--- title: Incremental Input-State Stability (δ-ISS) url: https://www.emergentmind.com/topics/incremental-input-to-state-stability-delta-iss type: topic --- # Incremental Input-State Stability (δ-ISS) 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 [2412.03901], [2210.09721], [2507.00695], [2602.18382]. ## 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 [2412.03901]. 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 [2210.09721]. 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 \(\gamma_{\delta}\in\mathcal{K}_\infty\) [2305.19235]. For deep LSTM networks, the corresponding property is written as \[ \| x_{a, k} - x_{b, k} \|_2 \leq \beta(\| x_{a, 0} - x_{b, 0} \|_2, k) + \gamma(\| u_{a, 0:k} - u_{b, 0:k} \|_{2, \infty}), \] with respect to an invariant state set \(\mathcal X\) and a bounded input set \(\mathcal U\) [2304.02975]. The central distinction from ordinary ISS is stated explicitly in multiple sources: ISS bounds one trajectory with respect to the origin, whereas \(\delta\)-ISS bounds the distance between two trajectories. This makes \(\delta\)-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 [2305.19235], [2412.03901], [2304.02975]. A recurring implication is that if the two inputs are equal, then \(\delta\)-ISS reduces to an incremental asymptotic stability property. In the polynomial data-driven setting, this is stated as incremental asymptotic stability, denoted \(\delta\)-GAS [2412.03901]. In neural network papers, the same point is phrased as convergence of trajectories when inputs match [2305.19235], [2304.02975]. ## 2. Lyapunov characterizations A standard route to \(\delta\)-ISS is via an incremental Lyapunov function \(V(x,\hat x)\) defined on a pair of states. For discrete-time systems, one characterization is: \[ \xi_1(\|x_1-x_2\|)\le V(x_1,x_2)\le \xi_2(\|x_1-x_2\|) \] and \[ V(f(x_1,u_1),f(x_2,u_2))-V(x_1,x_2) \le -\xi(\|x_1-x_2\|)+\sigma(\|u_1-u_2\|), \] with \(\xi_1,\xi_2,\xi\in\mathcal K_\infty\) and \(\sigma\in\mathcal K\); existence of such a Lyapunov function implies \(\delta\)-ISS [2210.09721]. An analogous formulation is used for unknown discrete-time systems: \[ \alpha_1(|x-\hat x|)\le V(x,\hat x)\le \alpha_2(|x-\hat x|) \] and \[ V(f(x,u),f(\hat x,\hat u)) - V(x,\hat x) \le -\alpha_3(|x-\hat x|)+\sigma(|u-\hat u|), \] with \(\alpha_1,\alpha_2,\alpha_3\in\mathcal K_\infty\) and \(\sigma\in\mathcal K\) [2501.05778]. For continuous-time systems, the dissipation inequality takes the derivative form \[ \frac{\partial V}{\partial x} f(x,g(x,w)) + \frac{\partial V}{\partial \hat x} f(\hat x,g(\hat x,\hat w)) \le -\kappa V(x,\hat x)+\sigma(|w-\hat w|), \] together with positive definiteness bounds in \(|x-\hat x|\) [2504.18330]. This is the continuous-time \(\delta\)-ISS control Lyapunov function formulation used for unknown systems. Several papers introduce control-oriented variants. A \(\delta\)-ISS control Lyapunov function (\(\delta\)-ISS-CLF) is defined as a function \(V\) that satisfies the incremental positivity and dissipation inequalities under some controller \(g\); if the closed-loop system admits such a \(\delta\)-ISS-CLF, then it is incrementally input-to-state stable [2503.04129], [2504.18330]. In the discrete-time unknown-system setting, the controller and the \(\delta\)-ISS-CLF are both learned from data and then formally verified [2503.04129]. In the continuous-time analogue, both are parameterized by neural networks and certified through a Lipschitz-based validity condition [2504.18330]. Hybrid locomotion introduces a specialized Lyapunov notion for disturbed Poincaré maps. There, a robust Lyapunov function \(V:B_\rho(x^*)\to\mathbb R_{\ge 0}\) satisfies \[ k_1\|x-x^*\|^c \le V(x)\le k_2\|x-x^*\|^c, \] and \[ \|x-x^*\|\ge \chi d \quad \implies \quad \Delta V(x,d):=V(\mathcal