_0$.

**Definition II.-2 (Ball).** Let $(\mathcal{X}, d)$ be a metric space, and let $x_0 \in \mathcal{X}$ be a point of this space. The open ball of radius $r > 0$ centered at $x_0$ is the set:

$$B_r(x_0) = \{x \in \mathcal{X} : d(x, x_0) < r\}. \quad \llap{^\blacksquare}$$

**Remark II.1** In the context of dynamical systems, the term *local* refers to properties that hold within a neighborhood of a point $x_0 \in \mathcal{X}$. Such neighborhoods are often taken to be balls $B_r(x_0)$ with sufficiently small radius $r$.

The definition of distance also allows us to define a contraction map.

**Definition II.-3 (Contraction map [45, Sec. 5.2].)** Let $(\mathcal{X}, d)$ be a metric space. A function $f: \mathcal{X} \to \mathcal{X}$ is a contraction if there exists a constant $\alpha \in [0, 1)$ such that

$$d(f(x), f(y)) \le \alpha d(x, y) \quad \forall x, y \in \mathcal{X}. \quad \llap{^\blacksquare}$$

The contraction property of $f$ ensures that repeated applications of $f$ bring any two points progressively closer together. Consider the sequence $\{x_k\}_{k \in \mathbb{N}}$ generated by

$$x_{k+1} = f(x_k), \quad x_0 \in \mathcal{X} \text{ arbitrary.}$$

By the contraction property,

$$d(x_{k+1}, x_k) \le \alpha d(x_k, x_{k-1}) \le \dots \le \alpha^k d(x_1, x_0).$$

Since $\alpha \in [0, 1)$ the sequence $\{x_k\}_{k \in \mathbb{N}}$ is a *Cauchy sequence*<fnref n="1" /> in the metric space $(\mathcal{X}, d)$. Thus, if the space is complete, the sequence converges to a unique *fixed point* of $f$ in $\mathcal{X}$.

In a vector space $\mathcal{X}$, an inner product $\langle \cdot, \cdot \rangle$ induces a norm via $\|x\| = \sqrt{\langle x, x \rangle}$, and any norm induces a metric $d$ through $d(x, y) = \|x - y\|$. The converses do not hold in general: not every norm arises from an inner product, and not every metric is induced by a norm.

**Example II.1 (Riemannian metric)** Let $\mathcal{X}$ be an $n$-dimensional real vector space, and for each $x \in \mathcal{X}$, let $P(x) \in \mathbb{R}^{n \times n}$ be a continuous symmetric positive definite matrix, i.e., $P \in \mathcal{C}^0$ and $P(x) = P(x)^\top > 0$. Then the point-dependent inner product

$$\langle v, w \rangle_x = v^\top P(x) w, \quad \forall v, w \in \mathcal{X}. \quad (1)$$

defines a smoothly varying assignment $x \mapsto \langle \cdot, \cdot \rangle_x$, constituting a *Riemannian metric* on $\mathcal{X}$.

In applications, one often works with integrable and square-integrable functions defined on compact intervals. For our purposes, in the following definition, it is sufficient to assume that the functions under consideration admit the usual Riemann integral, so that expressions such as $\int_{\mathcal{I}} f(x) dx$ are well defined and finite for any compact interval $\mathcal{I} \subseteq \mathbb{R}$.

**Definition II.-4 (Space $\mathcal{L}^1(\mathcal{I})$).** The space $\mathcal{L}^1(\mathcal{I})$ is the set of all integrable functions on $\mathcal{I}$,

$$\mathcal{L}^1(\mathcal{I}) := \{ f : \mathcal{I} \to \mathbb{C} \mid \int_{\mathcal{I}} |f(x)| dx < \infty \},$$

equipped with the norm

$$\|f\|_1 := \int_{\mathcal{I}} |f(x)| dx. \quad \llap{^\blacksquare}$$

The following classical result will be used later when analyzing the integrability condition of a Lyapunov function.

**Theorem II.-1 (Dominated Convergence Theorem [46]).** Let $\mathcal{I} \subset \mathbb{R}$ be a compact interval and let $\{f_t\}_{t \ge t_0}$ be a family of functions $f_t: \mathcal{I} \to \mathbb{R}$ satisfying:

a) $f_t(s) \to 0$ pointwise for all $s \in \mathcal{I}$ as $t \to \infty$,

b) there exists an integrable function $F \in \mathcal{L}^1(\mathcal{I})$ such that

$$|f_t(s)| \le F(s), \quad \forall s \in \mathcal{I}, \forall t \ge t_0.$$

Then $f_t \in \mathcal{L}^1(\mathcal{I})$ for all $t \ge t_0$, and

$$\lim_{t \to \infty} \int_{\mathcal{I}} f_t(s) ds = 0. \quad \llap{^\blacksquare}$$

### A. Differentiable curve on manifolds

In the following, we introduce differentiable double-parameter curves $\gamma$ on a manifold $\mathcal{M}$ without explicitly relying on local charts. Because $\mathcal{M}$ is smoothly immersed in $\mathbb{R}^n$, we can represent curves using their coordinate expressions in the ambient Euclidean space, which simplifies presentation (see [47] for a rigorous development).

**Definition II-A.1 (Differentiable double-parameter curve).** Let $\mathcal{M}$ be an $m$-dimensional differentiable manifold of class $\mathcal{C}^k$, smoothly immersed in $\mathbb{R}^n$, i.e., $\mathcal{M} \subseteq \mathbb{R}^n$. A differentiable double-parameter curve in $\mathcal{M}$ is a map

$$\gamma : \mathcal{I} \times \mathbb{R} \to \mathcal{M}, \quad \gamma : (s, t) \mapsto (\gamma_1, \gamma_2, \dots, \gamma_n)(s, t),$$

defined for all $t \in \mathbb{R}$ on an interval $\mathcal{I}^2$, which is of class $\mathcal{C}^k$ as a map into $\mathbb{R}^n$.

Let $\Gamma_t(p, q)$ be the set of all continuous double-parameter curves $\gamma(s, t) : [a, b] \times \mathbb{R} \to \mathcal{M}$ and $\gamma \in \mathcal{C}^0$ lying on a manifold $\mathcal{M}$, satisfying the boundary conditions $\gamma(a, t) = p(t)$ and $\gamma(b, t) = q(t)$, where $p, q : \mathbb{R} \to \mathcal{M}$ are continuous time-dependent trajectories on $\mathcal{M}$. We write $\Gamma_t^k(p, q)$, for $k \in \mathbb{N}$, for the subset of curves $\gamma$ that are of class $\mathcal{C}^k$ in $(s, t) \in [a, b] \times \mathbb{R}$. **Remark II.2** The requirement $\gamma \in \Gamma_t^2$ implicitly imposes that the boundary trajectories satisfy $p, q \in \mathcal{C}^2$.

**Definition II-A.2 (Spatial and temporal velocities).** Let $\gamma \in \Gamma_t(p, q)$ and let $\mathcal{T}_{\gamma(s,t)}\mathcal{M}$ denote the tangent space to $\mathcal{M}$ at $\gamma(s, t)$. The *spatial velocity* of $\gamma$ at $(s, t) \in [a, b] \times \mathbb{R}$ is the tangent vector

$$\gamma'(s,t) = \frac{\partial \gamma}{\partial s}(s,t) \\ = \left( \frac{\partial \gamma_1}{\partial s}, \frac{\partial \gamma_2}{\partial s}, \dots, \frac{\partial \gamma_n}{\partial s} \right) (s,t) \in \mathcal{T}_{\gamma(s,t)}\mathcal{M} \subseteq \mathbb{R}^n.$$

<footnote n="1">¹For $k \to \infty$, the sequence converges to a fixed point $x^*$ that belongs to the metric space $\mathcal{X}$, if the space is complete.</footnote>

<footnote n="2">²It is convenient to think of $\mathcal{I}$ as $\mathcal{I} = [a, b]$, with $a, b \in \mathbb{R}$, or in its normalized form $\mathcal{I} = [0, 1]$.</footnote>
Similarly, the temporal velocity of $\gamma$ at $(s, t)$ is the vector

$$ 
\begin{aligned}
\dot{\gamma}(s,t) &= \frac{\partial \gamma}{\partial t}(s,t) \\
&= \left( \frac{\partial \gamma_1}{\partial t}, \frac{\partial \gamma_2}{\partial t}, \dots, \frac{\partial \gamma_n}{\partial t} \right) (s,t) \in \mathcal{T}_{\gamma(s,t)} \mathcal{M} \subseteq \mathbb{R}^n.
\end{aligned}
$$

Since the spatial and temporal parameters, respectively $s$ and $t$, are independent, it follows that $\frac{\partial \gamma}{\partial t} = \frac{d\gamma}{dt}$ for $i = s, t$.

We now extend the classical definition of the Riemannian distance to the setting of time-dependent metrics and double-parameter curves.

**Definition II-A.3** (Time-varying Riemannian distance and geodesic curve). Let $(\mathcal{M}, g_{\gamma,t})$ be a manifold endowed with a time-varying Riemannian metric $g_{\gamma,t}$. Let $\Gamma_t(p, q)$ be the family of double-parameter curves introduced above. The time-varying Riemannian distance $d_{g_{\gamma,t}}: \mathcal{M} \times \mathcal{M} \to \mathbb{R}_{\ge 0}$ between two trajectories $p, q: \mathbb{R} \to \mathcal{M}$ is defined as

$$ d_{g_{\gamma,t}}(p,q) = \inf_{\gamma \in \Gamma_t(p,q)} \mathcal{L}_{g_{\gamma,t}}(\gamma,t), \quad (2) $$

where

$$ \mathcal{L}_{g_{\gamma,t}}(\gamma,t) := \int_a^b \sqrt{g_{\gamma,t}(\gamma'(s,t), \gamma'(s,t))} \, ds $$

is called the time-varying Riemannian length of $\gamma$ with respect to $g_{\gamma,t}$. A curve $\gamma$ achieving the infimum in (2) is called a time-varying geodesic with respect to $g_{\gamma,t}$.

**Remark II.3** If the manifold $\mathcal{M} \subseteq \mathbb{R}^n$ is endowed with a time-varying and position-dependent metric of the form

$$ g_{x,t}(u,v) = u^{\top} P(x,t)v, $$

where $P(x,t) = P(x,t)^{\top} > 0$ for all $(x,t) \in \mathcal{M} \times \mathbb{R}$, then the Riemannian length explicitly reads as

$$ \mathcal{L}_{g_{\gamma,t}}(\gamma,t) = \int_a^b \sqrt{\gamma'(s,t)^{\top} P(\gamma(s,t),t) \gamma'(s,t)} \, ds. $$

In this setting, the induced distance $d_{g_{\gamma,t}}(p,q)$ depends on the time-varying positive definite metric field $P(x,t)$ and reflects the associated time-varying non-Euclidean geometry of the ambient space.

