Incremental Global Asymptotic Stability (δ-GAS)

• δ-GAS is a trajectory-based stability property that guarantees all solutions driven by the same input converge, characterized by KL decay bounds in Euclidean or invariant metrics.
• It unifies various analysis methods, including Lyapunov functions, contraction metrics, and dissipativity, to certify stability in nonlinear, hybrid, and delay systems.
• Applications range from controller synthesis and piecewise-affine system verification to data-driven compositional methods in large-scale networks and stochastic frameworks.

Searching arXiv for recent and foundational papers on incremental global asymptotic stability ($\delta$-GAS). Searching arXiv for "incremental global asymptotic stability delta-GAS". Incremental Global Asymptotic Stability, usually denoted $\delta$-GAS, is a trajectory-to-trajectory stability property: instead of asking whether solutions converge to a fixed equilibrium, it asks whether any two solutions driven by the same admissible input converge to one another according to a $\mathcal{KL}$ estimate. In the Euclidean formulation this takes the form of a decay bound on $\|x(t)-\tilde x(t)\|$; in coordinate-invariant formulations it is stated with respect to an arbitrary metric $d$. Closely related terminology includes “incremental asymptotic stability,” “global incremental stability” (GIS), “convergence of all solutions,” and, in stronger cases, incremental exponential stability. Across the literature represented here, $\delta$-GAS appears as a state-space property for nonlinear control systems, as a contraction property for nonautonomous disturbed systems, and as a compositional certificate for large-scale networks (Zamani et al., 2012, Waitman et al., 2016, Vrabel, 2022, Zaker et al., 24 Jul 2025).

1. Formal notion and principal variants

For control systems $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$, the coordinate-invariant formulation introduced as $\delta_{\exists}$-GAS requires forward completeness and the existence of a metric $d$ and a function $\beta\in\mathcal{KL}$ such that, for any $\delta$0, any $\delta$1, and any common input $\delta$2,

$\delta$3

The same-input requirement is structural: the property compares trajectories under identical exogenous signals, whereas input-mismatch belongs to incremental ISS-type notions. The paper introducing $\delta$4-GAS is explicit that Angeli’s original Euclidean $\delta$5-GAS is generally not coordinate invariant, and that allowing an arbitrary metric restores coordinate invariance (Zamani et al., 2012).

A Euclidean state-space formulation is used for general nonlinear and piecewise-affine systems in the form

$\delta$6

for all $\delta$7, all $\delta$8, and any common input $\delta$9. When $\mathcal{KL}$0, the property is called incrementally globally asymptotically stable. The same source distinguishes regional and global versions, and identifies incremental exponential stability as the stronger case in which $\mathcal{KL}$1 for some $\mathcal{KL}$2 (Waitman et al., 2016).

For nonlinear time-varying perturbed systems

$\mathcal{KL}$3

the logarithmic-norm treatment uses the term GIS and defines incremental asymptotic stability on a positively invariant set $\mathcal{KL}$4 by the existence of $\mathcal{KL}$5 such that

$\mathcal{KL}$6

When $\mathcal{KL}$7, this is GIS; in standard control language, this corresponds to $\mathcal{KL}$8-GAS in the norm-induced metric under consideration. That source also distinguishes the stronger estimate

$\mathcal{KL}$9

which is standard incremental global uniform exponential stability (Vrabel, 2022).

2. Lyapunov, diagonal-set, and dissipativity characterizations

A central theme in the modern treatment of $\|x(t)-\tilde x(t)\|$0-GAS is that incremental stability can be reduced to ordinary stability of a lifted system. The coordinate-invariant framework constructs the product system

$\|x(t)-\tilde x(t)\|$1

and studies the diagonal set

$\|x(t)-\tilde x(t)\|$2

With the product metric $\|x(t)-\tilde x(t)\|$3, the identity

$\|x(t)-\tilde x(t)\|$4

turns inter-trajectory distance into distance-to-set. Under compactness of $\|x(t)-\tilde x(t)\|$5 and continuity of the metric fibers $\|x(t)-\tilde x(t)\|$6, $\|x(t)-\tilde x(t)\|$7-GAS is equivalent to the existence of a $\|x(t)-\tilde x(t)\|$8-GAS Lyapunov function $\|x(t)-\tilde x(t)\|$9 satisfying

$d$0

and

$d$1

for all $d$2 and $d$3. This converse Lyapunov theorem is one of the strongest structural results in the cited literature (Zamani et al., 2012).

A related but distinct route uses dissipativity on an augmented system. For general nonlinear systems $d$4, $d$5, incremental $d$6-gain is treated as dissipativity of the augmented dynamics with state $d$7, output difference $d$8, and supply rate

$d$9

When the compared inputs are equal, the dissipation inequality yields monotonicity of the incremental storage. To recover $\delta$0-GAS from incremental $\delta$1-gain, however, the paper requires additional uniform observability and uniform reachability assumptions. Under those assumptions, the storage admits lower and upper bounds in the state difference and strict finite-horizon decay, which is converted into a $\delta$2 estimate. Conversely, incremental asymptotic stability implies incremental $\delta$3-gain only under differentiability and local Lipschitz regularity of the Jacobian. This establishes that incremental $\delta$4-gain and $\delta$5-GAS are closely related but not interchangeable without auxiliary hypotheses (Waitman et al., 2016).