P(x,d))-V(x)\le -k_3\|x-x^*\|^c. \] An equivalent relaxed form is \[ V(\mathcal P(x,d)) - V(x) \le -k_4\|x-x^*\|^c + \frac12\sigma |d|^c. \] This is explicitly identified as an ISS-type Lyapunov pattern for the disturbed return map of a walking gait [2303.10231]. ## 3. Core analytical frameworks Recent work develops \(\delta\)-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 \[ x(k+1)=f(Ax(k)+Bu(k)), \] a sufficient condition is the existence of a symmetric positive definite matrix \(P=P^T\succ0\) with a structural sparsity constraint on nonlinear coordinates such that \[ \widetilde A^T P \widetilde A - P \prec 0, \] where \(\widetilde A=WA\) and \(W\) contains componentwise Lipschitz constants of the activation map [2210.09721]. The proof uses the quadratic incremental Lyapunov candidate \[ V(x_1,x_2)=\|x_1-x_2\|_P^2. \] This condition is presented as less conservative than earlier norm-based criteria and can be converted into LMIs in controller and observer design settings [2210.09721]. 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 \(\delta\)-ISS is \[ \mathcal{A}_{\delta}\le 1, \] with \[ \mathcal{A}_{\delta} \triangleq \sigma_{\hat{q}}\|\bar{S}\|_\infty \|A\|_\infty + \frac{1}{4}\|\bar{S}\|_\infty^2\|\hat{A}\|_\infty\|A\|_\infty + \frac{1}{4}\|\bar{S}\|_\infty^2\|\tilde{A}\|_\infty\|B\|_\infty. \] This bound arises from the \(1\)-Lipschitz property of \(\tanh\), the \(1/4\)-Lipschitz property of sigmoid, and the decomposition of state, input, and graph-support differences [2305.19235]. For deep GGNNs, each layer must satisfy its own condition \(\mathcal{A}^i_{\delta}\le 1\) [2305.19235]. A third line extends layerwise contraction reasoning to deep LSTMs. For each layer \(l\), the paper derives nonlinear inequalities involving recurrent matrices \(U_f^{(l)},U_i^{(l)},U_z^{(l)},U_r^{(l)}\), gate bounds, and invariant-set bounds. The sufficient condition is compactly written as \[ \nu(\Phi)<0, \] where \(\nu(\Phi)\) collects \(2L\) nonlinear inequalities for an \(L\)-layer network [2304.02975]. The proof stacks per-layer difference inequalities into a global block system and shows the resulting global matrix is Schur stable [2304.02975]. A fourth line derives \(\delta\)-ISS for positive Lur’e systems through linear dissipativity. Under the incremental gain condition \[ \esssup_{t\ge 0}|f(t,\zeta_1)-f(t,\zeta_2)| \le \Delta|\zeta_1-\zeta_2|, \] and the positivity hypothesis, the paper proves weighted one-norm trajectory-pair estimates such as \[ |(x_a-x_b)(t_1)|_p \le e^{-\xi(t_1-t_0)} |(x_a-x_b)(t_0)|_p + \alpha_s \|w_a-w_b\|_{L^s(t_0,t_1\,B_2^\top p)}, \] and states that for \(s=\infty\) this is an incremental exponential ISS estimate [2402.03955]. 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 \(\pi\), the paper studies a local \(\delta\)-ISS condition \[ \|x'_t-x_t\| \le \beta(\|x_0'-x_0\|,t) + \gamma\!\left(\max_{0\le k0\) [2411.01872]. The cited papers explicitly note that if \(c=0\), this becomes standard \(\delta\)-ISS [2411.01872], [2510.10450]. The practical term appears because Gaussian-process approximations of unknown dynamics introduce unavoidable model uncertainty [2411.01872], [2510.10450]. In strict-feedback nonlinear systems with unknown drift terms, Gaussian-process learning and backstepping are used to obtain \(\delta\)-ISpS with respect to an external incremental input \(\hat\upsilon\). The composite Lyapunov function \[ V(z,z')=\sum_{k=1}^h (z_k-z'_k)^\top(z_k-z'_k) \] satisfies \[ \dot V(z,z')\leq -kV(z,z')+\sigma(\|\hat{u}-\hat{u}'\|)+\tilde c, \] which yields \(\delta\)-ISpS rather than exact \(\delta\)-ISS because \(\tilde c\) is induced by GP approximation errors [2411.01872]. A feedback-linearization-based variant combines \(\delta\)-ISpS with control barrier functions and robust