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Input-to-state stability of infinite-dimensional systems: recent results and open questions

Andrii Mironchenko, Christophe Prieur

arXiv:1910.01714v3math.OCmath.APmath.DS

TL;DR

The survey addresses how ISS can provide robust stability tools for infinite-dimensional systems, where PDEs, boundary inputs, and large-scale interconnections create substantial analysis challenges. It synthesizes superposition, Lyapunov, semigroup, admissibility, and small-gain methods across linear, nonlinear, delayed, and networked models. The resulting framework characterizes ISS in broad system classes while identifying unresolved relations between ISS and integral ISS and other open problems.

  • Problem

    The survey addresses the need for systematic robust-stability methods for infinite-dimensional systems affected by initial states, disturbances, PDE dynamics, and interconnections.

  • Method

    The paper synthesizes ISS superposition and Lyapunov results with semigroup, admissibility, PDE, boundary-control, delay, and network small-gain techniques.

  • Results

    The survey provides criteria and characterizations for ISS across broad classes of infinite-dimensional systems, including linear systems with unbounded input operators and coupled PDE, ODE, and delay systems.

  • Takeaways & Limitations

    ISS offers a unified basis for analyzing robust stability of distributed-parameter systems and networks with heterogeneous components and coupling types.

  • Takeaways & Limitations

    Open questions remain about whether ISS implies integral ISS for linear systems with unbounded input operators and about limitations of Lyapunov methods in that setting.

Abstract

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In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this paper, starting from classic results for nonlinear ordinary differential equations, we motivate the study of ISS property for distributed parameter systems. Then fundamental properties are given, as an ISS superposition theorem and characterizations of (global and local) ISS in terms of Lyapunov functions. We explain in detail the functional-analytic approach to ISS theory of linear systems with unbounded input operators, with special attention devoted to ISS theory of boundary control systems. The Lyapunov method is shown to be very useful for both linear and nonlinear models, including parabolic and hyperbolic partial differential equations. Next, we show the efficiency of the ISS framework to study the stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows reducing the stability analysis of complex networks, by considering the stability properties of its components and the interconnection structure between the subsystems. An extra section is devoted to ISS theory of time-delay systems with the emphasis on techniques, which are particularly suited for this class of systems. Finally, numerous applications are considered in this survey, where ISS properties play a crucial role in their study. This survey suggests many open problems throughout the paper.

1. Introduction.

The survey develops a unified ISS framework for infinite-dimensional systems, motivated by robust control problems for PDEs and other distributed-parameter models. It combines Lyapunov, functional-analytic, and network methods across linear, nonlinear, delayed, and coupled systems.

  • Motivation: ISS extends robustness analysis to both initial states and external disturbances in infinite-dimensional systems.The survey frames this extension as a merger of infinite-dimensional ISS theory with PDE control methods.
  • Foundations: The framework builds on ISS Lyapunov functions, superposition results, and small-gain tools developed for nonlinear control.These tools connect stability without disturbances to robustness under inputs and support analysis of interconnected systems.
  • Networks: ISS provides a basis for studying coupled systems containing PDE, ODE, and delay components with in-domain or boundary couplings.The survey formulates network stability through small-gain conditions for ISS and integral ISS components.
  • Linear systems: The survey covers linear systems with unbounded input operators, including boundary control systems, using semigroup and admissibility theories.It also applies these criteria to linear boundary control systems and finite-dimensional input spaces.
  • PDEs: Lyapunov methods are applied to parabolic and hyperbolic PDEs with both in-domain and boundary inputs.The approach combines Lyapunov functions with classical inequalities to derive ISS criteria.
  • Finite-dimensional motivation: For linear systems, bounded inputs produce bounded asymptotic deviations, while global asymptotic stability without inputs alone does not ensure robustness to large inputs.The survey also notes that ISS implies 0-UGAS, but the converse can fail and may not guarantee forward completeness.
  • Basic properties: ISS is invariant under replacing one finite-dimensional norm with another, and ISS trajectories remain bounded for bounded inputs.These are practical consequences of the ISS estimate used throughout the survey.