3. Contraction, logarithmic norms, and Demidovich-type criteria

For nonautonomous disturbed systems, $\delta$6-GAS can be certified through matrix measures. In the perturbed system

$\delta$7

with $\delta$8 continuously differentiable in $\delta$9, continuous in $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$0, $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$1 continuous, and solutions uniquely determined for all $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$2, the key assumption is the global Jacobian bound

$\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$3

where $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$4 is the logarithmic norm associated with a chosen vector norm. The logarithmic norm is defined by

$\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$5

For the Euclidean norm,

$\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$6

and for a weighted Euclidean norm $\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$7,

$\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$8

Under the uniform negativity condition above, any two solutions satisfy

$\Sigma=(\mathbb{R}^n,U,\mathcal{U},f)$9

which is stronger than abstract $\delta_{\exists}$0-incremental stability and yields global uniform exponential incremental stability (Vrabel, 2022).

The proof proceeds by exact linearization of the trajectory difference through the mean-value identity

$\delta_{\exists}$1

which produces a linear time-varying system $\delta_{\exists}$2 for $\delta_{\exists}$3. Convexity of the matrix measure implies $\delta_{\exists}$4, and standard transition-matrix estimates then yield exponential contraction. A weaker assumption,

$\delta_{\exists}$5

still gives

$\delta_{\exists}$6

which ensures asymptotic pairwise convergence but is generally not uniform and not exponential (Vrabel, 2022).

That same treatment generalizes the classical Demidovich criterion both “horizontally and vertically.” The horizontal generalization extends the contraction test from weighted Euclidean norms to arbitrary norms and corresponding matrix measures. The vertical generalization adds a result that identifies a common limit, namely the origin, under the additional asymptotic smallness condition

$\delta_{\exists}$7

Under this condition every solution satisfies $\delta_{\exists}$8, even when $\delta_{\exists}$9 is not an equilibrium of the perturbed system and may fail to be a solution of the nominal system. The paper states that the logarithmic-norm approach makes stability “a topological notion”; technically, this refers to formulation with respect to arbitrary norm-induced topologies rather than a restriction to quadratic Euclidean metrics, not to coordinate invariance in the differential-geometric sense (Vrabel, 2022).

4. Computational verification and compositional certification

For piecewise-affine systems, $d$0-GAS can be certified by finite-dimensional convex conditions. The continuous-time PWA model

$d$1

is analyzed through an augmented system on $d$2 and a continuous piecewise-quadratic incremental Lyapunov or storage function. On diagonal cells $d$3, the storage must reduce to a quadratic form in $d$4; on off-diagonal cells $d$5, it may depend piecewise-quadratically on $d$6. The main theorem gives LMIs involving regional matrices $d$7, $d$8, S-procedure multipliers, continuity-enforcing matrices, and positive scalars $d$9. These LMIs enforce lower and upper quadratic bounds and strict decrease along same-input trajectories, yielding an incremental exponential estimate and therefore $\beta\in\mathcal{KL}$0-GAS when $\beta\in\mathcal{KL}$1. The method is explicitly presented as less conservative than single-quadratic certificates, but it remains sufficient, assumes no Zeno behavior, requires a common feedthrough matrix $\beta\in\mathcal{KL}$2, and may fail when local subsystem matrices $\beta\in\mathcal{KL}$3 are non-Hurwitz (Waitman et al., 2016).

A more recent verification direction is fully data-driven and compositional. For an interconnected discrete-time network of unknown nonlinear homogeneous subsystems

$\beta\in\mathcal{KL}$4

with each $\beta\in\mathcal{KL}$5 continuous and homogeneous of degree one, subsystem-level incremental stability is treated as $\beta\in\mathcal{KL}$6-ISS with respect to the internal inputs $\beta\in\mathcal{KL}$7. Each subsystem Lyapunov candidate is parameterized as

$\beta\in\mathcal{KL}$8

with homogeneous basis functions. Because the dynamics are unknown, the exact robust optimization program is replaced by a scenario optimization program based on one-step transition data normalized to the unit sphere. If the optimizer $\beta\in\mathcal{KL}$9 satisfies the deterministic correctness condition

$\delta$00

where $\delta$01 is a covering radius on the unit sphere and $\delta$02 is a Lipschitz bound for the constraint functions, then $\delta$03 is a valid subsystem $\delta$04-ISS Lyapunov function (Zaker et al., 24 Jul 2025).

Network-level $\delta$05-GAS is then obtained through a small-gain composition. Defining

$\delta$06

the compositional condition is

$\delta$07

Under that condition,

$\delta$08

is an incremental Lyapunov function for the full interconnection and certifies $\delta$09-GAS. The paper emphasizes that, for fixed subsystem dimensions, the compositional approach yields sample complexity growing linearly with subsystem count, whereas the monolithic approach exhibits exponential growth. Its large-scale example treats a homogeneous ring network with $\delta$10 subsystems and reports successful certification with a correctness guarantee (Zaker et al., 24 Jul 2025).