forward invariance, again with a nonzero practical residual [2510.10450]. Locomotion theory introduces a related but domain-specific notion, \(\delta\)-robustness, for periodic gaits on uneven terrain. Ground-height variation is modeled as a disturbance entering through the guard condition \[ S_d = \{x\in\mathcal X \mid h(x)=d,\ \dot h(x)<0\}, \] and the extended Poincaré map yields a discrete-time system with input \(d_k\): \[ x_{k+1} = \mathcal P(x_k,d_k). \] The resulting bound \[ \|x_k-x^*\| \le M\alpha^k\|x_0-x^*\| + \gamma\,\delta \] is an exponential ISS estimate for the gait evolution under bounded terrain disturbances [2303.10231]. This suggests that, in hybrid locomotion, \(\delta\)-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 \(\sigma_x^2\) [2602.18382]. The paper interprets this as a stochastic generalization of deterministic \(\delta\)-ISS. ## 5. Learning and data-driven synthesis A major current theme is the synthesis or certification of \(\delta\)-ISS from data without an explicit model. For unknown continuous-time input-affine nonlinear polynomial systems \[ \dot x(t)=AF(x(t))+Bu(t), \] a direct data-driven method uses only two measured input-state trajectories for the \(\delta\)-ISS part, plus a zero-input trajectory, under a full row-rank condition on monomial data matrices [2412.03901]. The candidate Lyapunov function is quadratic in the state difference, \[ V(x,\tilde x)=(x-\tilde x)^\top P(x-\tilde x), \qquad P\succ 0, \] and the controller has the form \[ u = \mathbf K(x)x+\hat u, \qquad \mathbf K(x)=\mathbf U_{0,T}\mathds Y(x)P x. \] The synthesis conditions are expressed as polynomial matrix equalities and the LMI-like inequality \[ \Sigma+\Sigma^\top+\vartheta I_n \preceq -\epsilon P^{-1}, \qquad \epsilon,\vartheta>0, \] and are implemented as a sum-of-squares optimization problem [2412.03901]. The method is described as using only two sufficiently exciting trajectories plus a zero-input dataset and requiring no model identification [2412.03901]. For unknown discrete-time systems, one approach learns a neural Lyapunov-like function \(V_{\theta,b}(x,\hat x)\) directly from black-box samples \(\{(x_s,u_p,f(x_s,u_p))\}\). The sampled constraints correspond to the lower bound, upper bound, and one-step decrease inequalities, and training uses hinge-like losses \(L_0\), \(L_1\), and \(L_2\) together with a validity loss \[ L_v(\eta)=\max\big(0,(\mathcal{L}\varepsilon+\eta)\big). \] The key certificate is the validity condition \[ \eta_S^*+\mathcal{L}\varepsilon\le 0, \] which guarantees that satisfaction on sampled data extends to the full continuous state-input domain [2501.05778]. The paper states that if the learned network satisfies the sampled constraints, the Lipschitz condition, and the validity condition, then it is a valid \(\delta\)-ISS Lyapunov function and the system is \(\delta\)-ISS [2501.05778]. A control-synthesis extension jointly learns a neural controller \(g_{\bar\theta,\bar b}(x,w)\) and a neural \(\delta\)-ISS-CLF \(V_{\theta,b}(x,\hat x)\) for unknown discrete-time systems on compact domains [2503.04129]. The robust optimization formulation includes both the incremental CLF inequalities and a control barrier function condition \[ h(f(x,g(x,w)))-\kappa(h(x))\le \eta, \] to ensure forward invariance of the compact state set [2503.04129]. The formal verification condition is again \[ \eta_S^* + \mathcal{L}\epsilon \le 0, \] with \(\mathcal L\) constructed from Lipschitz constants of the unknown dynamics, the neural controller, the neural CLF, the class-\(\mathcal K\) functions, and the barrier function [2503.04129]. The continuous-time counterpart introduces a neural \(\delta\)-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 \[ \eta_S^*+\mathcal{L}\varepsilon \le 0 \] for lifting sampled inequalities to the full compact domain [2504.18330]. 