Incremental Input-State Stability (δ-ISS)

• δ-ISS is a robustness property for dynamical systems that bounds the difference between any two trajectories based on initial state and input differences.
• It employs incremental Lyapunov functions and contraction metrics to certify stability, supporting applications in neural networks, hybrid systems, and control design.
• Data-driven synthesis and neural certification methods have advanced the practical implementation of δ-ISS in reinforcement learning and distributed control.

Searching arXiv for recent and relevant papers on incremental input-to-state stability to ground the article in the literature. Incremental input-to-state stability, denoted $\delta$-ISS, is an incremental robustness property for dynamical systems in which the distance between two trajectories is bounded by a term that decays with time and depends on the difference in initial states, together with a term depending on the difference between the two input signals. Unlike standard ISS, which compares a trajectory to an equilibrium or the origin, $\delta$-ISS compares two arbitrary trajectories of the same system driven by possibly different inputs. Across recent work, $\delta$-ISS appears as a unifying concept in nonlinear control, hybrid locomotion, recurrent neural networks, data-driven control synthesis, stochastic contraction theory, and reinforcement-learning-oriented stability analysis (Zaker et al., 2024, D'Amico et al., 2022, Pfrommer et al., 1 Jul 2025, Kawano et al., 20 Feb 2026).

1. Definition and conceptual scope

For continuous-time nonlinear systems, $\delta$-ISS is stated by requiring the existence of functions $\beta\in\mathcal{KL}$ and $\gamma\in\mathcal{K}_\infty$ such that, for any initial conditions and any two input signals, the state mismatch satisfies

$$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$

This formulation is used explicitly for continuous-time polynomial systems in the data-driven synthesis setting (Zaker et al., 2024). For discrete-time systems, the same structure appears in the form

$$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$

which is the standard definition adopted for broad recurrent neural network classes (D'Amico et al., 2022).

A closely related formulation is used for graph neural networks: $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$ with $\beta_{\delta}\in\mathcal{KL}$ and $\delta$0 (Marino et al., 2023). For deep LSTM networks, the corresponding property is written as

$\delta$1

with respect to an invariant state set $\delta$2 and a bounded input set $\delta$3 (Bonassi et al., 2023).

The central distinction from ordinary ISS is stated explicitly in multiple sources: ISS bounds one trajectory with respect to the origin, whereas $\delta$4-ISS bounds the distance between two trajectories. This makes $\delta$5-ISS a stronger trajectory-to-trajectory robustness property and one that is especially suited to synchronization, abstraction and compositional verification, nonlinear circuits, cyclic feedback systems, robust identification, and control-oriented learning (Marino et al., 2023, Zaker et al., 2024, Bonassi et al., 2023).

A recurring implication is that if the two inputs are equal, then $\delta$6-ISS reduces to an incremental asymptotic stability property. In the polynomial data-driven setting, this is stated as incremental asymptotic stability, denoted $\delta$7-GAS (Zaker et al., 2024). In neural network papers, the same point is phrased as convergence of trajectories when inputs match (Marino et al., 2023, Bonassi et al., 2023).

2. Lyapunov characterizations

A standard route to $\delta$8-ISS is via an incremental Lyapunov function $\delta$9 defined on a pair of states. For discrete-time systems, one characterization is: $\delta$0 and

$\delta$1

with $\delta$2 and $\delta$3; existence of such a Lyapunov function implies $\delta$4-ISS (D'Amico et al., 2022). An analogous formulation is used for unknown discrete-time systems: $\delta$5 and

$\delta$6

with $\delta$7 and $\delta$8 (Basu et al., 10 Jan 2025).

For continuous-time systems, the dissipation inequality takes the derivative form

$\delta$9

together with positive definiteness bounds in $\delta$0 (Basu et al., 25 Apr 2025). This is the continuous-time $\delta$1-ISS control Lyapunov function formulation used for unknown systems.

Several papers introduce control-oriented variants. A $\delta$2-ISS control Lyapunov function ($\delta$3-ISS-CLF) is defined as a function $\delta$4 that satisfies the incremental positivity and dissipation inequalities under some controller $\delta$5; if the closed-loop system admits such a $\delta$6-ISS-CLF, then it is incrementally input-to-state stable (Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025). In the discrete-time unknown-system setting, the controller and the $\delta$7-ISS-CLF are both learned from data and then formally verified (Basu et al., 6 Mar 2025). In the continuous-time analogue, both are parameterized by neural networks and certified through a Lipschitz-based validity condition (Basu et al., 25 Apr 2025).

Hybrid locomotion introduces a specialized Lyapunov notion for disturbed Poincaré maps. There, a robust Lyapunov function $\delta$8 satisfies

$\delta$9

and

$\beta\in\mathcal{KL}$0

An equivalent relaxed form is

$\beta\in\mathcal{KL}$1

This is explicitly identified as an ISS-type Lyapunov pattern for the disturbed return map of a walking gait (Tucker et al., 2023).