We conclude the subsection recalling Schwarz's theorem applied to the double-parameter curves $\gamma$.

**Theorem II-A.1** (Schwarz's theorem for double-parameter curves). Let $\gamma \in \Gamma_t^2(p,q)$, where $\Gamma_t^2(p,q)$ is the family of $C^2$ double parameter curves between the trajectories $p, q: \mathbb{R} \to \mathcal{M}$. Then the mixed partial derivatives of $\gamma$ commute, i.e.,

$$ \frac{\partial \gamma'}{\partial t}(s,t) = \frac{\partial \dot{\gamma}}{\partial s}(s,t), \quad \forall (s,t) \in [a,b] \times \mathbb{R}. $$

In double-parameter curves context, it is thus relevant to distinguish between spatial and temporal velocities.

**Remark II.4** From this point onward, for notational simplicity, we identify the smooth manifold $\mathcal{M}$ with its image in $\mathbb{R}^n$ and denote it by $\mathcal{X}$. Accordingly, the dynamical system $\dot{x} = f(x,t)$ is understood to be defined on $\mathcal{X} \subseteq \mathbb{R}^n$, while implicitly assuming that $\mathcal{X}$ is a smooth immersed submanifold of $\mathbb{R}^n$.

### III. THE DYNAMICAL SYSTEM CONTEXT

In this work, we consider a forward complete<fnref n="3" /> dynamical system formulated as the following initial value problem (IVP). Let $\mathcal{X} \subseteq \mathbb{R}^n$ be a smooth manifold and let $f: \mathcal{X} \times \mathbb{R} \to \mathcal{T}\mathcal{X}$ be a $C^1$ map in $(x,t)$. The system under consideration is

$$ \Sigma: \begin{cases} \dot{x}(t) = f(x(t),t), \\ x(t_0) = x_0, \end{cases} \quad (3) $$

where $x: \mathbb{R} \to \mathcal{X} \subseteq \mathbb{R}^n$ and $t \in [t_0, \infty) \subseteq \mathbb{R}$. The map $f$ is the system's vector field, and $x(t)$ denotes a local coordinate representation of a trajectory. System (3) is called inhomogeneous (or non-autonomous). If the dynamics does not depend explicitly on time, i.e.,

$$ \dot{x} = f(x), \quad x(t_0) = x_0, \quad (4) $$

then the system is referred to as homogeneous (or autonomous).

Given the IVP (3), we denote by $\varphi(x_0, t; t_0) \in \mathcal{X}$ the unique solution $x(t) = \varphi(x_0, t; t_0)$ starting at $x(t_0) = x_0 \in \mathcal{X}$ at time $t_0 \ge 0$. The map $\varphi: \mathcal{X} \times \mathbb{R} \to \mathcal{X}$ is referred to as the system flow.

The flow enjoys the following fundamental properties:

a) **Identity:** $\varphi(x_0, t_0; t_0) = x_0$ for all $x_0 \in \mathcal{X}$.

b) **Semigroup property:** for all $t_0 \le t_1 \le t_2$,

$$ \varphi(x_0, t_2; t_0) = \varphi(\varphi(x_0, t_1; t_0), t_2; t_1). $$

c) **Differentiability:** if $f(x,t)$ is $C^k$ in $x$, then $\varphi(x_0, t; t_0)$ is $C^k$ with respect to $x_0$.

The definition of flow allows to describe the (positively) invariant property of a subset.

**Definition III.-1** (Invariant and positively invariant subset). A subset $\bar{\mathcal{X}} \subseteq \mathcal{X} \subseteq \mathbb{R}^n$ is said to be invariant under the flow $\varphi$ of system (3) if, for every point $x_0 \in \bar{\mathcal{X}}$, the trajectory remains in $\bar{\mathcal{X}}$ for all backward and forward times, i.e.,

$$ \varphi(x_0, t; t_0) \in \bar{\mathcal{X}}, \quad \forall t \in \mathbb{R}. $$

Similarly, the set $\bar{\mathcal{X}}$ is said to be positively invariant if

$$ \varphi(x_0, t; t_0) \in \bar{\mathcal{X}}, \quad \forall t \ge t_0. $$

In other words, trajectories starting in a positively invariant set remain inside it for all future times.

In the following, $\mathcal{X}$ is regarded as forward invariant.

a) **Variational State:** To study the sensitivity of trajectories with respect to perturbations in the initial condition, we introduce a smooth family of solutions. For a reference initial state $x_0 \in \mathcal{X}$, consider a family of parameterized trajectories $\chi: (-\delta_{\chi}, \delta_{\chi}) \times \mathbb{R}_{\ge 0} \to \mathcal{X}$ such that, for each $\varepsilon \in (-\delta_{\chi}, \delta_{\chi})$, with $\delta_{\chi} \in \mathbb{R}_{>0}$ sufficiently small, the curve $t \mapsto \chi(\varepsilon, t)$ satisfies

$$ \dot{\chi}(\varepsilon, t) = f(\chi(\varepsilon, t), t), \quad \chi(\varepsilon, t_0) = x_0 + \varepsilon\nu, \quad (5) $$

where $\nu \in \mathbb{R}^n$ is an initial perturbation direction. The nominal trajectory is recovered for $\varepsilon = 0$, i.e., $\chi(0, t) = \varphi(x_0, t; t_0) = x(t)$.

<footnote n="3">³A system is called forward complete if, for every initial condition, the corresponding solution is defined for all forward times [48].</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Parametrized trajectories $\chi(\varepsilon, t)$ in the neighborhood of the nominal trajectory $x(t) = \chi(0, t)$. In both cases, the time evolution of $x(t)$ is shown together with two perturbed trajectories, $\chi(\delta\chi, t)$ and $\chi(-\delta\chi, t)$. The variational displacements $\delta x^{\pm}(t)$ are depicted as vectors orthogonal to the time axis, representing infinitesimal deviations along the family of trajectories.</figcaption></figure>

We can now introduce the notion of variational state and the associated variational dynamics. Starting from the parametrized family of trajectories $\chi: (-\delta\chi, \delta\chi) \times \mathbb{R}_{\ge 0} \to \mathcal{X}$, where $\chi(\varepsilon, t)$ denotes the solution of (5) corresponding to the perturbed initial condition $x_0 + \varepsilon\nu$, we define the variational state as

$$ \delta x^{\pm}(t) := \lim_{\varepsilon \to 0^{\pm}} \frac{\partial \chi(\varepsilon, t)}{\partial \varepsilon} = \frac{\partial \chi(\varepsilon, t)}{\partial \varepsilon} \Big|_{\varepsilon=0^{\pm}}. \quad (6) $$

Figure 1 provides a graphical illustration of the variational state, highlighting its role within the family of parameterized trajectories.

b) Variational and prolonged dynamics: The evolution of $\delta x(t)^4$ is governed by a linear time-varying system known as the variational dynamics, which captures system (3) infinitesimal behavior around the nominal solution $\varphi(x_0, t; t_0)$, i.e.,

$$ \mathcal{D}\Sigma: \left\{ \begin{array}{l} \dot{\delta x}(t) = \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \delta x(t), \\ \delta x(t_0) = \nu, \end{array} \right. \quad (7) $$

where $\nu \in \mathcal{T}_{x_0} \mathcal{X} \subseteq \mathbb{R}^n$ denotes an admissible perturbation direction at time $t_0$. Classically, and directly from the definition of variational state (6), this linear time-varying and state-dependent system characterizes the evolution of infinitesimal deviations from the reference trajectory $\varphi(x_0, t; t_0)$ and plays a fundamental role in the stability and local sensitivity analysis of dynamical systems [43], [44], [49].

In the general case, we consider the parallel configuration of systems (3) and (7), commonly referred to as the prolonged dynamics [50], [51]. The prolonged (also termed “extended” or “lifted” as in [52]) system associated with $\Sigma$ is the augmented 2n-dimensional system defined on the tangent bundle $\mathcal{T}\mathcal{X}$, consisting of $\Sigma$ and its variational dynamics $\mathcal{D}\Sigma$ in parallel. More precisely, the prolonged system dynamics is given by

$$ \mathcal{T}\Sigma: \left\{ \begin{array}{l} \dot{x} = f(x, t), \quad x(0) = x_0, \\ \dot{\delta x}(t) = \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \delta x(t), \quad \delta x(0) = \nu. \end{array} \right. \quad (8) $$

In principle, all subsequent analyses and stability properties should be formulated with respect to the prolonged system (8), since the variational dynamics (7) is well defined only along trajectories generated by the nonlinear flow $\varphi$. However, in the remainder, the evolution of the base trajectory $x(t) = \varphi(x_0, t; t_0)$ is always assumed to be implicitly governed by (3). Accordingly, for notational simplicity, we will refer exclusively to the variational dynamics, with the understanding that all quantities are evaluated along solutions of the underlying nonlinear system.

## IV. DIFFERENT NOTIONS OF STABILITY

In this section, we gather various notions of stability to contextualize the framework in which contraction theory is positioned. The first group includes classical Lyapunov-based concepts such as simple, asymptotic and exponential. The second group gathers more recent notions emphasizing the relative behavior between trajectories, such as incremental stability, convergent dynamics, and contractive systems. Finally, we discuss their differential counterparts, namely differential regional stability, differential regional asymptotic stability, and differential regional exponential stability.

### A. Classical notions of stability

Classical notions of stability, such as Lyapunov, asymptotic, and exponential stability, form the foundation for analyzing the local behavior of nonlinear dynamical systems around equilibrium points. For time-varying (inhomogeneous) systems, stronger notions such as uniform stability are often required to ensure that the stability properties hold uniformly with respect to the initial time $t_0$. References for classical Lyapunov, asymptotical, exponential stability can be found in [53, Sec. 4.1-4.3] and [54, Sec. 5.1]. Uniform stability is treated in [53, Sec. 4.5], [54, Sec. 5.2], and [55, Ch.5].

All classical notions of stability are based on the definition of an equilibrium point.

**Definition IV-A.1 (Equilibrium point).** A point $x^* \in \mathcal{X}$ is an equilibrium of (3) if $f(x^*, t) = 0$, or, equivalently, for the associated flow $\varphi(x^*, t; t_0) = x^*$, for each $t \in \mathbb{R}$.

**Definition IV-A.2 (Simple stability).** An equilibrium point $x^*$ is said to be stable in the sense of Lyapunov (or simply stable) if, for every $\varepsilon > 0$, there exists $\delta = \delta_{t_0, \varepsilon} > 0$ such that

$$ \|x_0 - x^*\| < \delta \implies \|\varphi(x_0, t; t_0) - x^*\| < \varepsilon, \forall t \ge t_0. \quad (9) $$

<footnote n="4">⁴In the following, the notation $\pm$ is omitted, and we simply consider the limiting case $\varepsilon \to 0$.</footnote>
In the above definition, the notation $\delta_{t_0, \varepsilon}$ indicates that $\delta$ implicitly depends on the initial time $t_0$ and the chosen $\varepsilon$.

