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Characterizations of input-to-state stability for infinite-dimensional systems

Andrii Mironchenko, Fabian Wirth

arXiv:1701.08952v2math.OCmath.APmath.DS

TL;DR

The paper asks which ISS characterizations for ODEs remain valid for infinite-dimensional systems. It develops uniformity-based criteria across several system classes, establishes strong ISS and non-coercive Lyapunov results under stated conditions, and uses counterexamples to separate invalid ODE-era equivalences.

  • Problem

    Finite-dimensional ISS characterizations do not generally extend to infinite-dimensional systems, where uniformity affects equivalence between stability properties.

  • Method

    The paper introduces ULIM, sLIM, sAG, and strong ISS, then proves characterization theorems and analyzes counterexamples across infinite-dimensional system classes.

  • Results

    ISS is characterized by uniform asymptotic gain and by local stability together with ULIM, while several ODE characterizations fail for infinite-dimensional systems.

  • Takeaways & Limitations

    Strong ISS recovers ISS for nonlinear ODEs, and non-coercive ISS Lyapunov functions imply ISS for a broad class of Banach-space evolution equations under restrictions.

Abstract

from arXiv · show

We prove characterizations of input-to-state stability (ISS) for a large class of infinite-dimensional control systems, including some classes of evolution equations over Banach spaces, time-delay systems, ordinary differential equations (ODE), switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang for ODE systems. For the special case of differential equations in Banach spaces we prove even broader criteria for ISS and apply these results to show that (under some mild restrictions) the existence of a non-coercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for the sISS property. At the same time, we show by means of counterexamples, that many characterizations, which are valid in the ODE case, are not true for general infinite-dimensional systems.

I. INTRODUCTION

The paper examines which finite-dimensional ISS characterizations extend to infinite-dimensional systems and develops a broader framework covering several system classes. It introduces uniformity-based notions and strong ISS to recover valid criteria while identifying failures of ODE equivalences.

  • Motivation: ISS characterizations central to finite-dimensional nonlinear control motivate extending these results to infinite-dimensional systems.Existing ISS theory supports robust stabilization, observer design, network analysis, and related applications.
  • Scope: The paper answers which ISS characterizations survive, under what conditions, and how stability properties can be classified across infinite-dimensional systems.The considered class includes ODEs, Banach-space differential equations, time-delay systems, and switched systems.
  • Challenges: Infinite-dimensional systems require uniformity because finite-dimensional equivalences may fail, and even forward completeness or global asymptotic stability may not bound finite-time reachability sets.A counterexample demonstrates the reachability-set difficulty.
  • New stability notions: The authors introduce ULIM, sLIM, and sAG as stability notions extending limit and asymptotic-gain concepts to address infinite-dimensional difficulties.ULIM supplies a uniform limit formulation, while the strong notions support a distinct strong ISS property.
  • ISS criteria: ULIM underpins generalized ISS characterizations, including criteria combining local stability with ULIM and results linking non-coercive ISS Lyapunov functions to ISS under restrictions.The latter result is established for a class of evolution equations with Lipschitz continuous nonlinearities.
  • Strong ISS and failures of equivalence: Strong ISS is characterized through sLIM and sAG, while finite-dimensional equivalence with the usual limit property recovers the ODE characterizations.For infinite-dimensional linear systems, ULIM is strictly stronger than sLIM or LIM, and nonuniform combinations such as AG ∧ GS are not generally equivalent to ISS.

A. Stability notions for undisturbed systems

The paper distinguishes stability of systems without inputs by whether trajectories remain bounded, approach the origin, and do so uniformly across initial states. In infinite-dimensional systems, nonuniform and uniform notions can differ substantially, unlike in the ODE case.

  • 0-ULS bounds trajectories uniformly for initial states within a neighborhood of the origin.
  • 0-LIM requires trajectories to approach the origin arbitrarily closely, whereas 0-UGATT requires uniform convergence times over initial states.
  • In infinite-dimensional systems, 0-GAS is weaker than 0-UGAS, whereas ODE systems make these properties equivalent.
  • For systems without inputs, 0-GATT implies 0-LIM, and sLIM and LIM coincide while remaining weaker than ULIM even for some linear systems.