2. Fundamental properties of ISS systems.

The section develops fundamental ISS characterizations for infinite-dimensional systems, beginning with superposition criteria and continuing through Lyapunov-based tests for global, local, and integral ISS. These results clarify the combination of stability, attractivity, boundedness, and input-output behavior required for ISS.

  • ISS superposition: The bULIM property requires uniformly bounded-input trajectories from bounded initial sets to repeatedly approach an input-dependent neighborhood within a uniform time.ULIM strengthens this by allowing arbitrary inputs rather than only inputs from a bounded set.
  • ISS superposition: The ISS superposition theorem characterizes ISS equivalently through UAG, CEP, and BRS, or through bULIM, ULS, and BRS.The theorem applies to forward complete control systems.
  • Criteria for ISS: For ODEs, ISS is equivalent to forward completeness, ULS, and a LIM property, whereas infinite-dimensional systems generally require stronger uniformity conditions.The ODE equivalence relies on BRS being equivalent to forward completeness and bULIM being equivalent to LIM; these relaxations do not generally extend to infinite dimensions.
  • Lyapunov characterizations: A coercive ISS Lyapunov function implies ISS under the BIC property, while a non-coercive one implies ISS for forward complete systems that are CEP and BRS.A non-coercive ISS Lyapunov function also yields the ULIM property.
  • Local and converse Lyapunov results: For the considered nonlinear system, 0-UAS, existence of a coercive locally Lipschitz LISS Lyapunov function, and LISS are equivalent.Under bi-Lipschitz continuity on bounded balls, ISS is likewise equivalent to existence of a coercive ISS Lyapunov function that is Lipschitz on bounded balls.
  • Integral ISS: iISS implies 0-UGAS and BECS, and a coercive iISS Lyapunov function implies iISS, but iISS can coexist with unbounded trajectories under constant inputs.The cited example is iISS but not input-convergent-state and admits unbounded trajectories for sufficiently large initial states under arbitrarily small constant inputs.

3. ISS of linear systems.

For linear infinite-dimensional systems, ISS can be analyzed through semigroup stability and input-operator admissibility, with Lyapunov characterizations available in several settings. The survey develops criteria for bounded and unbounded input operators, including boundary-control and diagonal systems, while identifying unresolved ISS–iISS and Lyapunov-function questions.

  • Unbounded input operators: For unbounded input operators, ISS analysis reduces to exponential semigroup stability together with admissibility of the input operator.This functional-analytic framework is used for linear systems with inputs in Lp spaces and for boundary disturbances.
  • Bounded input operators: For bounded input operators, ISS, iISS, zero-input uniform global asymptotic stability, exponential semigroup stability, and ISS Lyapunov-function conditions are equivalent.The equivalences include both non-coercive and globally Lipschitz coercive ISS Lyapunov functions.
  • Admissibility and well-posedness: q-admissibility ensures forward completeness for Lq inputs, while ∞-admissibility does so for L∞ inputs when mild solutions are norm-continuous in time.Under suitable analytic-semigroup and finite-dimensional-input assumptions, every input operator into X−1 is ∞-admissible.
  • ISS and iISS: For Lp input spaces with finite p, ISS and iISS are equivalent; for L∞ inputs, iISS implies ISS under norm-continuity of the solution map.Whether L∞-ISS implies L∞-iISS remains an open problem, and a negative answer would exclude coercive ISS Lyapunov functions for such systems.
  • Lyapunov methods: Non-coercive ISS Lyapunov functions imply ISS under admissibility and continuity assumptions, and explicit constructions apply to ∞-admissible operators and negative definite self-adjoint generators.The latter setting includes heat equations with Dirichlet boundary input.
  • Diagonal systems: For diagonal systems with a sectorial eigenvalue basis and scalar input operator, the resulting system is both L∞-iISS and L∞-ISS.The criterion assumes a q-Riesz basis of eigenvectors with eigenvalues in the open left half-plane and B ∈ L(C, X−1).