**Definition IV-A.3 (Asymptotic stability).** Let $x^* \in \mathbb{R}^n$ be an equilibrium point of system (3). The equilibrium point $x^*$ is said to be *asymptotically stable* if the following two conditions hold:

a) **Stability:** For every $\varepsilon > 0$, there exists $\delta = \delta_{t_0, \varepsilon} > 0$ such that the implication (9) holds, i.e., the equilibrium point is stable in the sense of Lyapunov,

b) **Attractivity:** There exists $\delta_0 = \delta_{0, t_0} > 0$ such that

$$ \|x_0 - x^*\| < \delta_0 \implies \lim_{t \to \infty} \|\varphi(x_0, t; t_0) - x^*\| = 0. \quad (10) $$

If these properties hold for a sufficiently small $\delta_0$, then the equilibrium is said to be *Local Asymptotic Stable (LAS)*. If they hold for all initial conditions in a region $\mathcal{X}_{x^*} \subset \mathcal{X}$ with $x^* \in \mathcal{X}_{x^*}$, that is

$$ x_0 \in \mathcal{X}_{x^*} \implies \lim_{t \to \infty} \|\varphi(x_0, t; t_0) - x^*\| = 0, \quad (11) $$

then $x^*$ is *Regionally Asymptotically Stable (RAS)*. If the region of initial conditions, such that (11) holds, extends to the entire space, i.e., $\mathcal{X}_{x^*} \equiv \mathcal{X} \subseteq \mathbb{R}^n$, then the equilibrium $x^*$ is said to be *Globally Asymptotically Stable (GAS)*.

**Definition IV-A.4 (Exponential stability).** An equilibrium point $x^* \in \mathcal{X}$ of system (3) is said to be *locally exponentially stable* (LES) if it is asymptotically stable (i.e., Lyapunov stability with attractivity) and there exist real constants $\kappa = \kappa_{t_0} > 0$, $\alpha = \alpha_{t_0} > 0$, and $\delta_0 = \delta_{0, t_0} > 0$ such that, for all $t \ge t_0$,

$$ \|x_0 - x^*\| < \delta_0 \implies \|\varphi(x_0, t; t_0) - x^*\| \le \kappa e^{-\alpha(t-t_0)} \|x_0 - x^*\|. \quad (12) $$

If property (12) holds on a region $\mathcal{X}_{x^*} \subset \mathcal{X}$ around the equilibrium point $x^* \in \mathcal{X}_{x^*}$ then the equilibrium point $x^*$ is *Regionally Exponentially Stable (RES)*. If the region of initial conditions, such that (12) holds, extends to the entire space, i.e., $\mathcal{X}_{x^*} \equiv \mathcal{X} \subseteq \mathbb{R}^n$, then the equilibrium $x^*$ is said to be *Globally Exponentially Stable (GES)*.

The notions above form a hierarchy of increasingly stronger stability properties. Lyapunov (simple) stability ensures boundedness of trajectories near an equilibrium. Asymptotic stability strengthens this notion by enforcing convergence to the equilibrium. Exponential stability further strengthen it by ensuring an exponential rate of convergence.

For completeness, a stronger notion of stability relevant to both time-varying (inhomogeneous) systems (3) and to time-invariant (homogeneous) systems (4) is *uniform stability*. In this setting, all constants appearing in the definitions of stability and attractivity must be independent of the initial time $t_0$. To indicate this, we simply omit the dependence on $t_0$ and prepend the term *Uniform* to the corresponding definition.

**Remark IV.1 (On the role of uniformity).** Uniformity plays a fundamental role in the analysis of time-varying systems. For inhomogeneous dynamics, the dependence on the initial time

$t_0$ may influence the rate or even the existence of convergence, since the behavior of the system can explicitly vary with $t$. Uniform stability ensures that the same bounds and the convergence rates apply uniformly for all (or independently of) $t_0$, excluding pathological cases where the stability depends on when the motion is initiated. This property is crucial for robustness analyses involving bounded time-varying perturbations, for the characterization of input-to-state stability (ISS), and for Lyapunov-based arguments requiring time-invariant estimates. Intuitively, uniform stability guarantees that the system behaves “consistently” over time, enabling stronger results such as uniform asymptotic or uniform exponential stability.

1) *Lyapunov theory for time-varying nonlinear systems:*

Consider an equilibrium point $x^* \in \mathcal{X}$ for (3). To establish Lyapunov stability of $x^*$, the second method of Lyapunov (or the direct method) employs an auxiliary scalar function, called a Lyapunov function, whose evolution along trajectories captures the stability properties of the equilibrium.

**Definition IV-A.5 (Lyapunov function).** Let $\mathcal{X}_{x^*} \subseteq \mathcal{X}$ be a region containing $x^*$, i.e. $x^* \in \mathcal{X}_{x^*}$. A continuously differentiable function $V: \mathcal{X}_{x^*} \times \mathbb{R} \to \mathbb{R}_{\ge 0}$, $V \in C^1(\mathcal{X}_{x^*} \times \mathbb{R})$, is called a (regional) Lyapunov function for $x^*$ if the following conditions hold:

a) **Positive definite:** $V(x^*, t) = 0$ for all $t$ and $V(x, t) > 0$ for all $x \in \mathcal{X}_{x^*} \setminus \{x^*\}$ and $\forall t \in \mathbb{R}$;

b) **Uniformly bounded in $x$:** there exists class-$\mathcal{K}$ functions $\underline{\alpha}$, $\overline{\alpha}$ such that, for all $x \in \mathcal{X}_{x^*}$ and all $t$,

$$ \underline{\alpha}(\|x - x^*\|) \le V(x, t) \le \overline{\alpha}(\|x - x^*\|); \quad (13) $$

c) **Non-increasing in $t$:** the total derivative of $V$ along the trajectories of (3) satisfies $\dot{V}(x, t) \le 0$ for all $x \in \mathcal{X}_{x^*}$ and all $t \ge t_0$.

**Theorem IV-A.1 (Lyapunov stability for time-varying systems).** Let $x^* \in \mathcal{X}$ be an equilibrium of system (3). Suppose that there exists a Lyapunov function $V \in C^1(\mathcal{X}_{x^*} \times \mathbb{R})$ defined on a region $\mathcal{X}_{x^*}$ around $x^*$, i.e., it satisfies $\dot{V}(x, t) \le 0$ for all $x \in \mathcal{X}_{x^*}$ and $t \ge t_0$ with bounds (13). Then the equilibrium $x = x^*$ is (locally) stable in the sense of Lyapunov.

To obtain global stability results, one additional requires that the Lyapunov function $V$ is *radially unbounded* (or proper), meaning that there exists a class-$\mathcal{K}_\infty$ function $\underline{\alpha}$ such that for all $x \in \mathcal{X}$ and all $t$, $\underline{\alpha}(\|x - x^*\|) \le V(x, t)$. If $\dot{V}$ is defined on $\mathcal{X}_{x^*} \equiv \mathcal{X}$ and is radially unbounded, it is also called a global Lyapunov function.

**Theorem IV-A.2 (Uniform asymptotic stability - Lyapunov criterion).** Let $x^* \in \mathcal{X}$ be an equilibrium of system (3). Suppose that there exists a Lyapunov function $V$ defined on a region $\mathcal{X}_{x^*}$ around $x^*$, satisfying (13). Assume further that there exists a class-$\mathcal{K}_\infty$ function $\beta$ such that, for all $x \in \mathcal{X}_{x^*}$ and all $t \ge t_0$,

$$ \dot{V}(x, t) \le -\beta(\|x - x^*\|). \quad (14) $$

Then the equilibrium $x = x^*$ is uniformly asymptotically stable on $\mathcal{X}_{x^*}$. If $\mathcal{X}_{x^*} \equiv \mathcal{X}$ and $V$ is radially unbounded,
the equilibrium is uniformly globally asymptotically stable (UGAS).

**Remark IV.2** If V is positive definite but its upper bound depends on t, the asymptotic stability property may still be obtained, but not uniformly with respect to the initial time t₀. Having a time-independent upper bound for V and ˙V is therefore essential for uniform stability statements.

**Theorem IV-A.3 (Exponential stability - Lyapunov condition).** Assume there exist positive constants $\underline{\alpha}$, $\bar{\alpha}$, $\alpha > 0$ and a function $V \in C^1(\mathcal{X}_{x^*} \times \mathbb{R})$ such that, for all $x \in \mathcal{X}_{x^*}$ and for all t,

$$ \underline{\alpha} \cdot \|x - x^*\|^2 \leq V(x, t) \leq \bar{\alpha} \cdot \|x - x^*\|^2, \quad (15) $$

$$ \dot{V}(x, t) \leq -2\alpha V(x, t). \quad (16) $$

Then the equilibrium point $x^* \in \mathcal{X}_{x^*}$ is regionally uniformly exponentially stable, and every solution $x(t)$ with $x(t_0) \in \mathcal{X}_{x^*}$ satisfies

$$ \|x(t) - x^*\| \leq \sqrt{\frac{\bar{\alpha}}{\underline{\alpha}}} e^{-\alpha(t-t_0)} \|x(t_0) - x^*\|. $$

If the inequalities hold globally, i.e., $\mathcal{X}_{x^*} \equiv \mathcal{X}$, the exponential stability is global.

## B. Incremental and convergent dynamics

While classical notions of stability are defined with respect to an equilibrium point, modern system analysis often requires understanding how trajectories evolve relative to one another. This motivates the introduction of *incremental* and *differen- tial* notions of stability, which generalize Lyapunov stability by characterizing convergence between system trajectories. Incremental stability was first formalized in the context of Lyapunov theory in [6], while differential stability and con- traction analysis were developed in [7], [52]. These concepts form the theoretical foundation for convergent dynamics and contraction theory, providing powerful tools for the analysis of stability of nonlinear dynamical systems.

As in the classical definitions of stability, the extent of the set $\mathcal{X}$ determines whether the corresponding stability prop- erty holds locally, regionally, or globally. Moreover, uniform versions of the definitions can be obtained by removing the explicit dependence on the initial time $t_0$. In the remainder of this section, these aspects are not restated for each definition, although they are implicitly assumed to apply to all the stability notions introduced below.

1) **Incremental stability:** Incremental stability describes the property that any two trajectories $x_a(t) := \varphi(x_{0a}, t; t_0)$ and $x_b(t) := \varphi(x_{0b}, t; t_0)$ of system (3) converge to each other.