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 [2503.04129], [2504.18330]. ## 6. Recurrent neural networks, control design, and applications \(\delta\)-ISS has become a central analytical tool for recurrent architectures used in control and system identification. For generic RNNs of the form \[ x(k+1)=f(Ax(k)+Bu(k)),\qquad y(k)=Cx(k)+Du(k), \] the incremental stability condition \(\widetilde A^T P \widetilde A-P\prec 0\) is used not only for analysis but also for controller synthesis, observer design, and explicit integral-action control [2210.09721]. The paper derives LMI reformulations when the closed-loop matrix is parameterized as \(A=F+GJ\) or \(A=F+JG\), enabling static state-feedback and dynamic output-feedback design [2210.09721]. It also gives an observer \[ \hat x(k+1)=f(A\hat x(k)+Bu(k)+L(y(k)-\hat y(k))) \] with convergence guaranteed by \[ \lambda_{\max}(W)\|A-LC\|<1 \] or an equivalent LMI [2210.09721]. For GGNNs, the \(\delta\)-ISS condition \(\mathcal A_\delta\le 1\) is incorporated into training through a regularizer \[ \Pi = \sum_{i=0}^{L} \rho_{-}\min(0,\delta\mathcal{A}_i-1-\epsilon) + \rho_{+}\max(0,\delta\mathcal{A}_i-1-\epsilon), \] with \(\rho_+=1\) and \(\rho_-=0.01\) [2305.19235]. 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 [2305.19235]. The paper’s conclusion is that enforcing the stability condition increases robustness in distributed control tasks [2305.19235]. For deep LSTMs, stability is enforced through a regularized training objective \[ \mathcal L(\Phi)=\mathrm{MSE}(\mathcal I_{tr}^{\{b\};\Phi)+\rho(\nu(\Phi)), \] where \(\rho(\nu)\) penalizes violations of \(\nu(\Phi)<0\) [2304.02975]. Training uses TBPTT, mini-batch gradient-based optimization, RMSProp in the experiment, and early stopping based on validation MSE [2304.02975]. On a real brake-by-wire apparatus, a two-layer deep LSTM with eight cells per layer learns a model satisfying the \(\delta\)-ISS inequalities and achieves FIT \(=88.6\%\) on test data [2304.02975]. The reported interpretation is that the \(\delta\)-ISS property contributes mainly through robustness and reliable recurrent behavior rather than raw fit alone [2304.02975]. Regional \(\delta\)-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 \(q(v)\), and a local incremental sector condition \[ \Delta q(v+\Delta v,v)^\top S\big(\Delta v-H\Delta q(v+\Delta v,v)\big)\ge 0 \] holds on a set \(\mathcal V(H)\) [2506.20334]. 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 [2506.20334]. The main theorem states recursive feasibility and convergence of the output to a bounded neighborhood determined by the observer and controller invariant sets [2506.20334]. ## 7. Broader interpretation and current directions The literature presents \(\delta\)-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 [2303.10231]. In stochastic dynamics, it becomes a mean-square or Wasserstein-distance contractivity statement with explicit terms for diffusion intensity and input mismatch [2602.18382]. 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 [2507.00695]. 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 [2412.03901], [2501.05778], [2503.04129], [2504.18330]. A recurring technical theme is that \(\delta\)-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 [2305.19235], or replace exact \(\delta\)-ISS by \(\delta\)-ISpS when model uncertainty prevents exact cancellation [2411.01872], [2510.10450]. 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 [2303.10231], [2503.04129], [2504.18330], [2506.20334]. Taken together, these developments indicate that \(\delta\)-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. Source: https://www.emergentmind.com/topics/incremental-input-to-state-stability-delta-iss