Delta-ISS Lyapunov Functions

• Delta-ISS Lyapunov functions are relational certificates that compare pairs of trajectories to assess incremental input-to-state stability in dynamic systems.
• They employ comparison bounds and dissipation inequalities to ensure exponential decay of trajectory differences despite input mismatches in both continuous and discrete time.
• Recent advances integrate data-driven and neural methods with test-function approaches to synthesize these certificates using optimization and regularity over rich reward families.

Delta-ISS Lyapunov functions are pairwise stability certificates for incremental input-to-state stability: they compare two trajectories, rather than one trajectory against an equilibrium, and quantify how their separation evolves under mismatched initial conditions and inputs. In the classical incremental framework, the certificate is a function $V(x,\tilde x)$ satisfying comparison bounds with $|x-\tilde x|$ and a dissipation inequality along paired trajectories. Recent work also identifies a distinct line of analysis in which local incremental stability is characterized through uniform Hölder regularity of reward-induced value and $Q$-functions over a sufficiently rich reward family, without interpreting the value function itself as a classical Lyapunov certificate (Zaker et al., 2024, Pfrommer et al., 1 Jul 2025).

1. Classical pairwise formulation

In the standard continuous-time setting, $\delta$-ISS is expressed by a trajectory-to-trajectory estimate of the form

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

with $\beta\in\mathcal{KL}$ and $\gamma\in\mathcal{K}_\infty$. The corresponding $\delta$-ISS Lyapunov function is a nonnegative two-point function $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$ satisfying comparison bounds and a dissipation inequality. For the quadratic candidate

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$

the required bounds are

$|x-\tilde x|$0

with $|x-\tilde x|$1 and $|x-\tilde x|$2, while the incremental dissipation condition is

$|x-\tilde x|$3

As recalled in the continuous-time data-driven synthesis paper, this is the standard incremental Lyapunov framework from Angeli, and the dissipation inequality is the key certificate because it states that the distance between trajectories decays exponentially up to an input mismatch term (Zaker et al., 2024).

In the discrete-time setting, the same structure appears in one-step form. A $|x-\tilde x|$4-ISS control Lyapunov function for the closed loop satisfies

$|x-\tilde x|$5

and

$|x-\tilde x|$6

This is explicitly used to show that the closed-loop system is incrementally input-to-state stable. The same pairwise logic also appears in the neural Lyapunov formulation for unknown discrete-time systems, where the function $|x-\tilde x|$7 is trained directly on pairs of states and input pairs rather than on a single-state equilibrium problem (Basu et al., 6 Mar 2025).

2. Local, global, and detectability-oriented variants

The literature does not use a single uniform $|x-\tilde x|$8-ISS notion. One important variant is the local closed-loop notion for a fixed deterministic static policy $|x-\tilde x|$9, studied for deterministic full-information dynamics $Q$0 with additive exogenous disturbances $Q$1. The property is formulated as

$Q$2

where

$Q$3

This is called $Q$4-locally incremental input-to-state stabilizing, or local-$Q$5ISS. In the main equivalence theorem, the policy is assumed $Q$6-Lipschitz, the state space is finite-dimensional, and compactness is added in the second theorem. The result is explicitly local, not global, and the converse direction depends on a reward class rich enough to separate states (Pfrommer et al., 1 Jul 2025).

A closely related, but distinct, two-point Lyapunov notion is $Q$7-IOSS. In the moving-horizon estimation literature, the relevant certificate is $Q$8, which satisfies

$Q$9

and

$\delta$0

with $\delta$1. In the exponential case used for moving horizon estimation, $\delta$2 becomes an $\delta$3-step Lyapunov function for the estimator once the horizon is sufficiently large, and the paper provides LMI conditions for constructing such functions for nonlinear detectable systems (Schiller et al., 2022).

These variants show that the two-point Lyapunov idea serves more than one role. In ordinary $\delta$4-ISS it quantifies incremental robustness to input mismatch; in $\delta$5-IOSS it quantifies incremental detectability through disturbance and output mismatch. The underlying object remains relational and two-state.