**Definition IV-B.1 (Incremental asymptotic stability [6]).** System (3) is said to be incrementally asymptotically stable in a forward invariant set $\bar{\mathcal{X}} \subset \mathbb{R}^n$ if there exists a class $\mathcal{KL}$ function $\beta_{t_0}$ such that, for any $x_{0a}, x_{0b} \in \bar{\mathcal{X}}$ and $t \ge t_0$,

$$ \|x_a(t) - x_b(t)\| \le \beta_{t_0} (\|x_{0a} - x_{0b}\|, t - t_0). $$

A coordinate-independent version uses a distance function $d$ defined on the manifold $\mathcal{X}$.

**Definition IV-B.2 (Incremental stability on manifolds [52]).** Consider system (3) on a forward invariant manifold $\mathcal{X}$. The system is incrementally stable on $\bar{\mathcal{X}} \subseteq \mathcal{X}$ with respect to a continuous distance $d: \bar{\mathcal{X}} \times \bar{\mathcal{X}} \to \mathbb{R}_{\ge 0}$, if there exists a class $\mathcal{K}$ function $\bar{\alpha}_{t_0}$ such that

$$ d(x_a(t), x_b(t)) \le \alpha_{t_0} (d(x_{0a}, x_{0b})), $$

and, for all $x_{0a}, x_{0b} \in \bar{\mathcal{X}}$,

$$ \lim_{t \to \infty} d(x_a(t), x_b(t)) = 0. $$

2) **Convergent dynamics:** A particular case of incremental stability occurs when there exists a unique limiting trajectory $x^*(t) := \varphi(x_0^*, t; t_0)$, attractive for all trajectories in a region $\bar{\mathcal{X}} \subseteq \mathcal{X}$ and bounded for all time.

**Definition IV-B.3 (Convergent dynamics [3]).** System (3) is convergent in a forward invariant set $\bar{\mathcal{X}} \subseteq \mathcal{X}$ if:

* there exists a unique solution $x^*(t)$ in $\bar{\mathcal{X}}$ defined and bounded for all $t \in \mathbb{R}$;

* $x^*(t)$ is attractive in $\bar{\mathcal{X}}$, i.e., there exists a class $\mathcal{KL}$ function $\beta_{t_0}$ such that for any trajectory $x(t)$ initialized at $(t_0, x_0) \in \mathbb{R} \times \bar{\mathcal{X}}$,

$$ \|x(t) - x^*(t)\| \le \beta_{t_0} (\|x_0 - x_0^*\|, t - t_0), \quad \forall t \ge t_0. $$

## C. Differential stability and contractive dynamics

1) **Differential stability:** The concept of differential stability originates from the analysis of the variational dynamics associated with the nonlinear system (3). This variational system (7) is a time-varying and trajectory-dependent linear system whose stability properties depend simultaneously on the state trajectory $x(t)$ and on the time variable $t$. Consequently, any stability property defined for this linear, time-varying, and state-dependent system can be equivalently interpreted as a differential stability property of the nonlinear dynamics (3).

**Definition IV-C.1 (differential Regional Stability (dRS)).** System (3) is said to be differentially (regionally) stable in a forward invariant region $\bar{\mathcal{X}} \subseteq \mathcal{X}$ if, for any trajectory $\varphi(x_0, t; t_0)$ with $x_0 \in \bar{\mathcal{X}}$, the solution $\delta x(t)$ of the associated variational dynamics remains bound for all forward time, i.e., for every $\varepsilon > 0$ there exists a $\delta = \delta_{\varepsilon, x_0, t_0} > 0$ such that, any initial infinitesimal displacement $\delta x(t_0) \in \mathcal{T}_{x_0, \mathcal{X}}$ satisfying

$$ \|\delta x(t_0)\| < \delta \implies \|\delta x(t)\| < \varepsilon, \forall t \ge t_0. \quad (17) $$

**Definition IV-C.2 (differential Regional Asymptotic Stability (dRAS)).** System (3) is said to be differentially (regionally) asymptotically stable in a forward invariant region $\bar{\mathcal{X}}$ if, for any initial condition $x_0 \in \bar{\mathcal{X}}$, it is differentially (regionally) stable and the solution $\delta x(t)$ of the variational system satisfies

$$ \lim_{t \to \infty} \|\delta x(t)\| = 0, \quad \forall x_0 \in \bar{\mathcal{X}}. $$

**Definition IV-C.3 (differential Regional Exponential Stability (dRES)).** System (3) is differentially (regionally) exponentially stable in a forward invariant region $\bar{\mathcal{X}}$ if there exist real
constants $\kappa = \kappa_{x_0, t_0} > 0$ and $\alpha = \alpha_{x_0, t_0} > 0$ such that, for all initial conditions $x_0 \in \bar{\mathcal{X}}$ and $\delta x(t_0) \in \mathcal{T}_{x_0} \mathcal{X}$,

$$ \|\delta x(t)\| \le \kappa e^{-\alpha(t-t_0)} \|\delta x(t_0)\|, \quad \forall t \ge t_0. \quad (18) $$

In this framework, the classical notions of stability for the nonlinear system (3) naturally extend to its variational dynamics. Classically, the resulting differential stability properties capture the local, infinitesimal behavior of the system along its trajectories. This viewpoint forms the foundation for the forthcoming concepts of *contractive dynamics* and *differential Lyapunov functions*.

**2) Contractive dynamics:** The contraction approach was originally introduced as a tool for analyzing the stability properties of nonlinear systems, serving as a differential counterpart to the classical Lyapunov framework, see [7]. The theory was later extended to systems endowed with nonlinear, state-dependent metrics in [52], and further developed in [6], [11].

**Definition IV-C.4 (Contractive time-varying system).** An inhomogeneous system (3) on a smooth manifold $\mathcal{X}$ is said to be contractive in a region $\mathcal{X}_* \subseteq \mathcal{X}$ if there exists a $C^1$ matrix-valued function $P: \mathcal{X} \times [t_0, +\infty) \to \mathbb{R}^{n \times n}$ with a well-defined time derivative, and two strictly positive real numbers $\underline{p}$ and $\bar{p}$ such that $P$ satisfies $\forall x \in \mathcal{X}_* \setminus \{x^*\}$ and $t \ge t_0$,

$$ \underline{p}I \le P(x, t) \le \bar{p}I, \\ \frac{dP}{dt}(x, t) + P(x, t) \frac{\partial f}{\partial x}(x, t) + \frac{\partial f^\top}{\partial x}(x, t) P(x, t) < 0. \quad (19) $$

In such a case, $P$ is classically called a *contractive metric*. ▼

The contractive metric $P$ naturally induced the differential Lyapunov function $V: \mathcal{T}\mathcal{X} \times \mathcal{X} \times \mathbb{R} \to \mathbb{R}$, with $V(\delta x, x, t) := \delta x^\top P(x, t)\delta x$, which recalls the Riemannian metric introduced through the inner product (1).

**Remark IV.3** The concept of contractive dynamics is closely related, although conceptually independent of the notion of *incremental stability*, which quantifies the *uniform convergence* between arbitrary pairs of trajectories. It has been shown in [11, Property P1] that if the vector field $f(x)$ is globally Lipschitz with bound second derivatives, then the notions of *contractivity* and *incremental stability* are equivalent. Under these assumptions, both properties are also equivalent to the *global exponential attractiveness* of the manifold

$$ \mathcal{E} = \{(x, \delta x) : \delta x = 0\} $$

for the prolonged dynamics associated with the homogeneous system (4), which is reproduced here for completeness,

$$ \dot{x} = f(x), \quad \dot{\delta x} = \frac{\partial f}{\partial x}(x) \delta x. $$

Interpreting $\delta x = f(x)$ as a specific direction of the variational state in the tangent bundle $\mathcal{T}\mathcal{X}$, the manifold $\mathcal{E}$ introduced in [11, Property P2] corresponds to the set of equilibrium points of the autonomous dynamics. Since the contraction property in [11] holds globally, the system admits either a

single asymptotically stable equilibrium point or a continuum of such points:

$$ \mathcal{E} = \{(\bar{x}, 0)\} \quad \text{with} \quad f(\bar{x}) = 0. $$

Moreover, because all trajectories converge exponentially one another, one concludes by contradiction that within the contraction region the system admits a unique exponentially stable equilibrium point. ▼

Using definitions IV-C.3 (with $\bar{\mathcal{X}} \equiv \mathcal{X}$) and IV-C.4, we can state the following.

**Lemma IV-C.1.** A contractive dynamics (3) whose associated metric $P(x)$ satisfies, for all $x \in \mathcal{X}$ and $t \ge t_0$

$$ 0 < \underline{p}I \le P(x, t) \le \bar{p}I, \quad \underline{p}, \bar{p} \in \mathbb{R} $$

$$ \frac{dP}{dt}(x, t) + P(x, t) \frac{\partial f}{\partial x}(x, t) + \frac{\partial f}{\partial x}(x, t)^\top P(x, t) \le -2\alpha P(x, t) \quad (20) $$

with $\alpha > 0$, is differentially *Globally Exponentially Stable with* $\kappa = \sqrt{\bar{p}/\underline{p}}$. ▼

Note that (20) is a strictly stronger condition than (19).

**Proof.** The proof follows directly from the definition of contractive system together with standard Lyapunov bounds. Consider the differential Lyapunov function $V(\delta x, x, t) = \delta x^\top P(x, t)\delta x$ for the variational system (7). For all $x \in \mathcal{X}$ and $t \ge t_0$

$$ \underline{p} \|\delta x\|^2 \le V(\delta x, x, t) \le \bar{p} \|\delta x\|^2. $$

From inequality (20), the time derivative of $V$ satisfies, for $t \ge t_0$

$$ \frac{dV}{dt}(\delta x, x, t) = \frac{d\delta x^\top P(x, t)\delta x}{dt} \le -2\alpha V(\delta x, x, t). $$

Separating the variables and integrating the inequality

$$ \int_{t_0}^{t} \frac{\dot{V}(\delta x(\tau), x(\tau), \tau)}{V(\delta x(\tau), x(\tau), \tau)} d\tau \le - \int_{t_0}^{t} 2\alpha d\tau $$

we obtain

$$ \ln \left( \frac{V(\delta x(t), x(t), t)}{V(\delta x(t_0), x(t_0), t_0)} \right) \le -2\alpha (t - t_0) $$

and therefore

$$ V(\delta x, x, t) \le e^{-2\alpha(t-t_0)} V(\delta x(t_0), x_0, t_0). $$

Using the bounds on $V$, i.e., $\bar{p}\|\delta x(t_0)\|^2 \ge V(\delta x(0), x_0, t_0)$ and $V(\delta x, x, t) \ge \underline{p}\|\delta x\|^2$, we conclude that

$$ \|\delta x\| \le \sqrt{\frac{\bar{p}}{\underline{p}}} e^{-\alpha(t-t_0)} \|\delta x(t_0)\|, $$

which proves the claim. □

In a sense, the standard GES definition can be viewed as an integral counterpart of its differential version. Remark IV.3 implies that the system (4) converges to its unique equilibrium point $x = x^*$ regardless of its initial condition at
time $t_0$. Moreover, the transient behavior is exponential when examined from the variational dynamics point of view.

