Incremental Input-to-State Stability

• Incremental Input-to-State Stability is a property that quantifies how differences between system trajectories evolve over time in nonlinear and uncertain environments.
• It underpins robust observer design, optimization-based state estimation, and compositional analysis by establishing explicit bounds and Lyapunov-based conditions.
• Its characterization through Lyapunov functions and small-gain theorems facilitates the analysis of interconnected, stochastic, and learning-based control systems.

Incremental input-to-state stability (incremental ISS, or abbreviated iISS/δ-ISS) is a property of dynamical systems that quantifies how the difference between two system trajectories—driven by possibly different initial states and inputs—evolves over time. This property extends the classical notion of input-to-state stability by focusing on trajectory-to-trajectory convergence rather than convergence to a single equilibrium or reference trajectory. Incremental ISS is fundamental for robust observer design, optimization-based state estimation, and compositional analysis of complex, possibly interconnected or uncertain systems.

1. Formal Definitions and Core Concepts

Incremental input-to-state stability is defined for both continuous- and discrete-time nonlinear systems. Let $X$ denote the state space and $U$ the input space, with appropriate norm or metric. For discrete-time systems $x(t+1) = f(x(t),u(t),w(t))$, a system is incrementally input/output-to-state stable if there exist comparison functions $\beta, w, v, u, y$ in class KL (i.e., strictly increasing in the first argument, decreasing to zero in the second) such that for all $t \ge 0$:

$$|x(t),\chi(t)| \leq \max \left\{ \beta(|x_0, \chi_0|, t),\ \max_{1 \leq \tau \leq t} w(|w(t-\tau), \omega(t-\tau)|, \tau),\ \cdots, \max_{1 \leq \tau \leq t} y(|y(t-\tau), \zeta(t-\tau)|, \tau) \right\}$$

where $(x,u,w,v,y)$ and $(\chi, \upsilon, \omega, \nu, \zeta)$ are trajectory pairs of the system and disturbance/measurement processes (Knuefer et al., 2020). In continuous time, the time-discounted integral variant or classical version is analogously formulated with possibly different classes of comparison functions (Schiller et al., 2023).

Specializations include (a) the case of additive disturbances, which reduces the number of disturbance terms, and (b) the time-discounted variant, such as exponentially-weighted decay $\beta(r,t) = c\eta^t r$ with $\eta\in(0,1)$ (Knuefer et al., 2020, Schiller et al., 2023). If the additive constant is included (i.e., distance between solutions is bounded up to a constant), the property is called incremental input-to-state practical stability (δ-ISpS) (Sundarsingh et al., 2024, Sangeerth et al., 12 Oct 2025).

The ISS property is often characterized in terms of existence of a bivariate Lyapunov function $U$0 satisfying suitable dissipation (decrement) and positivity conditions.

2. Lyapunov Characterizations

One principal approach to verifying incremental ISS is via construction of a dissipation-form incremental Lyapunov function. For a continuous-time system with state $U$1:

$U$2

satisfies, for some class $U$3 functions $U$4 and $U$5:

• Positivity/Bounding: $U$6
• Dissipation: $U$7

For discrete-time systems, the analogous decrement is

$U$8

This yields the ISS-type bound on trajectory pairs:

$U$9

Existence of such a Lyapunov function is both necessary and sufficient for incremental input-to-state stability under mild technical conditions, including forward completeness and regularity (Zamani et al., 2012, Sangeerth et al., 12 Oct 2025).

For the most general, time-discounted i-IOSS, a two-point Lyapunov function is shown to be both sufficient and, under further construction, necessary—especially for observer design and detectability (Knuefer et al., 2020).

3. Explicit Bounds and Specializations

The iISS property can be instantiated in several fundamental system classes:

Linear Systems: For $x(t+1) = f(x(t),u(t),w(t))$0, $x(t+1) = f(x(t),u(t),w(t))$1, the time-discounted i-IOSS property is equivalent to detectability of $x(t+1) = f(x(t),u(t),w(t))$2. There are explicit exponentially-decaying bounds:

$x(t+1) = f(x(t),u(t),w(t))$3

with appropriately constructed weighting matrices and system parameters (Knuefer et al., 2020, Gatke et al., 8 Apr 2026).

Positive Lur’e Systems: For forced Lur’e systems, if the incremented nonlinearity satisfies a sector condition and the augmented linear part is Hurwitz (in the positive sense), the class admits a 1-norm storage function proving exponential δ-ISS as well as linear incremental input-output gain:

$x(t+1) = f(x(t),u(t),w(t))$4

(Piengeon et al., 2024).