3. The test-function approach and the re-interpretation of Lyapunov-like certificates

A major conceptual shift is introduced by the test-function approach to incremental stability. The paper studies a closed-loop system under a fixed deterministic static policy and proves a two-way equivalence between local-$\delta$6ISS and regularity of RL-style value functions. For an $\delta$7-Lipschitz policy $\delta$8, a discount schedule $\delta$9, and a $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$0-sensitive reward class $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$1, local-$$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$2ISS with bounds

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

is equivalent to the statement that for every reward $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$4 and every proper discount schedule, the value function $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$5 is $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$6-Hölder continuous in $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$7, and the action-value function $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$8 is locally $$|x(t)-\tilde x(t)| \le \beta(|x(0)-\tilde x(0)|,t)+\gamma(|u-\tilde u|_\infty),\qquad t>0,$$9-Hölder continuous around $\beta\in\mathcal{KL}$0. The stronger theorem replaces a single reward by all time-varying reward sequences $\beta\in\mathcal{KL}$1, which is important because it avoids accidental cancellations that can occur with a single fixed reward (Pfrommer et al., 1 Jul 2025).

This framework explicitly recalls the classical local-$\beta\in\mathcal{KL}$2ISS Lyapunov condition

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

together with

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

Its novelty is not to reinterpret the RL value function as such a certificate. RL-style value functions are constructed by exponentially decaying a Lipschitz reward function and may be non-smooth and unbounded on both sides; accordingly, the paper states that they cannot be directly understood as Lyapunov certificates. Stability is inferred instead from uniform regularity over a sufficiently rich class of rewards. A reward class is $\beta\in\mathcal{KL}$5-sensitive when every reward is $\beta\in\mathcal{KL}$6-Hölder continuous in $\beta\in\mathcal{KL}$7 and the class separates states through

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

The resulting certificate is therefore family-based and relational rather than single-function and pointwise. The paper also notes that even $\beta\in\mathcal{KL}$9 need not produce a Lyapunov function (Pfrommer et al., 1 Jul 2025).

A common misunderstanding is that this result identifies value functions with Lyapunov functions in the classical sense. It does not. The point is precisely that local incremental stability can be characterized without requiring a decrease inequality of the form $\gamma\in\mathcal{K}_\infty$0. This suggests a broader notion of Lyapunov-like certification for incremental stability, one based on adversarial test functions and regularity rather than monotone decay.

4. Data-driven and neural synthesis for unknown systems

Recent work turns $\gamma\in\mathcal{K}_\infty$1-ISS Lyapunov analysis into an overview problem for unknown systems. For unknown continuous-time input-affine nonlinear systems with polynomial dynamics

$\gamma\in\mathcal{K}_\infty$2

a direct data-driven method constructs both a quadratic $\gamma\in\mathcal{K}_\infty$3-ISS Lyapunov function and a $\gamma\in\mathcal{K}_\infty$4-ISS controller without identifying the full dynamics $\gamma\in\mathcal{K}_\infty$5. The method uses two sets of input-state trajectories from sufficiently excited dynamics, as introduced by Willems et al.'s fundamental lemma, together with an additional zero-input trajectory. Under the full-row-rank condition on $\gamma\in\mathcal{K}_\infty$6 and $\gamma\in\mathcal{K}_\infty$7, the controller is parameterized as

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

and the synthesis is cast as a sum-of-squares program over $\gamma\in\mathcal{K}_\infty$9, polynomial matrices $\delta$0, $\delta$1, and $\delta$2, with the key inequality

$\delta$3

From this, the learned function

$\delta$4

satisfies

$\delta$5

The paper demonstrates the construction on a rotating rigid spacecraft using maximum monomial degree $\delta$6, $\delta$7, and sampling time $\delta$8, and then simulates 1000 arbitrary pairs of initial conditions under equal excitation inputs so that the theory predicts $\delta$9-GAS (Zaker et al., 2024).

For unknown discrete-time systems, a neural network $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$0 can be trained as a Lyapunov-like certificate directly from black-box samples. The robust optimization problem is approximated by a sampled scenario problem, and the key verification condition is

$$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$1

Under Lipschitz assumptions on the unknown dynamics and on the neural Lyapunov function, this condition upgrades sample-based feasibility to global feasibility over the compact state and input sets. The paper formulates the comparison functions as

$$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$2

and validates the method on a scalar nonlinear system and a permanent magnet DC motor (Basu et al., 10 Jan 2025).

The neural-controller literature extends this idea from certification to joint synthesis. In the discrete-time controller case, a $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$3-ISS-CLF $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$4 and a controller $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$5 are learned jointly, together with a control barrier function condition for forward invariance on compact state sets. The formal verification step is again based on the scenario convex problem and the validity condition

$$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$6

with $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$7 assembled from Lipschitz constants of the unknown dynamics, the Lyapunov network, the controller, and the barrier function. The reported case studies include a scalar system with a non-affine non-polynomial structure, a one-link manipulator, a nonlinear Moore-Greitzer model of the jet engine, and a rotating rigid spacecraft model (Basu et al., 6 Mar 2025).