From this perspective, contraction analysis provides a differential generalization of classical Lyapunov stability: instead of requiring the value of a scalar potential function to decrease along trajectories, the classical contraction interpretation requires that infinitesimal displacements shrink over time. This implies incremental and exponential stability at the system trajectories level, although this has never been formally established for time-varying nonlinear systems (3). We established such a result in the next sections. Before doing so, we present a more general definition and an extension of the variational dynamics, which will serve as the main tool for proving contraction of nonlinear time-varying systems. The motivation for the first instead arises from the observation that, while the variational dynamics of a homogeneous system coincide with the dynamics of its vector field (see Remark IV.3), the same does not hold for an inhomogeneous system.

## V. A GENERALIZED VARIATIONAL DYNAMICS

As introduced in section III, the flow $\varphi$ of the system satisfies the IVP (3), which can equivalently be expressed as

$$ \varphi(x_0, t; t_0) = x_0 + \int_{t_0}^{t} f(\varphi(x_0, \tau; t_0), \tau) d\tau. \quad (21) $$

To define a variational state, one typically constructs a parametrized family of trajectories $\chi(\varepsilon, t)$ that satisfy (5) for $t \ge t_0$ and $\varepsilon \in (-\delta_{\chi}, \delta_{\chi})$, with sufficiently small positive $\delta_{\chi}$, and boundary condition $\chi(0, t) = x(t) = \varphi(x_0, t; t_0)$. However, using the system flow $\varphi$ and fixing $\nu \in \mathbb{R}^n$ and $\mu \in \mathbb{R}$, we introduce an explicit and generalized parametrized family of trajectories, i.e.

$$ \chi_{\nu,\mu}(\varepsilon, t) := \varphi(x_0 + \varepsilon\nu, t + \varepsilon\mu; t_0), \quad (22) $$

or in integral form

$$ \chi_{\nu,\mu}(\varepsilon, t) = \varphi(x_0 + \varepsilon\nu, t_0 + \varepsilon\mu; t_0) + \int_{t_0}^{t} f(\varphi(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0), \tau + \varepsilon\mu) \, d\tau. \quad (23) $$

Here, the parameter $\varepsilon$ appears in both the initial condition and the time arguments of the flow.

Following the definition of variational state (6), from $\chi_{\nu,\mu}$ in (22), the generalized variational state is defined as

$$ \delta x_{\nu,\mu}(t) := \frac{\partial \chi_{\nu,\mu}}{\partial \varepsilon}(\varepsilon, t) \Big|_{\varepsilon=0}. \quad (24) $$

**Theorem V.-1.** Consider the dynamical system (3) on the forward invariant smooth manifold $\mathcal{X}$, and let $x(t) = \varphi(x_0, t; t_0)$. The parameterized trajectory (22) defines an admissible family satisfying (5), with $\chi_{\nu,\mu}(0, t) = x(t)$. Furthermore, for fixed $\nu \in \mathbb{R}^n$ and $\mu \in \mathbb{R}$, the associated variational dynamics of the family (22) is given by

$$ \frac{d}{dt} \delta x_{\nu, \mu}(t) = \frac{\partial f}{\partial x}(x(t), t) \delta x_{\nu, \mu} + \frac{\partial f}{\partial t}(x(t), t) \mu \quad (25) $$

with initial condition

$$ \delta x_{\nu, \mu}(t_0) = \nu + f(x_0, t_0) \mu \in \mathbb{R}^n. \quad \llap{\vphantom{\delta x_{\nu, \mu}(t_0)}} $$

**Proof.** We first show that (22) defines an admissible family of parametrized trajectories satisfying (5).

By the definition of $\chi_{\nu,\mu}$ in (22), we immediately obtain $\chi_{\nu,\mu}(0, t) = \varphi(x_0, t; t_0) = x(t)$. Since $\varphi$ is the flow of system (3), we have

$$ \begin{aligned} \frac{d\chi_{\nu,\mu}}{dt}(\varepsilon, t) &= f(\varphi(x_0 + \varepsilon\nu, t + \varepsilon\mu; t_0), t + \varepsilon\mu) \\ &= f(\chi_{\nu,\mu}(\varepsilon, t), t + \varepsilon\mu). \end{aligned} $$

Setting $\sigma = t + \varepsilon\mu$ and $\chi_{\sigma}(\varepsilon, \sigma) = \chi_{\nu,\mu}(\varepsilon, \sigma - \varepsilon\mu)$, we rewrite the previous identity as

$$ \frac{d\chi_{\nu,\mu}}{dt}(\varepsilon, t) = \frac{d\chi_{\sigma}}{d\sigma}(\varepsilon, \sigma) = f(\chi_{\sigma}(\varepsilon, \sigma), \sigma), $$

which shows that $\chi_{\nu,\mu}(\varepsilon, t)$ is indeed an admissible family of parametrized curves.

Next, the generalized variational dynamics (25) derives from the partial derivative of (22) with respect to $\varepsilon$, which yields

$$ \begin{aligned} \frac{\partial \chi_{\nu, \mu}}{\partial \varepsilon}(t, \varepsilon) &= \frac{\partial \varphi(x_0 + \varepsilon\nu, t_0 + \varepsilon\mu; t_0)}{\partial \varepsilon} + \\ & \int_{t_0}^{t} \frac{\partial f(\varphi(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0), \tau + \varepsilon\mu)}{\partial \varepsilon} ds \\ &= \frac{\partial \varphi}{\partial x_0}(x_0 + \varepsilon\nu, t_0 + \varepsilon\mu; t_0)\nu + \frac{\partial \varphi}{\partial t}(x_0 + \varepsilon\nu, t_0 + \varepsilon\mu; t_0)\mu + \\ & \int_{t_0}^{t} \frac{\partial f}{\partial x}(\varphi(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0), \tau + \varepsilon\mu) \cdot \\ & \qquad \frac{\partial \varphi}{\partial \varepsilon}(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0) + \\ & \qquad \frac{\partial f}{\partial t}(\varphi(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0), \tau + \varepsilon\mu) \, \mu ds. \end{aligned} $$

Thus, the variational state is

$$ \begin{aligned} \delta x_{\nu,\mu}(t) &= \left. \frac{\partial \chi_{\nu,\mu}}{\partial \varepsilon}(\varepsilon, t) \right|_{\varepsilon=0} = \\ & \qquad \frac{\partial \varphi}{\partial x_0}(x_0, t_0; t_0)\nu + \left. \frac{\partial \varphi}{\partial t}(x_0, t_0; t_0)\mu + \right. \\ & \qquad \left. \int_{t_0}^{t} \frac{\partial f}{\partial x}(\varphi(x_0, \tau; t_0), \tau) \frac{\partial \varphi}{\partial \varepsilon}(x_0 + \varepsilon\nu, \tau + \varepsilon\mu; t_0) \right|_{\varepsilon=0} + \\ & \qquad \frac{\partial f}{\partial t}(\varphi(x_0, \tau; t_0), \tau) \, \mu ds \\ &= \frac{\partial \varphi}{\partial x_0}(x_0, t_0; t_0)\nu + \left. \frac{\partial \varphi}{\partial t}(x_0, t_0; t_0)\mu + \right. \\ & \qquad \left. \int_{t_0}^{t} \frac{\partial f}{\partial x}(\varphi(x_0, \tau; t_0), \tau) \cdot \delta x_{\nu,\mu}(\tau) + \right. \\ & \qquad \left. \frac{\partial f}{\partial t}(\varphi(x_0, \tau; t_0), \tau) \, \mu ds, \right. \end{aligned} \quad (26) $$

from which the initial condition follows:

$$ \delta x_{\nu,\mu}(t_0) = \nu + f(x_0, t_0)\mu. \quad (27) $$

Differentiating with respect to $t$ then yields

$$ \begin{aligned} \frac{d\delta x_{\nu,\mu}}{dt}(t) &= \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \delta x_{\nu,\mu}(t) + \\ & \frac{\partial f}{\partial t}(\varphi(x_0, t; t_0), t) \mu \end{aligned} \quad (28) $$

where $x(t) = \varphi(x_0, t; t_0)$ and the proof is complete. $\square$
From the parametrized trajectories (22), the closed form expression of the generalized variational state is

$$ 
\begin{aligned}
\delta x_{\nu, \mu}(t) &= \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0)\nu + \frac{\partial \varphi}{\partial t}(x_0, t; t_0)\mu \\
&= \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0)\nu + f(\varphi(x_0, t; t_0), t)\mu.
\end{aligned} 
\quad (29) $$

The family of parameterized trajectories (22) also captures purely initial condition perturbations or purely time perturbations. Given $\nu \in \mathbb{R}^n$ and $\mu \in \mathbb{R}$, we define

$$ \chi_{\nu}(\varepsilon, t) = \varphi(x_0 + \varepsilon\nu, t; t_0) = x_0 + \varepsilon\nu + \int_0^t f(\varphi(x_0 + \varepsilon\nu, \tau; t_0), \tau) d\tau \quad (30) $$

and

$$ \chi_{\mu}(\varepsilon, t) = \varphi(x_0, t + \varepsilon\mu; t_0) = \varphi(x_0, \mu\varepsilon; t_0) + \int_0^t f(\varphi(x_0, \tau + \varepsilon\mu; t_0), \tau + \varepsilon\mu) d\tau. \quad (31) $$

Here, $\chi_{\nu}(\varepsilon, t) = \chi_{\nu,0}(\varepsilon, t)$ and $\chi_{\mu}(\varepsilon, t) = \chi_{0,\mu}(\varepsilon, t)$. Thus, the associated forms of variational states are

$$ \delta x_{\nu}(t) := \left. \frac{\partial \chi_{\nu}}{\partial \varepsilon}(\varepsilon, t) \right|_{\varepsilon=0}, \quad \delta x_{\mu}(t) := \left. \frac{\partial \chi_{\mu}}{\partial \varepsilon}(\varepsilon, t) \right|_{\varepsilon=0}. $$

As a direct consequence of Theorem V.-1, we obtain the following corollaries. Their proofs follow immediately by specializing the general case of the theorem.

**Corollary V.-1.** Let the dynamical system (3) evolve on the forward invariant smooth manifold $\mathcal{X}$, and let $x(t) = \varphi(x_0, t; t_0)$ be the corresponding trajectory. For a fixed direction $\nu \in \mathbb{R}^n$, the variational dynamics associated with the parameterized trajectories (30) is, for all $t \ge t_0$,

$$ \frac{d}{dt} \delta x_{\nu}(t) = \frac{\partial f}{\partial x}(x(t), t) \delta x_{\nu}(t), \quad (32) $$

with initial condition $\delta x_{\nu}(t_0) = \nu \in \mathbb{R}^n$.