B. Stability notions for systems with inputs

For systems with inputs, the paper defines boundedness, limit, and asymptotic-gain properties according to how uniformly trajectories reach neighborhoods determined by input size. The distinctions mainly concern how the required time depends on states and inputs.

  • UGB provides a global trajectory bound through functions of the initial-state and input norms, and it is equivalent to BND.
  • AG, sAG, and UAG all imply convergence to the ball of radius γ(∥u∥U), but differ in whether convergence time depends on the state norm, state, or input.
  • For AG, convergence time depends on state and input; for sAG it depends on state but not input; for UAG it depends only on state norm.
  • The paper introduces additional properties to formalize reachability of ε-neighborhoods around input-dependent balls.
  • LIM requires each state-input pair to reach an input-dependent neighborhood in finite time, while sLIM makes the time bound independent of the input.
  • ULIM strengthens LIM by using one time bound for all states within a norm ball and all inputs.

D. Input-to-state stability

The paper defines ISS and related Lyapunov notions, then contrasts finite-dimensional characterizations with infinite-dimensional failures and introduces stronger uniformity-based concepts.

  • Definitions: ISS requires a class KL state bound and a class K input-gain bound for all states, inputs, and times.The local counterpart, LISS, restricts the same inequality to bounded state and input neighborhoods.
  • Lyapunov characterizations: Lyapunov functions provide a principal tool for analyzing ISS and LISS, including continuous functions with bounds and Dini-derivative conditions.The paper defines LISS Lyapunov functions locally and ISS Lyapunov functions through corresponding dissipation inequalities.
  • Finite-dimensional results: For forward-complete finite-dimensional systems, ISS characterizations include the equivalences depicted in Figure 1 and the existence of a Lipschitz continuous ISS Lyapunov function.These results also identify ISS with uniform properties and with combinations involving limit properties and local stability.
  • Infinite-dimensional obstacles: In infinite dimensions, the finite-dimensional equivalences can fail because uniformity is essential and forward completeness or global asymptotic stability need not ensure bounded reachability sets.The paper also notes that additional uniformity assumptions are required for some Banach-space characterizations.
  • Counterexample: The example system is 0-UGAS, sAG, AG with zero gain, UGS with zero gain, and LISS with zero gain, but not ISS.This counterexample invalidates several ISS characterizations based on AG or LIM combined with UGS or 0-UGAS.
  • New stability notions: The paper introduces ULIM, sLIM, sISS, and sAG to distinguish stability properties and generalize finite-dimensional ISS criteria.These notions extend LIM, AG, and UAG while separating the uniformity levels that matter in infinite-dimensional systems.

A. Main result and structure of the paper

The paper establishes a hierarchy of ISS-related properties for infinite-dimensional systems, identifies equivalences under explicit assumptions, and uses counterexamples to mark failed or unresolved implications.

  • General framework: For forward-complete systems with BRS and CEP, the paper establishes the stability-property relations represented in Figure 2.The figure distinguishes general implications, assumption-dependent equivalences, failed implications, and unresolved converses.
  • Uniformity: ISS is also equivalent to ULIM combined with LS, showing that uniformity of attractivity and reachability is especially important in infinite dimensions.The paper contrasts this with combinations involving nonuniform AG or LIM properties, which are not generally equivalent to ISS.
  • ISS characterizations: ISS is equivalent to UAG with CEP and BRS, to ULIM with ULS and BRS, and to ULIM with UGS.These equivalences constitute the main characterization theorem for general forward-complete control systems.
  • Strong ISS: Strong ISS satisfies sISS ⇔ sAG ∧ UGS ⇔ sLIM ∧ UGS, while remaining distinct from ISS in infinite-dimensional systems.For nonlinear ODE systems, sISS is equivalent to ISS; for linear systems without inputs, it corresponds to strong stability of the associated semigroup.
  • Semilinear systems: For semilinear systems under Assumption 1 and property ♦, ISS is equivalent to UAG and BRS, ULIM with ULS and BRS, ULIM with UGS, and ULIM with 0-ULS and BRS.The proof uses BRS to obtain CEP and specializes the general characterization to system (4).