Integral and Discounted Variants: Integral forms (i-iIOSS) provide time-discounted integral estimates coupling state differences to input and output mismatches, crucial for observer analysis and robust estimation:

$x(t+1) = f(x(t),u(t),w(t))$5

(Schiller et al., 2023).

4. Observer Design and Detectability

Incremental ISS is fundamentally linked to the design and analysis of robust state observers. For instance, robust global asymptotic stability (RGAS) of full-order nonlinear observers requires that the underlying process be incrementally IOSS (specifically, the time-discounted variant or its integral counterpart, depending on the observer's structure):

$x(t+1) = f(x(t),u(t),w(t))$6

and the observer must satisfy an output-injection consistency condition with the process dynamics (Knuefer et al., 2020, Schiller et al., 2023). This necessity is critical in the analysis and design of optimization-based state estimation schemes, such as moving horizon estimation and full-information observers.

5. Incremental ISS in Large-Scale, Interconnected, and Stochastic Systems

The incremental ISS property admits powerful compositionality and robustness analysis in networks of interconnected or distributed subsystems. For nonlinear large-scale systems, small-gain theorems establish conditions under which the coupling of exponentially i-IOSS subsystems yields overall exponential i-IOSS for the full distributed system. The small-gain condition is formulated in terms of a spectral radius bound for a matrix of coupling gains:

$x(t+1) = f(x(t),u(t),w(t))$7

where $x(t+1) = f(x(t),u(t),w(t))$8 is constructed from local subsystem interconnection gains (Gatke et al., 8 Apr 2026). Both trajectory-based and Lyapunov-function-based small-gain theorems are available.

Stochastic variants of incremental ISS have been established for Itô-SDEs with contracting drifts and Lipschitz inputs. Under suitable assumptions, mean-square exponential convergence of the inter-trajectory discrepancy is guaranteed, with noise and input-contribution terms made explicit. The property further extends to Fokker-Planck equations and Wasserstein-metric contraction (Kawano et al., 20 Feb 2026).

6. Data-Driven, Learning-Based, and Neural Approaches

Incremental ISS has motivated learning-based and data-driven certification and controller synthesis techniques for unknown or partially unknown systems. Recent work leverages scenario-based neural network training, scenario convex programming, and sum-of-squares optimization to fit bivariate Lyapunov functions and controllers using only trajectories collected from the real system. Certified neural Lyapunov functions can be obtained with formal guarantees by enforcing Lipschitz bounds, loss function constraints encoding the incrementally dissipative inequalities, and verifying optimality margins on the dataset (Basu et al., 10 Jan 2025, Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025, Zaker et al., 2024).

For example, in the fully data-driven setting with polynomial dynamics, two sufficiently exciting trajectories are used per Willems' lemma, followed by a sum-of-squares program to recover both the incremental Lyapunov function and controller (Zaker et al., 2024). Neural-network and LMI-based characterizations are established for recurrent neural network models, enabling scalable controller and observer designs (D'Amico et al., 2022).

7. Variants, Extensions, and Theoretical Implications

Variants such as practical incremental ISS (δ-ISpS) allow for additive error bounds in the presence of uncertainty (e.g., from GP-based model learning), with Lyapunov characterization augmented by a constant term and corresponding modification in the definition:

$x(t+1) = f(x(t),u(t),w(t))$9

(Sundarsingh et al., 2024, Sangeerth et al., 12 Oct 2025). This relaxation is essential for safe reinforcement-learning-based or feedback-linearization designs using learned models and is compatible with invariance-based safety filters.

Incremental ISS is also equivalent to the regularity of families of value functions arising in reinforcement learning under certain test function frameworks, bridging the gap between Lyapunov-contraction and RL-based perspectives (Pfrommer et al., 1 Jul 2025).

In summary, incremental input-to-state stability provides a mathematically and practically robust framework for reasoning about trajectory separation in nonlinear, stochastic, distributed, and data-driven systems. It underpins observer design, compositional analysis, and learning-based control, with a unified Lyapunov-theoretic, optimization-based, and system-theoretic foundation (Zamani et al., 2012, Knuefer et al., 2020, Schiller et al., 2023, Piengeon et al., 2024, Sundarsingh et al., 2024, Basu et al., 10 Jan 2025, Basu et al., 6 Mar 2025, Basu et al., 25 Apr 2025, Pfrommer et al., 1 Jul 2025, Sangeerth et al., 12 Oct 2025, Kawano et al., 20 Feb 2026, Gatke et al., 8 Apr 2026).