**Corollary V.-2.** Let the dynamical system (3) evolve on the forward invariant smooth manifold $\mathcal{X}$, and let $x(t) = \varphi(x_0, t; t_0)$ be the corresponding trajectory. For a fixed time-direction parameter $\mu \in \mathbb{R}$, the variational dynamics associated with the parameterized trajectories (31) is, for all $t \ge t_0$,

$$ \frac{d}{dt} \delta x_{\mu}(t) = \frac{\partial f}{\partial x}(x(t), t) \delta x_{\mu}(t) + \frac{\partial f}{\partial t}(x(t), t) \mu, \quad (33) $$

with initial condition $\delta x_{\mu}(t_0) = f(x_0, t_0) \mu \in \mathcal{T}_{x_0} \mathcal{X}$.

**Remark V.1** The initialization of the variational dynamics is informative. Given two distinct initial conditions $x_{0a}, x_{0b} \in \mathcal{X}$, one may consider the perturbation direction $\nu = x_{0b} - x_{0a}$. Such a vector, however, does not necessarily lie in the tangent space at $x_{0a}$, i.e., $\nu \notin \mathcal{T}_{x_{0a}} \mathcal{X}$.

From the above analysis, we observe that the variational state $\delta x$, defined in (6) and evolving according to (7), shares structural similarities with the family of parameterized trajectories (30) corresponding to the case $\mu = 0$. Within the context of this section, the perturbation direction $\nu$ is allowed to lie in the ambient space $\mathbb{R}^n$, without the requirement of belonging to

the tangent space $\mathcal{T}_{x_0} \mathcal{X}$. In contrast, in the classical setting of variational dynamics, as recalled in Section III, it is essential for the analysis that the direction of the perturbation satisfies $\nu \in \mathcal{T}_{x_0} \mathcal{X}$.

**Remark V.2** The classes of parameterized trajectories introduced in this section are not unique; they must instead be chosen according to the context so as to be intuitive, workable, and informative for development.

## VI. EXTENDED VARIATIONAL DYNAMICS

Inspired by the idea, as discussed above, of linking two initial conditions $x_{0a} \in \mathcal{X}$ and $x_{0b} \in \mathcal{X}$ via the initial perturbation direction $\nu = x_{0b} - x_{0a} \in \mathbb{R}^n$, we construct an extension of the variational state.

Let $\mathcal{X}$ be a connected $\mathbb{C}^2$ manifold that is forward-invariant with respect to the flow $\varphi$ of system (3). Endow the manifold with a time-varying, position-dependent Riemannian metric of the form

$$ g_{x,t}(u, v) = u^{\top} P(x, t)v, \quad u, v \in \mathcal{T}_{x} \mathcal{X} \quad (34) $$

where $P(x, t)$ is symmetric and positive definite for all $(x, t) \in \mathcal{X} \times \mathbb{R}$, so that the metric $g_{x,t}$ defines a proper inner product on each tangent space $\mathcal{T}_{x} \mathcal{X}$.

Let $\mathcal{X}_0 \subseteq \mathcal{X}$ denote the submanifold of all admissible initial conditions for system (3), i.e.,

$$ \mathcal{X}_0 := \{x_0 \in \mathcal{X} : \varphi(x_0, t; t_0) \in \mathcal{X}, \forall t \ge t_0\}. $$

For any two initial conditions $x_{0a}, x_{0b} \in \mathcal{X}_0$, the properties of $\mathcal{X}_0$ ensure that they generate two well-defined trajectories $x_a(t) = \varphi(x_{0a}, t; t_0)$ and $x_b(t) = \varphi(x_{0b}, t; t_0)$ in $\mathcal{X}$.

Before introducing the connecting curve $\gamma(s, t)$ between the two system trajectories, we define a static connecting curve $\gamma_0 \in \Gamma^2(x_{0a}, x_{0b})^5$ between two initial conditions $x_{0a}$ and $x_{0b}$. For all $s \in [0, 1]$, the curve satisfies $\gamma_0(s) \in \mathcal{X}_0 \subseteq \mathcal{X}$ and $\gamma_0(0) = x_{0a}$ and $\gamma_0(1) = x_{0b}$, see Fig.2 for a graphical interpretation. Using the properties of the system flow $\varphi$, for all $x_0 \in \mathcal{X}$,

$$ \frac{\partial \varphi}{\partial t}(x_0, t; t_0) = f(\varphi(x_0, t; t_0), t), $$

we define a time-dependent connection curve between the trajectories $x_a(t)$ and $x_b(t)$ as the curve $\gamma \in \Gamma^2_t(x_a, x_b)$ given by

$$ \gamma(s, t) = \varphi(\gamma_0(s), t; t_0). \quad (35) $$

For each $s \in [0, 1]$, the time evolution satisfies

$$ \frac{\partial \gamma}{\partial t}(s, t) = \frac{\partial \varphi}{\partial t}(\gamma_0(s), t; t_0) = f(\gamma(s, t), t), $$

so $\gamma(s, t)$ for each $s \in [0, 1]$, is a trajectory of the system, which remains in the invariant manifold $\mathcal{X}$ because $\gamma_0(s) \in \mathcal{X}_0$, for each $s \in [0, 1]$. See Fig. 3 for an illustration.

**Remark VI.1** This construction parallels the family of parametrized curves introduced in (5), but extends it from

<footnote>$^5$For simplicity, throughout this section the domain of the classes $\Gamma = \Gamma_{t_0}$ for a fixed $t_0 \in \mathbb{R}$, and $\Gamma_t$ is taken to be $[0, 1] \subset \mathbb{R}$ rather than a generic interval $\mathcal{I} = [a, b] \subseteq \mathbb{R}$.</footnote>
a sufficiently small neighborhood of initial conditions to the whole region of interest $\mathcal{X}$. Thus, it extends the concept of parametrized curves to an infinite dimensional family of curves connecting any two trajectories through all intermediate system's trajectories.

Given the spatial derivative $\gamma'(s, t)$ and the Reimannian metric $g_{x,t}$, the associate Riemannian length is defined as

$$ \mathcal{L}_{g_{\gamma,t}}(\gamma, t) = \int_{0}^{1} \sqrt{\gamma'(s,t)^{\top} P(\gamma(s,t), t) \gamma'(s,t)} ds, \quad (36) $$

which varies with time due to both the time-varying nature of the metric matrix $P$ and the double-parameter dependence of $\gamma$.

Under the condition that the flow $\varphi$ is sufficiently smooth, i.e., $\varphi \in C^2(\mathcal{X} \times \mathbb{R})$, Schwarz's Theorem (Th. II-A.1) yields the identity

$$ \begin{aligned} \frac{\partial}{\partial t} \frac{\partial \gamma}{\partial s}(s,t) &= \frac{\partial}{\partial s} \frac{\partial \gamma}{\partial t}(s,t) = \frac{\partial f}{\partial s}(\gamma(s,t), t) \\ &= \frac{\partial f}{\partial x}(\gamma(s,t), t) \cdot \frac{\partial \gamma}{\partial s}(s,t), \quad \forall s \in [0, 1], t \ge t_0 \end{aligned} $$

and therefore the extended variational dynamics satisfies, $\forall s \in [0, 1]$ and $t \ge t_0$

$$ \frac{\partial \gamma'}{\partial t}(s,t) = \frac{\partial f}{\partial x}(\gamma(s,t), t) \cdot \gamma'(s,t). \quad (37) $$

We are now ready to present the main result of the paper.

**Theorem VI.-1.** Let $\mathcal{X}$ be endowed with the Riemannian metric (34), and let it be the forward invariant manifold under the flow $\varphi$ of system (3). Consider $\gamma$, constructed in (35), as a connecting curve. Under the following assumptions:

a) **Vector field smoothness:** the vector field $f(x, t)$ is $C^2$ over $\mathcal{X} \times \mathbb{R}$;

b) **Smoothness of the metric:** $P(x, t) \in C^1(\mathcal{X} \times \mathbb{R})$ is symmetric and positive definite, and for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$

$$ 0 < \lambda_{\min}(x, t)I \le P(x, t) \le \lambda_{\max}(x, t)I $$

where $\lambda_{\min}$, $\lambda_{\max} \in C^1(\mathcal{X} \times \mathbb{R})$ are the minimum and maximum eigenvalues of $P$;

c) **Attractivity:** there exists a positive function $\beta: \mathcal{X} \times \mathbb{R} \to \mathbb{R}_{\ge 0}$, with $\beta \in C^1(\mathcal{X} \times \mathbb{R})$, such that for all $t \ge t_0$ and $s \in (0, 1]$,

$$ \frac{dP}{dt}(\gamma, t) + P(\gamma, t) \frac{\partial f}{\partial x}(\gamma, t) + \frac{\partial f}{\partial x}(\gamma, t)^{\top} P(\gamma, t) \le -2\beta(\gamma, t)I \quad \forall \gamma(s,t) \in \mathcal{X} \quad (38) $$

and for $s \in (0, 1]$

$$ \lim_{t \to \infty} \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau = +\infty. \quad (39) $$

d) **Integrable initial derivative:** The initial connecting curve $\gamma_0$ satisfies $\|\gamma_0'\| \in L^1([0, 1])$.