3. Core analytical frameworks

Recent work develops $\beta\in\mathcal{KL}$2-ISS through several distinct but mathematically related analytical frameworks.

One line uses contraction or weighted quadratic metrics. For a generic discrete-time recurrent neural network

$\beta\in\mathcal{KL}$3

a sufficient condition is the existence of a symmetric positive definite matrix $\beta\in\mathcal{KL}$4 with a structural sparsity constraint on nonlinear coordinates such that

$\beta\in\mathcal{KL}$5

where $\beta\in\mathcal{KL}$6 and $\beta\in\mathcal{KL}$7 contains componentwise Lipschitz constants of the activation map (D'Amico et al., 2022). The proof uses the quadratic incremental Lyapunov candidate

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

This condition is presented as less conservative than earlier norm-based criteria and can be converted into LMIs in controller and observer design settings (D'Amico et al., 2022).

A second line develops graph-aware, weight-based conditions for gated graph neural networks. A GGNN is modeled as a distributed discrete-time nonlinear dynamical system with graph filters and gates, and a sufficient condition for single-layer $\beta\in\mathcal{KL}$9-ISS is

$\gamma\in\mathcal{K}_\infty$0

with

$\gamma\in\mathcal{K}_\infty$1

This bound arises from the $\gamma\in\mathcal{K}_\infty$2-Lipschitz property of $\gamma\in\mathcal{K}_\infty$3, the $\gamma\in\mathcal{K}_\infty$4-Lipschitz property of sigmoid, and the decomposition of state, input, and graph-support differences (Marino et al., 2023). For deep GGNNs, each layer must satisfy its own condition $\gamma\in\mathcal{K}_\infty$5 (Marino et al., 2023).

A third line extends layerwise contraction reasoning to deep LSTMs. For each layer $\gamma\in\mathcal{K}_\infty$6, the paper derives nonlinear inequalities involving recurrent matrices $\gamma\in\mathcal{K}_\infty$7, gate bounds, and invariant-set bounds. The sufficient condition is compactly written as

$\gamma\in\mathcal{K}_\infty$8

where $\gamma\in\mathcal{K}_\infty$9 collects $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$0 nonlinear inequalities for an $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$1-layer network (Bonassi et al., 2023). The proof stacks per-layer difference inequalities into a global block system and shows the resulting global matrix is Schur stable (Bonassi et al., 2023).

A fourth line derives $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$2-ISS for positive Lur’e systems through linear dissipativity. Under the incremental gain condition

$$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$3

and the positivity hypothesis, the paper proves weighted one-norm trajectory-pair estimates such as

$$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$4

and states that for $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$5 this is an incremental exponential ISS estimate (Piengeon et al., 2024). The mechanism is a positive-systems dissipativity argument rather than a quadratic Euclidean one.

A fifth line reformulates incremental stability in reinforcement-learning terms. For a deterministic policy $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$6, the paper studies a local $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$7-ISS condition

$$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$8

and proves an equivalence between such local incremental stability and uniform Hölder regularity of value and $$|x(t)-\tilde x(t)| \le \beta\big(|x(0)-\tilde x(0)|,t\big)+\gamma\big(|u-\tilde u|_\infty\big),\qquad t>0.$$9-functions over a sufficiently sensitive reward class (Pfrommer et al., 1 Jul 2025). This shifts the certificate from a decrease condition on one function to regularity of a family of reward-to-go functions.

4. Variants, relaxations, and adjacent notions

A prominent relaxation is incremental input-to-state practical stability, denoted $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$0-ISpS. This notion permits a nonzero residual term: $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$1 for some metric $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$2, $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$3, $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$4, and constant $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$5 (Sundarsingh et al., 2024). The cited papers explicitly note that if $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$6, this becomes standard $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$7-ISS (Sundarsingh et al., 2024, Sangeerth et al., 12 Oct 2025). The practical term appears because Gaussian-process approximations of unknown dynamics introduce unavoidable model uncertainty (Sundarsingh et al., 2024, Sangeerth et al., 12 Oct 2025).

In strict-feedback nonlinear systems with unknown drift terms, Gaussian-process learning and backstepping are used to obtain $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$8-ISpS with respect to an external incremental input $$\|x(k,x_{01},\vec u_1)-x(k,x_{02},\vec u_2)\| \le \beta(\|x_{01}-x_{02}\|,k)+\gamma(\|\vec u_1-\vec u_2\|_\infty),$$9. The composite Lyapunov function

$$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$0

satisfies

$$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$1

which yields $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$2-ISpS rather than exact $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$3-ISS because $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$4 is induced by GP approximation errors (Sundarsingh et al., 2024). A feedback-linearization-based variant combines $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$5-ISpS with control barrier functions and robust forward invariance, again with a nonzero practical residual (Sangeerth et al., 12 Oct 2025).