The continuous-time neural version introduces a smooth $$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$8-ISS control Lyapunov function

$$\mathbf V:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$$9

satisfying

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$0

and

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$1

Here too, a validity condition of the form

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$2

provides the formal certificate, and the training additionally enforces forward invariance through a control barrier function on the compact state set (Basu et al., 25 Apr 2025).

5. Hybrid, delayed, interconnected, and nonsmooth analogues

Several ISS frameworks are not stated as $$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$3-ISS theorems, but they provide methodological components that are structurally close to incremental Lyapunov analysis. For switching retarded systems, a converse theorem shows that ISS is equivalent to the existence of a relaxed common Lyapunov-Krasovskii functional. The strongest result permits a continuous functional $$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$4 whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere along trajectories,

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$5

and this is equivalent to both measurable-input ISS and piecewise-constant ISS. For impulsive systems, ISS Lyapunov functions require dwell-time restrictions: a general ISS Lyapunov function yields ISS under the nonlinear fixed dwell-time condition

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$6

while an exponential ISS Lyapunov function yields uniform ISS under the generalized average dwell-time condition

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$7

Related results for interconnected impulsive systems with and without delays combine exponential Lyapunov-Razumikhin or Lyapunov-Krasovskii objects with small-gain and average dwell-time conditions (Haidar et al., 2022, Dashkovskiy et al., 2012, Dashkovskiy et al., 2010).

For large-scale interconnections, the infinite-network small-gain theorem constructs a global ISS Lyapunov function of max type,

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$8

from subsystem ISS Lyapunov functions and a path of strict decay for the infinite-dimensional gain operator. For upper semicontinuous differential inclusions, the nonsmooth theory of nonpathological ISS-Lyapunov functions replaces the Clarke-based condition by a Lie generalized derivative and builds composite Lyapunov functions for interconnections through

$$\mathbf V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x),\qquad P\succ 0,$$9

Neither framework states a direct $|x-\tilde x|$00-ISS theorem, but both isolate reusable ingredients for incremental analysis: subsystem-wise certificates, gain operators, max-type aggregation, and nonsmooth derivative notions that remain meaningful across switching or inclusion boundaries (Kawan et al., 2021, Rossa et al., 2021).

6. Interpretation, misconceptions, and current directions

A first misconception is that a $|x-\tilde x|$01-ISS Lyapunov function is merely an ordinary Lyapunov function written in different coordinates. The classical certificates summarized here are intrinsically two-point objects, either on $|x-\tilde x|$02 or on an error state derived from two trajectories. Their defining inequalities quantify contraction up to input mismatch and are therefore relational rather than equilibrium-centered (Zaker et al., 2024, Basu et al., 6 Mar 2025).

A second misconception is that any discounted value function may be treated as a Lyapunov certificate. The test-function approach explicitly rejects that identification. RL-style value functions may be non-smooth, non-coercive, and even unbounded on both sides; what certifies local incremental stability is not a pointwise decrease condition on one value function but uniform Hölder regularity of value and $|x-\tilde x|$03-functions over a discriminative reward family. The same paper also states that a full global $|x-\tilde x|$04ISS converse would require stronger machinery and is left as future work, so the present equivalence is deliberately local (Pfrommer et al., 1 Jul 2025).

A third misconception is that data-driven or neural $|x-\tilde x|$05-ISS certificates are merely empirical. The recent neural and data-driven papers attach explicit proof obligations to the learned object: sum-of-squares feasibility in the polynomial continuous-time setting, or Lipschitz-based validity conditions such as

$|x-\tilde x|$06

in the discrete-time and continuous-time neural settings. This suggests that current research is not abandoning Lyapunov theory; it is re-encoding it in optimization, scenario verification, and function approximation frameworks suited to unknown dynamics (Zaker et al., 2024, Basu et al., 10 Jan 2025, Basu et al., 25 Apr 2025).

Taken together, the recent literature presents two complementary meanings of “Delta-ISS Lyapunov functions.” In the classical sense, they are pairwise functions satisfying comparison bounds and dissipation inequalities. In the broader sense introduced by the test-function approach, they are part of a larger family of relational certificates in which incremental stability is encoded by regularity of reward-induced functionals rather than by monotone decrease of a single scalar quantity.