Then, all trajectories of the system initialized in $\mathcal{X}$ converge to one another. Equivalently, the length of every admissible

<figure><img src="image_3.png" /><figcaption>Fig. 2: Class of oriented curves $\gamma_0$ connecting any two initial conditions $x_{0a}$ and $x_{0b}$ in the phase plane at initial time $t = t_0$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Evolution of the oriented connecting curves $\gamma(s, t)$ between the trajectories $x_a(t) = \varphi(x_{0a}, t; t_0)$ and $x_b(t) = \varphi(x_{0b}, t; t_0)$ in the phase plane, at a generic time $t > t_0$.</figcaption></figure>
connecting curve between any two trajectories tends to zero as $t \to \infty$, i.e., the Riemannian length of $\gamma$ satisfies

$$ \mathcal{L}_{g_{\gamma,t}}(\gamma, t) := \int_{0}^{1} \sqrt{\gamma'^{\top} P(\gamma, t) \gamma'} ds \xrightarrow{t \to \infty} 0. \quad \] ]

*Proof.* The smoothness assumption $f \in C^2(X \times \mathbb{R})$ guarantees that the flow $\varphi(x_0, t; t_0)$ is $C^2$ in both $x_0$ and $t$. Thus, the connecting curves defined in (35) are sufficiently smooth, and their spatial velocity $\gamma'(s, t)$ satisfies (37) for all $s \in [0, 1]$ and $t \ge t_0$. Define the Lyapunov function

$$ V(s, t) = \gamma'(s, t)^{\top} P(\gamma(s, t), t) \gamma'(s, t). $$

By the smoothness property of $P$, $V$ admits upper and lower bounds through two continuous class-$\mathcal{K}$ functions, i.e.,

$$ \lambda_{\min}(\gamma, t) \|\gamma'(s, t)\|^2 \le V(s, t) \\ \le \lambda_{\max}(\gamma, t) \|\gamma'(s, t)\|^2. \quad (40) $$

From (38), for $t \ge t_0$ and $\forall \gamma(s, t) \in \mathcal{X}, s \ne 0$

$$ \dot{V} = \gamma'^{\top} \left( \dot{P} + P \frac{\partial f}{\partial x} + \frac{\partial f^{\top}}{\partial x} P \right) \gamma' \le -2\beta \|\gamma'\|^2. $$

Using the upper bound in (40), since $\lambda_{\max} > 0$ on the entire domain and $-\|\gamma'\|^2 \le -V/\lambda_{\max}$, we obtain

$$ \dot{V} = \gamma'^{\top} \left( \dot{P} + P \frac{\partial f}{\partial x} + \frac{\partial f^{\top}}{\partial x} P \right) \gamma' \le -2 \frac{\beta}{\lambda_{\max}} V. $$

From separating the variable and integrating the inequality from $t_0$ to $t$, i.e.,

$$ \int_{t_0}^{t} \frac{\dot{V}(s, \tau)}{V(s, \tau)} d\tau \le -2 \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau, $$

we obtain

$$ \ln \left( \frac{V(s, t)}{V(s, t_0)} \right) \le -2 \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau, $$

and hence

$$ V(s, t) \le \exp \left( -2 \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau \right) V(s, t_0). \quad (41) $$

Assumption (39) implies that for each $s \in (0, 1]$, $V(s, t) \xrightarrow{t \to \infty} 0$.

Consider now the time-varying Riemannian length (36), and rewrite it in terms of the Lyapunov function $V$ as

$$ \mathcal{L}_{g_{\gamma,t}}(\gamma(s,t),t) = \int_{0}^{1} \sqrt{\gamma'(s,t)^{\top} P(\gamma(s,t),t) \gamma'(s,t)} ds, \\ = \int_{0}^{1} \sqrt{V(s,t)} ds. $$

The continuity property of the square root function and the Lyapunov function property (41), ensured that, for each fixed $s \in (0, 1]$,

$$ \lim_{t \to \infty} \sqrt{V(s, t)} = 0. $$

To prove that $\mathcal{L}_{g_{\gamma,t}} \to 0$, it remains to show the existence of an integrable dominating function. To this end, the integrability of the spatial derivative of the initial connection curve, i.e., $\gamma'_0 \in L^1([0, 1])$, allows us to determine an integrable upper bound as follows. Property (41) ensures that, for each $t \ge t_0$ and all $s \in [0, 1]$,

$$ \sqrt{V(s, t)} \le \sqrt{V(s, t_0)} $$

and

$$ \sqrt{V(s, t_0)} \le \sqrt{\lambda_{\max}(\gamma_0(s), t_0)} \|\gamma_0'(s)\|. $$

Define the dominating function

$$ \bar{V}(s) := \sqrt{\lambda_{\max}(\gamma_0(s), t_0)} \|\gamma_0'(s)\| $$

which is integrable by the assumption $\gamma'_0 \in L^1([0, 1])$, and by compactness of $[0, 1]$. In fact, the continuity property of $\lambda_{\max}(\gamma_0(s), t_0)$ for $s \in [0, 1]$ (implied by the continuity assumption of $\lambda_{\max}$ on the entire domain), ensures

$$ \exists M > 0 : \quad \lambda_{\max}(\gamma_0(s), t_0) \le M, \quad \forall s \in [0, 1]. $$

Therefore

$$ \int_0^1 \sqrt{\lambda_{\max}(\gamma_0(s), t_0)} \cdot \|\gamma_0'(s)\|, ds \le \sqrt{M} \int_0^1 \|\gamma_0'(s)\|, ds < \infty, $$

which guarantees $\bar{V}(s) \in L^1([0, 1])$. Then by the Dominated Convergence Theorem (Th. II.-1),

$$ \lim_{t \to \infty} \mathcal{L}_{g_{\gamma,t}}(\gamma, t) = \lim_{t \to \infty} \int_{0}^{1} \sqrt{V(s, t)} ds \\ = \int_{0}^{1} \lim_{t \to \infty} \sqrt{V(s, t)} ds = 0. $$

This concludes the proof. $\square$

<figure><img src="image_5.png" /></figure>

**Remark VI.2** It is not necessary for $P$ to be uniformly positive-definite, i.e., no uniform bound on $\lambda_{\min}$, $\lambda_{\max}$ is required. The key requirement is that the initial curve derivative is $L^1$, i.e., its length is finite under the standard Euclidean metric, which guarantees the existence of a uniform integrable function $\bar{V}(s)$ dominating $\sqrt{V(s, t)}$ for all $t$, allowing the use of Dominated Convergence Theorem (Th. II.-1). The eigenvalue $\lambda_{\max}$ can grow with $\|\gamma\|$, because the integral is taken over the compact interval $s \in [0, 1]$, where $\gamma_0(s)$ is well defined. Only continuity of $\lambda_{\max}(\gamma_0(s), t_0)$ is required for integrability. The smoothness of $\gamma$ and $P$ ensures that $V(s, t)$ is continuous in $s$ and $t$, preventing pathological spikes in $\sqrt{V}$ that could violate $L^1$ integrability. Furthermore, the pointwise decay of $V(s, t)$ is sufficient; no uniform exponential decay is necessary. ]

Thus, contraction on the variational dynamics implies that all connecting curves $\gamma$ between two trajectories asymptotically have zero length, so the trajectories converge towards each other.

At this stage, it is worth emphasizing that, from (41), the only contraction in the sense of Banach (Def.II.-3), is $V(\cdot, t)$ where

$$ V(s, t) = \gamma'(s, t)^{\top} P(\gamma(s, t), t) \gamma'(s, t), $$

<footnote n="6">⁶Note that we exclude the point $s = 0$, because the analysis takes place, without loss of generality, from the point of view of the trajectory $x_a(t) = \varphi(x_{0a}, t; t_0)$.</footnote>
rather than the vector field $f(x, t)$ as usually stated in the standard contraction theory literature. It would be more accurate to call $f$ a *contractive vector field* rather than a contraction.

Furthermore, the definition of the Riemannian metric (34) and the length (36), since $P(x, t)$ is positive definite on $\mathcal{X} \times \mathbb{R}$, induce a distance

$$d(x, y) = \inf_{\gamma \in \Gamma^2(x, y)} \mathcal{L}_{g_{\gamma, t}}(\gamma, t),$$

where $\Gamma^2(x, y)$ is the set of $\mathcal{C}^2$ curves lying in $\mathcal{X}$ connecting $x$ and $y$, with $x, y \in \mathcal{X}$. Hence $(\mathcal{X}, d)$ defines a metric space for each fixed $t$.

**Remark VI.3** The definition of $\gamma$ in (35) provides a link between the spatial velocities of the curves $\gamma$ and their initial conditions $\gamma_0$, i.e.,

$$\gamma'(s, t) = \frac{\partial \varphi}{\partial x_0}(\gamma_0(s), t; t_0) \cdot \gamma_0'(s), \quad (42)$$

which corresponds to the closed form solution of the variational state $\delta x_{\nu, \mu}$ in (29) for $\mu = 0$.

Furthermore, (35) links $\gamma'(s, t)$ to $\delta x(t)$ in (6), i.e.,

$$\lim_{s \to 0} \gamma'(s, t) = \delta x(t).$$

Thus, $\gamma'(s, t)$ acts as an *extended variational state*; for a given time $t$, each $s \in [0, 1]$ corresponds to the tangent vector at $\gamma(s, t)$. See Fig.3 for a graphical interpretation.

**Theorem VI.1** is also constructive in the sense that provides an upper bound for the norm of the spatial derivatives of the curve allowing us to obtain the following stability results.

**Corollary VI.1 (differential Regional Asymptotic Stability).** In the framework of Th. VI-.1, the curve $\gamma$ satisfies, for $s \in (0, 1]$,

$$\|\gamma'(s, t)\| \le \exp\left(-\int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau\right) \cdot \sqrt{\frac{\lambda_{\max}(\gamma(s, t_0), t_0)}{\lambda_{\min}(\gamma(s, t), t)}} \|\gamma'(s, t_0)\|. \quad (43)$$

Furthermore, if there exists $\lambda_{t_0} > 0$ such that $\lambda_{\min}(x(t), t) \ge \lambda_{t_0}$ for all $t \ge t_0$, then system (3) is *dRAS (Def.(IV-C.2)) on* $\mathcal{X} \times [t_0, \infty)$.

**Proof.** Using the two inequalities in (40), from (41) we have, for $s \ne 0$,

$$\begin{aligned} \lambda_{t_0} \|\gamma'(s, t)\|^2 \le \lambda_{\min}(\gamma(s, t), t) \|\gamma'(s, t)\|^2 \le \\ \exp\left(-2 \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau\right) \cdot \lambda_{\max}(\gamma(s, t_0), t_0) \|\gamma'(s, t_0)\|^2. \end{aligned}$$

Rearranging the terms, we obtain

$$\|\gamma'(s, t)\|^2 \le \frac{\lambda_{\max}(\gamma(s, t_0), t_0)}{\lambda_{t_0}} \exp\left(-2 \int_{t_0}^{t} \frac{\beta(\gamma(s, \tau), \tau)}{\lambda_{\max}(\gamma(s, \tau), \tau)} d\tau\right) \cdot \|\gamma'(s, t_0)\|^2.$$

And from (39) the result follows. □

A particular case of the above results is the differential exponential stability, requiring stronger conditions on $P$.

**Corollary VI-.2. (differential Regional Exponential Stability)** In the framework of Th.VI-.1, if there exist positive $\lambda, \bar{\lambda}, \underline{\beta} \in \mathbb{R}$ such that for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$

$$\lambda \le \lambda_{\min}(x, t), \quad \lambda_{\max}(x, t) \le \bar{\lambda}, \quad \underline{\beta} \le \beta(x, t)$$

then, for $s \in (0, 1]$, $\gamma'$ satisfies (18) with $\kappa = \sqrt{\lambda/\bar{\lambda}}$ and $\alpha = \beta/\bar{\lambda}$. Equivalently, system (3) is *dRES (Def.(IV-C.3)) on* $\mathcal{X} \times [t_0, \infty)$.