Locomotion theory introduces a related but domain-specific notion, $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$6-robustness, for periodic gaits on uneven terrain. Ground-height variation is modeled as a disturbance entering through the guard condition

$$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$7

and the extended Poincaré map yields a discrete-time system with input $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$8: $$\|\bm{x}(t)_1-\bm{x}(t)_2\|_\infty \le \beta_{\delta}\!\left(\|\bm{x}(0)_1-\bm{x}(0)_2\|_\infty,t\right) + \gamma_{\delta}\!\left(\|\bm{u}_1-\bm{u}_2\|_\infty\right),$$9 The resulting bound

$\beta_{\delta}\in\mathcal{KL}$0

is an exponential ISS estimate for the gait evolution under bounded terrain disturbances (Tucker et al., 2023). This suggests that, in hybrid locomotion, $\beta_{\delta}\in\mathcal{KL}$1-ISS-type reasoning can quantify robustness of nominal periodic orbits under guard perturbations.

Stochastic systems motivate another adjacent notion: incremental noise- and input-to-state stability. For an Itô SDE with uniformly contracting drift and uniformly Lipschitz input dependence, the mean-square inter-trajectory distance satisfies an exponential estimate with an input-convolution term and an additive noise term proportional to the diffusion bound $\beta_{\delta}\in\mathcal{KL}$2 (Kawano et al., 20 Feb 2026). The paper interprets this as a stochastic generalization of deterministic $\beta_{\delta}\in\mathcal{KL}$3-ISS.

5. Learning and data-driven synthesis

A major current theme is the synthesis or certification of $\beta_{\delta}\in\mathcal{KL}$4-ISS from data without an explicit model.

For unknown continuous-time input-affine nonlinear polynomial systems

$\beta_{\delta}\in\mathcal{KL}$5

a direct data-driven method uses only two measured input-state trajectories for the $\beta_{\delta}\in\mathcal{KL}$6-ISS part, plus a zero-input trajectory, under a full row-rank condition on monomial data matrices (Zaker et al., 2024). The candidate Lyapunov function is quadratic in the state difference,

$\beta_{\delta}\in\mathcal{KL}$7

and the controller has the form

$\beta_{\delta}\in\mathcal{KL}$8

The synthesis conditions are expressed as polynomial matrix equalities and the LMI-like inequality

$\beta_{\delta}\in\mathcal{KL}$9

and are implemented as a sum-of-squares optimization problem (Zaker et al., 2024). The method is described as using only two sufficiently exciting trajectories plus a zero-input dataset and requiring no model identification (Zaker et al., 2024).

For unknown discrete-time systems, one approach learns a neural Lyapunov-like function $\delta$00 directly from black-box samples $\delta$01. The sampled constraints correspond to the lower bound, upper bound, and one-step decrease inequalities, and training uses hinge-like losses $\delta$02, $\delta$03, and $\delta$04 together with a validity loss

$\delta$05

The key certificate is the validity condition

$\delta$06

which guarantees that satisfaction on sampled data extends to the full continuous state-input domain (Basu et al., 10 Jan 2025). The paper states that if the learned network satisfies the sampled constraints, the Lipschitz condition, and the validity condition, then it is a valid $\delta$07-ISS Lyapunov function and the system is $\delta$08-ISS (Basu et al., 10 Jan 2025).

A control-synthesis extension jointly learns a neural controller $\delta$09 and a neural $\delta$10-ISS-CLF $\delta$11 for unknown discrete-time systems on compact domains (Basu et al., 6 Mar 2025). The robust optimization formulation includes both the incremental CLF inequalities and a control barrier function condition

$\delta$12

to ensure forward invariance of the compact state set (Basu et al., 6 Mar 2025). The formal verification condition is again

$\delta$13

with $\delta$14 constructed from Lipschitz constants of the unknown dynamics, the neural controller, the neural CLF, the class-$\delta$15 functions, and the barrier function (Basu et al., 6 Mar 2025).

The continuous-time counterpart introduces a neural $\delta$16-ISS-CLF and neural controller for unknown continuous-time nonlinear systems, together with a continuous-time barrier-function constraint and a comparable Lipschitz-based validity condition

$\delta$17

for lifting sampled inequalities to the full compact domain (Basu et al., 25 Apr 2025). In both discrete- and continuous-time settings, the central methodological pattern is the same: incremental Lyapunov inequalities are turned into sampled optimization constraints, neural networks parameterize the certificate and controller, and Lipschitz margins bridge finite data and continuous-domain correctness (Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025).