5. Delay systems, stochastic analogues, and controller synthesis

Time-delay systems make the uniformity aspect of $\delta$11-GAS especially delicate. The paper on retarded time-delay systems does not define incremental stability directly, but it is relevant because incremental arguments are commonly reduced to ordinary GAS or UGAS of a product or error system. For

$\delta$12

with state in $\delta$13, it proves that $\delta$14. It also proves that, in the stronger spaces $\delta$15 or $\delta$16 with $\delta$17, forward completeness implies RFC, and consequently $\delta$18. The same source provides Lyapunov-Krasovskii characterizations of UGAS-X and, for $\delta$19, $\delta$20, a Dini-derivative characterization. A plausible implication is that once a $\delta$21-GAS problem for a delay system is converted into stability of a lifted retarded system, these results supply the “uniformization” step needed to obtain a $\delta$22 estimate; however, that paper does not itself formulate $\delta$23-GAS, does not state a diagonal-set theorem, and does not address neutral delay systems incrementally (Karafyllis et al., 2022).

For stochastic systems with diffusion and jumps, the cited literature replaces deterministic $\delta$24-GAS by a moment-based coordinate-invariant notion. The central definition is $\delta$25-ISS-$\delta$26: $\delta$27 When the compared inputs are equal, this becomes the stochastic analogue of $\delta$28-GAS in the $\delta$29-th moment. The corresponding incremental Lyapunov function $\delta$30 is defined on $\delta$31 and must satisfy metric bounds together with a generator inequality containing drift, diffusion, and jump terms. For stochastic Hamiltonian systems with jumps, an explicit backstepping controller and composite incremental Lyapunov function are constructed, yielding $\delta$32-ISS-$\delta$33, and hence the equal-input specialization gives incremental global asymptotic convergence in moment. The spring-pendulum example illustrates this mechanism in a noisy environment (Jagtap et al., 2016).

Deterministic controller synthesis also appears in a backstepping form. For systems

$\delta$34

if the virtual subsystem

$\delta$35

is $\delta$36-ISS with respect to $\delta$37, then the feedback

$\delta$38

renders the closed loop $\delta$39-ISS with respect to $\delta$40. Setting $\delta$41 yields a $\delta$42-GAS closed-loop system. The same framework provides recursive constructions of incremental Lyapunov functions and, in the smooth case, contraction metrics; it is also presented as applicable to some non-smooth systems (Zamani et al., 2012).

6. Interpretive issues, related notions, and limitations

A recurring interpretive issue is metric dependence. Euclidean $\delta$43-GAS, GIS defined through a chosen norm, and coordinate-invariant $\delta$44-GAS are closely related but not identical notions. The literature here is explicit that Euclidean incremental stability is generally not coordinate invariant, whereas $\delta$45-GAS is formulated with respect to some metric and is therefore coordinate invariant (Zamani et al., 2012). By contrast, the logarithmic-norm GIS criterion is norm-flexible but “not intrinsically coordinate-invariant in the modern differential-geometric sense”; its success may depend on the chosen norm or weighted norm (Vrabel, 2022).

Another common misconception is that any incremental energy or input-output property should imply $\delta$46-GAS. The dissipativity-based analysis shows otherwise: incremental $\delta$47-gain is an input-output bound, and it implies incremental asymptotic stability only under suitable observability and reachability assumptions; the converse likewise requires regularity of the state-space representation (Waitman et al., 2016).

Uniformity is also more subtle than finite-dimensional intuition suggests. For finite-dimensional time-invariant systems, GAS and UGAS are equivalent, but the delay-system analysis shows that this equivalence is nontrivial in retarded infinite-dimensional settings and may depend on RFC or on working in stronger Sobolev or Hölder state spaces. The same source notes that for neutral functional differential equations, GAS and UGAS are not equivalent, even in the linear time-invariant case (Karafyllis et al., 2022).

Finally, all verification frameworks discussed here are sufficient rather than necessary, and each carries structural restrictions. The logarithmic-norm criterion is global, finite-dimensional, and Jacobian-based (Vrabel, 2022). The PWA LMI method assumes no Zeno behavior and can miss incrementally stable systems with non-Hurwitz modes (Waitman et al., 2016). The data-driven compositional certificate hinges on degree-one homogeneity, the chosen Lyapunov parameterization, sufficient data coverage, Lipschitz estimation, and a sufficient small-gain inequality (Zaker et al., 24 Jul 2025). The stochastic backstepping results certify convergence in expected $\delta$48-th moments rather than almost-sure convergence (Jagtap et al., 2016).

Taken together, these works present $\delta$49-GAS as a unifying notion for pairwise trajectory convergence across nonlinear control theory, contraction analysis, dissipativity, hybrid and piecewise-affine computation, delay systems, stochastic dynamics, and data-driven certification. What changes from one framework to another is not the core idea—that all trajectories driven by the same signal should converge toward one another—but the metric, the certificate class, the form of uniformity, and the assumptions under which that convergence can be proved.