**Proof.** The proof follows directly from (43), where

$$-\frac{\beta(x, t)}{\lambda(x, t)} \le -\frac{\beta}{\bar{\lambda}}, \quad \forall (x, t) \in \mathcal{X} \times [t_0, \infty)$$

and

$$\frac{\lambda_{\max}(x, t_0)}{\lambda_{\min}(x, t)} \le \frac{\bar{\lambda}}{\underline{\lambda}}, \quad \forall (x, t) \in \mathcal{X} \times [t_0, \infty).$$

□

To include $s = 0$ in the domain of definition of the curves throughout the development in this section, it is sufficient to impose the additional assumption $\gamma_0'(0) = 0 \in \mathbb{R}^n$ and use (42). Note that, in the classical contraction framework as described in section III, this initialization would induce, by continuity of $\gamma_0'$, the trivial solution $\delta x(t) = 0$ (see also Remark VI.3), for all $t \ge t_0$, whereas in this setup it does not affect the results, in view of the continuity of the differential Lyapunov function in $s$.

## A. Reinterpreting Lyapunov functions

For an LTI system, the vector field is given by $f(x(t), t) = A x(t)$ with $x(t_0) = x_0 \in \mathbb{R}^n$. When $A \in \mathbb{R}^{n \times n}$ is Hurwitz, the origin is the unique equilibrium point. A natural choice for the initial connecting curve $\gamma_0$ is a straight line joining the initial condition $x_0 \in \mathbb{R}^n$ to the origin:

$$\gamma_0(s) = x_{0a} + s(x_{0b} - x_{0a}) = x_0 + s(0 - x_0) = (1 - s)x_0.$$

The flow of an LTI system is explicitly known to be $\varphi(x_0, t; t_0) = \exp(A(t - t_0))x_0$, therefore, the double-parameter curve $\gamma$ takes the form

$$\gamma(s, t) := \varphi(\gamma_0(s), t; t_0) = \exp(A(t - t_0))\gamma_0(s).$$

Its spatial derivative $\gamma'(s, t)$ is given by

$$\begin{aligned} \gamma'(s, t) &= \frac{\partial \varphi}{\partial x_0}((1-s)x_0, t; t_0) \frac{\partial (1-s)x_0}{\partial s} \\ &= -\exp(A(t-t_0))x_0 = -x(t). \end{aligned}$$

For intuition, Fig. 4 shows the vector field of a bidimensional LTI system with a “slow” repeated eigenvalue $\lambda = -0.01$, together with the initial connecting line $\gamma_0$ and its spatial derivative as $s = 0$, i.e., $\gamma_0'(0)$. Consider now a constant positive definite matrix $P = P^\top \in \mathbb{R}^n$. The length of the curve $\gamma(s, t)$ induced by the metric in $P$, as in (36), is

$$\begin{aligned} \mathcal{L}_g(\gamma, t) &= \int_0^1 \sqrt{\gamma'(s, t)^\top P \gamma'(s, t)} ds \\ &= \int_0^1 \sqrt{-x(t)^\top P (-x(t))} ds = \sqrt{x(t)^\top P x(t)}. \end{aligned}$$
<figure><img src="image_6.png" /><figcaption>Fig. 4: Vector field with repeated eigenvalues. The states' dynamics are given by $\dot{x}_1 = \lambda x_1 + x_2$ and $\dot{x}_2 = \lambda x_2$. The white line represents the straight segment $\gamma_0(s)$ connecting the initial condition $x_0$ to the origin. The red vector depicts the spatial derivative $\gamma_0'(0) = -x_0$, applied at $\gamma(0, t_0) = x_0$ and oriented in the direction of $-x_0$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5: Vector field in the transformed coordinates $z = \text{diag}(\lambda, \lambda^2)x = Hx$ as proposed in [56]. In these coordinates, the states' dynamics are given by $\dot{z}_1 = \lambda z_1 + \lambda z_2$ and $\dot{z}_2 = \lambda z_2$. The white curve denotes the straight line $\gamma_0(s)$ connecting the initial condition $z_0 = \text{diag}(\lambda, \lambda^2)x_0$ to the origin. The red vector, instead, represents the spatial derivative $\gamma_0'(0) = -z_0$ applied at $\gamma(0, t_0) = z_0$ and oriented in the direction of $-z_0$. The phase portrait is plotted with a scale factor of $10^2$ for visual clarity.</figcaption></figure>

Thus, the classical Lyapunov function $V(x) = x^\top P x$ for LTI systems is not an energy function dissipated in the physical sense, but rather the squared length of the curve connecting the trajectory $x(t)$ to the equilibrium at the origin under the metric in $P$.

A strictly decreasing Lyapunov function $V$, where $P$ satisfies the Lyapunov inequality $PA + A^\top P < 0$ corresponds to a strictly decreasing curve length $\mathcal{L}_g(\gamma, t)$.

Solving the Lyapunov inequality also provides a constant invertible matrix $H$, such that $P = H^\top H$, yielding a change of coordinates $z = Hx$ under which the Euclidean norm $\|z\|$ becomes strictly non-increasing. For visualization, Fig. 5 shows the same bidimensional vector field as in Fig. 4, but expressed in the new change of coordinates. The connecting line $\gamma_0$ and its spatial derivative $\gamma_0'(0)$ retain the same interpretation in transformed state space.

Finally, observe that the constant matrix $P$ defines both a metric and an inner product. Hence, for linear systems, the Lyapunov function $V = x^\top P x = z^\top z$ simultaneously represents the squared length of the connecting curve $\mathcal{L}_g(\gamma, t)$ and the square of the norm $\|x\|_P := \sqrt{x^\top P x} = \sqrt{z^\top z} = \|z\|$. **Remark VI.4** Conceptually, convergence to an equilibrium point, to an invariant trajectory, or mutual attraction among trajectories can be established by showing that the length $\mathcal{L}_{g,\gamma,t}$ is finite in forward time and converges to zero asymptotically. This weakens the classical requirement of a strictly decreasing (or non-increasing) Lyapunov function.

## VII. CONCLUSIONS AND FUTURE WORKS

In this work, we developed a comprehensive and geometrically grounded framework for the analysis of stability, convergence, and contractivity of nonlinear time-varying

dynamical systems. Starting from classical Lyapunov-based notions of stability, we progressively introduced incremental and differential perspectives, showing how these concepts naturally extend to the characterization of contractive behavior in nonlinear systems.

To support the developments, we presented a concise yet essential mathematical background, encompassing metric structures, parametrized curves on differentiable manifolds, and associated Riemannian length. We then formalized the dynamical context on manifolds with time-varying vector fields, revisiting the definitions of the flow map, variational states, and prolonged dynamics. Building on this, we introduced generalized variational dynamics, highlighting how traditional variational equations only capture variations in initial conditions, whereas time-varying systems require a broader description in which variations also arise from the time domain. To prove then the convergence between trajectories of the system, we leveraged the definition of double-parameterized curves, connecting such trajectories, to derive an extended variational dynamics. The vanishing length of these curves provided a unified and geometrically transparent condition for contraction within a region $\mathcal{X}$, allowing us to establish sufficient conditions for asymptotic and exponential convergence in time-varying nonlinear systems. Connections with the classical notion of Banach contraction were discussed, suggesting that the differential Lyapunov function defines a contraction.

Furthermore, the employment of the Riemannian length in the convergence proof implies a reinterpretation of the Lyapunov
punov function. In particular, differential Lyapunov functions naturally emerge as metrics on the underlying manifold rather than as energy-like quantities associated with the physical system, offering a deeper geometric interpretation of stability and convergence.

Overall, the results presented in this paper lay a rigorous foundation for the systematic study of contractivity in nonlinear time-varying dynamical systems. The proposed framework opens several promising directions for future research, including the development of constructive methods for designing time-varying contraction metrics, as proposed in the companion work of this paper [57].

## A. Future Works

The results suggest several extensions to different classes of dynamical systems, such as discrete-time, switching, hybrid, and non-smooth systems. Each of these poses unique challenges, particularly regarding the definition of generalized variations, contraction criteria, and discontinuities in the governing vector field.

Potential applications include observer design and distributed dynamical systems, where contraction-based metrics and operator-based decompositions may yield new insights into synchronization, estimation error dynamics, and collective stability in interconnected networks. Moreover, operator-based spectral decomposition opens further opportunities for a structured analysis of nonlinear systems and for the development of new control design methodologies. These include applications in the stabilization of nonlinear dynamical systems, in output regulation, and in the characterization of dominant dynamics that govern the behavior of complex nonlinear models.

The metric space observation in Section VI also raises the question of whether contractivity is primarily a dynamical property, a topological one, or an interplay of both. This observation opens a further line of research aimed at deepening the understanding of the coupling between the topology of invariant sets and the underlying system dynamics.

Another promising direction concerns the extension of the theory to forward-time contraction analysis, which aims to determine whether a system trajectory contracts onto itself over time, effectively eliminating any dependence on initial time shifts. Formalizing this notion requires a precise definition of steady-state trajectories and a detailed understanding of how such trajectories can emerge or persist in the nonlinear time-varying settings. Furthermore, the generalized variational approach introduced in this work naturally raises questions regarding the existence and uniqueness of solutions for the original nonlinear system. The generalized variational state is directly connected to the concept of system flows and provides a framework to study the forward-time evolution of the system when perturbations are applied along the time dimension. This property makes the generalized variational state, introduced in Sec. V, a potentially powerful tool for deepening the understanding of dynamical systems with a differentiable vector field.

Finally, an ambitious but natural direction is the extension to infinite-dimensional systems, including distributed parameter systems and systems governed by partial differential equations.

Overall, these directions highlight the richness of the proposed framework and point toward a broad range of theoretical, computational, and applied developments that can further advance the understanding of nonlinear systems dynamics, their differential stability and contraction properties.

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<figure><img src="image_8.png" /></figure>

Mario Spirito Mario Spirito received the B.S. and M.S. (Laurea) degrees cum laude in Automation Engineering from the University of Bologna, Bologna, Italy, in 2016 and 2018, respectively. He earned the Ph.D. degree in Systems Engineering (Automatic Control) from the same institution in 2022. Throughout 2022, he was a Research Fellow with the ACTEMA research group at the University of Bologna. From January 2023 to December 2024, he served as a Postdoctoral Researcher at Université Claude Bernard Lyon 1, Lyon, France, where he was funded by the ANR project IMPACT. Since March 2025, he has been a Postdoctoral Researcher with the Rehab Technologies Lab at the Italian Institute of Technology (IIT), Genoa, Italy. His current work involves control-oriented models for human-robot interaction systems, soft exoskeletons, and upper-limb prostheses.

His research interests include nonlinear control theory, output regulation, and observer design, including nonlinear identity observers and KKL observers. He also focuses on the control of non-minimum phase systems, energy-based modeling of complex multi-physics dynamics (such as port-Hamiltonian systems and SMA), and internal models in neuroscience and biological systems.