B. ISS via non-coercive ISS Lyapunov functions

The paper examines whether non-coercive ISS Lyapunov functions can establish ISS in infinite-dimensional systems despite lacking a coercive lower bound. It proves ULIM generally and ISS under additional structural assumptions.

  • Scope: Non-coercive Lyapunov functions are motivated by natural infinite-dimensional candidates that may be positive definite but non-coercive.The paper refers to prior work for their detailed advantages and limitations.
  • Definitions: Non-coercive ISS Lyapunov functions omit the usual lower bound ψ1(∥x∥X) ≤ V(x), retaining positivity and dissipation conditions.They may satisfy V(x)>0 for x≠0 without controlling the state norm from below.
  • ULIM result: A non-coercive ISS Lyapunov function implies the uniform limit property for any forward complete control system.The proof derives suitable input gain and convergence-time functions from the Lyapunov estimates.
  • ISS result: For systems satisfying the stated assumptions, bounded reachability sets plus a non-coercive ISS Lyapunov function imply ISS.The argument combines 0-UGAS, ULIM, and the paper’s ISS characterization theorem.

IV. CHARACTERIZATIONS OF ISS

This section develops stability-property characterizations for forward complete systems using comparison-function reformulations and intermediate properties. It establishes several equivalences and implications linking boundedness, stability, attractivity, and ISS.

  • BRS characterization: BRS is equivalent to the existence of a continuous increasing comparison function satisfying the corresponding trajectory bound.Lemma 3 provides this comparison-function restatement.
  • Stability reformulations: For forward complete systems, ULS together with UGB is equivalent to UGS.The result is stated as Lemma 4.
  • Attractivity implications: ISS implies UAG, while ULIM together with UGS also implies UAG.These are established separately in Lemma 5 and Lemma 7.
  • Local stability: UAG together with CEP implies ULS.The proof combines the uniform attractivity estimate with continuity at the equilibrium.
  • Boundedness implications: BRS plus the uniform limit property implies UGB.The proof constructs a continuous increasing bound from the uniform limit time and the BRS estimate.
  • ISS characterization: UAG together with UGS implies ISS.This closes the characterization chain by constructing a class KL estimate from the two properties.

V. STRONG ISS

The paper introduces strong ISS as an intermediate infinite-dimensional stability property and characterizes it through weaker attractivity and stability conditions. Strong ISS coincides with ISS for ODEs but may be strictly weaker in infinite dimensions.

  • Definition: Strong ISS is defined through a state-dependent decay term that is bounded by a function of the initial-state norm and an input gain.Its decay function need only belong to L for each nonzero initial state.
  • Infinite-dimensional scope: ISS implies sISS, but the converse fails in general for infinite-dimensional systems.The distinction is illustrated by the gap between GAS and UGAS in infinite-dimensional systems without inputs.
  • ODE case: For ODEs, sISS and ISS are equivalent.The paper derives this from the ODE characterization linking UGS and AG to ISS.
  • Characterizations: sISS is equivalent to the conjunction of sAG and UGS, and also to sLIM and UGS.The theorem gives both equivalent characterizations.

VI. COUNTEREXAMPLES

The counterexamples show that stability and attractivity characterizations valid for ODEs can fail in infinite-dimensional systems. They separate global nonuniform attractivity, bounded reachability, global stability, and ISS-related properties.