6. Recurrent neural networks, control design, and applications

$\delta$18-ISS has become a central analytical tool for recurrent architectures used in control and system identification.

For generic RNNs of the form

$\delta$19

the incremental stability condition $\delta$20 is used not only for analysis but also for controller synthesis, observer design, and explicit integral-action control (D'Amico et al., 2022). The paper derives LMI reformulations when the closed-loop matrix is parameterized as $\delta$21 or $\delta$22, enabling static state-feedback and dynamic output-feedback design (D'Amico et al., 2022). It also gives an observer

$\delta$23

with convergence guaranteed by

$\delta$24

or an equivalent LMI (D'Amico et al., 2022).

For GGNNs, the $\delta$25-ISS condition $\delta$26 is incorporated into training through a regularizer

$\delta$27

with $\delta$28 and $\delta$29 (Marino et al., 2023). The stable model, called sGGNN, is reported to outperform an unconstrained GGNN in flocking and multi-robot motion control tasks under reduced communication range, communication delay, sparse connectivity, and RRT-assisted evaluation (Marino et al., 2023). The paper’s conclusion is that enforcing the stability condition increases robustness in distributed control tasks (Marino et al., 2023).

For deep LSTMs, stability is enforced through a regularized training objective

$\delta$30

where $\delta$31 penalizes violations of $\delta$32 (Bonassi et al., 2023). Training uses TBPTT, mini-batch gradient-based optimization, RMSProp in the experiment, and early stopping based on validation MSE (Bonassi et al., 2023). On a real brake-by-wire apparatus, a two-layer deep LSTM with eight cells per layer learns a model satisfying the $\delta$33-ISS inequalities and achieves FIT $\delta$34 on test data (Bonassi et al., 2023). The reported interpretation is that the $\delta$35-ISS property contributes mainly through robustness and reliable recurrent behavior rather than raw fit alone (Bonassi et al., 2023).

Regional $\delta$36-ISS has also been used as the backbone of robust output-feedback control and model predictive control for recurrent equilibrium networks. In this setting, the plant is rewritten as a linear system plus a nonlinear residual $\delta$37, and a local incremental sector condition

$\delta$38

holds on a set $\delta$39 (Ravasio et al., 25 Jun 2025). Observer and controller LMIs produce robust positively invariant ellipsoids for the error dynamics and closed-loop state, and the resulting invariant tubes are used for constraint tightening in tube-based NMPC (Ravasio et al., 25 Jun 2025). The main theorem states recursive feasibility and convergence of the output to a bounded neighborhood determined by the observer and controller invariant sets (Ravasio et al., 25 Jun 2025).

7. Broader interpretation and current directions

The literature presents $\delta$40-ISS as a general trajectory-to-trajectory robustness principle that adapts naturally to different modeling paradigms.

In hybrid locomotion, the notion captures bounded recurrence around a nominal periodic orbit under step-to-step terrain uncertainty rather than exact periodicity (Tucker et al., 2023). In stochastic dynamics, it becomes a mean-square or Wasserstein-distance contractivity statement with explicit terms for diffusion intensity and input mismatch (Kawano et al., 20 Feb 2026). In reinforcement learning, it is characterized through the regularity of value functions over a discriminative reward class rather than through a single Lyapunov decrease condition (Pfrommer et al., 1 Jul 2025). In data-driven control, it becomes a property that can be synthesized or certified directly from samples through sum-of-squares programs, neural certificates, and sampled-to-continuous validity conditions (Zaker et al., 2024, Basu et al., 10 Jan 2025, Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025).

A recurring technical theme is that $\delta$41-ISS is stronger than nominal stability but often requires structural assumptions or conservative certificates. Several papers therefore emphasize that their criteria are sufficient, not necessary (Marino et al., 2023), or replace exact $\delta$42-ISS by $\delta$43-ISpS when model uncertainty prevents exact cancellation (Sundarsingh et al., 2024, Sangeerth et al., 12 Oct 2025). Another recurring theme is the importance of forward invariance: compact invariant sets, RPI ellipsoids, or invariant sublevel sets are repeatedly used to ensure that incremental estimates remain meaningful on the domain where the local analysis is valid (Tucker et al., 2023, Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025, Ravasio et al., 25 Jun 2025).

Taken together, these developments indicate that $\delta$44-ISS is no longer confined to a narrow nonlinear-control setting. It functions as a common language for incremental robustness in hybrid systems, learned controllers, neural dynamical models, distributed multi-agent policies, and stochastic systems. A plausible implication is that future work will continue to combine classical incremental Lyapunov theory with data-driven, optimization-based, and policy-centric formulations, while retaining the defining requirement that input differences induce proportionally bounded trajectory differences.