  • Linear systems: For linear undisturbed infinite-dimensional systems, sAG together with UGS does not imply LISS.The obstruction follows because strong semigroup stability need not imply exponential stability.
  • Counterexample construction: The paper constructs two nonlinear systems without inputs and two with inputs to refute several proposed implications.The examples satisfy the system axioms while violating the listed conclusions.
  • Invalid implications: The constructed systems invalidate implications from forward completeness, global attractivity, local stability, and ISS-related properties to BRS or UGS.The specific failures are summarized for systems S1 through S4.
  • Attractivity versus stability: Nonuniform global attractivity need not ensure bounded reachability sets or global stability in nonlinear infinite-dimensional systems.Systems S1 and S3 exhibit these separations even without inputs.
  • Broader failures: Even strong attractivity and local properties together may fail to imply ISS or BRS in infinite-dimensional systems.The paper notes failures involving 0-UGAS, sAG, AG, UGS, LISS, and related combinations.
  • System S1: A system can be forward complete, 0-GAS, and 0-UAS while lacking bounded reachability sets.In S1, coordinate peaks grow with the index despite each coordinate eventually converging to zero.

According to the previous arguments, ∑N−1

The examples construct infinite-dimensional systems separating bounded reachability, asymptotic stability, uniform stability, and input-to-state properties. Time rescaling and additional damping show how these properties can differ.

  • S1 is forward complete, 0-GAS, and 0-UAS, but it is not BRS.
  • S2 is 0-UGAS, forward complete, LISS, and AG with zero gain, but it is not BRS.
  • A nonuniform time transformation makes the modified system BRS while it remains 0-GAS and fails to be 0-UGS.
  • Adding the dynamics generated by ξ improves the modified system to 0-UAS, yet it still fails to be 0-UGS.
  • S4 is forward complete, BRS, 0-UGAS, LISS, and AG with zero gain, but it is not UGS.

VII. ROBUSTNESS OF EQUILIBRIA FOR DIFFERENTIAL EQUATIONS IN BANACH SPACES

For differential equations in Banach spaces, bounded reachability sets provide the regularity needed to establish continuity properties of the flow and continuity at the equilibrium.

  • BRS implies that the flow is Lipschitz continuous on compact intervals for uniformly bounded inputs.
  • BRS also implies that the flow is continuous at the equilibrium under Assumption 1.
  • The continuity proof combines flow Lipschitz estimates, continuity of f(0,·), bounded reachability, and Grönwall’s lemma.

VIII. CHARACTERIZATION OF ISS FOR ODES

For forward-complete finite-dimensional systems, the limit properties LIM and ULIM coincide, supporting the equivalence of the strong and ordinary ISS characterizations in the ODE setting.

  • For forward-complete finite-dimensional systems, LIM holds if and only if ULIM holds.
  • The proof converts LIM’s state-dependent convergence times into a uniform time bound over bounded initial states and inputs.
  • The construction handles large input norms by selecting t := 0 when the gain already bounds the state within the target tolerance.

IX. SYSTEMS WITHOUT INPUTS

For systems without inputs, the paper organizes stability relations through limit, local stability, global attractivity, and uniform properties. It also contrasts finite-dimensional equivalences with infinite-dimensional failures and introduces open questions for strong ISS.

  • For systems without inputs, 0-LIM combined with 0-ULS is equivalent to 0-GAS.
  • The characterization diagram applies to systems satisfying BRS and REP, with selected implications becoming equivalences for ODE and linear systems.
  • Strong stability of strongly continuous semigroups is weaker than exponential stability, illustrating an infinite-dimensional separation among stability notions.
  • The paper’s ISS results identify equivalence with uniform asymptotic gain and with local stability plus the uniform limit property.
  • Strong ISS is equivalent to ISS for nonlinear ODEs but corresponds to strong stability of C0-semigroups for linear systems with inputs.
  • Counterexamples show that properties equivalent to ISS for ODE systems become distinct in the infinite-dimensional setting.
  • For a broad class of Banach-space evolution equations, a non-coercive ISS Lyapunov function implies ISS.
  • Whether LIM implies sLIM, AG implies sAG, or AG ∧ UGS implies sAG ∧ UGS remains open for nonlinear infinite-dimensional systems.
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