4. Boundary control systems.

Boundary control systems model inputs acting through system boundaries and can be reformulated using extrapolation spaces and admissible input operators. This enables ISS criteria based on semigroup stability and supports applications to finite-dimensional-input Riesz-spectral systems.

  • System formulation: Boundary control systems represent dynamics with a formal operator, a boundary input operator, and a generator defined under homogeneous boundary conditions.The lifting operator maps input values into the state space and helps connect the boundary formulation with an abstract evolution system.
  • Solution representation: For sufficiently smooth inputs and compatible initial states, the boundary system has a unique classical solution represented by a variation-of-constants formula.The representation is established for u ∈ C2([0, τ], U) and x0 − Gu(0) ∈ D(A).
  • Extrapolation-space reformulation: A boundary control system can be reformulated as a linear system with a bounded input operator in the extrapolation space X−1.The reformulation uses the operator ˆAG − A−1G ∈ L(U, X−1), avoiding derivatives of the input in the mild-solution representation.
  • Admissibility: Admissibility is required for the integral term in the mild solution to remain in X when inputs have lower regularity.If B is q-admissible, the mild solution is defined for inputs in Lq and can be extended to larger Lp input spaces.
  • ISS criteria: Exponential semigroup stability together with q-admissibility of the input operator implies ISS for every input space Lp with p ∈ [q, +∞].For Riesz-spectral systems with exponentially stable analytic semigroups and finite-dimensional input spaces, admissibility and ISS follow under the stated assumptions.

5. Lyapunov methods for ISS analysis of PDE systems.

Lyapunov methods provide a central route to ISS analysis for distributed-parameter systems, including parabolic and hyperbolic PDEs. Their effectiveness depends on suitable function-space inequalities, boundary conditions, and assumptions ensuring solution existence and coercive decay estimates.

  • Lyapunov methodology: Constructing an ISS Lyapunov function is generally an efficient method for proving ISS of nonlinear PDE systems.Verification commonly uses Friedrichs, Poincare, Agmon, Jensen inequalities, and linear matrix inequalities.
  • Lyapunov notions: Weak Lyapunov functions can establish asymptotic stability of undisturbed systems, while exponential Lyapunov functions imply 0-UGAS.The weak-function conclusion uses the Barbashin-Krasovskii-LaSalle invariance principle.
  • Assumptions and limitations: The Lyapunov constructions require attention to boundary conditions and do not by themselves guarantee asymptotic stability or global solution existence.Neumann conditions can admit constant non-decaying solutions, and the analysis is valid only as long as solutions exist.

Then the function

Under the stated assumptions, the constructed function is a coercive ISS Lyapunov function satisfying a trajectory-wise dissipation inequality.

  • Result: The constructed function is a coercive ISS Lyapunov function when the condition on δ is satisfied.Its dissipation inequality holds with positive constants λ1 and λ2.

5.2. Lyapunov methods for semilinear parabolic systems with bound-

For semilinear parabolic systems with boundary inputs, Lyapunov estimates combined with sharp functional inequalities yield ISS conditions and decay information. The approach also exposes important limitations for Dirichlet boundary inputs and leaves unrestricted-input ISS for Burgers’ equation open.

  • Lyapunov estimates: The representative Ginzburg-Landau analysis uses a Lyapunov candidate and inequalities such as Jensen, Cauchy, Agmon, and Poincare to derive ISS conditions.The state is measured in L2(0, 1), while the classical-solution formulation uses smooth inputs with the supremum input norm.
  • ISS criteria: If condition (5.34) holds, the parabolic boundary-input system is ISS; under condition (5.35), it is not 0-UAS.The analysis provides fairly tight results and can compute the precise uniform decay rate of solutions.
  • Dirichlet-input limitation: The discussed method cannot handle parabolic systems with Dirichlet boundary inputs, because the required boundary derivative estimate fails.Functionally, the associated input operator is not 2-admissible, making the analysis more challenging; only a non-coercive Lyapunov function is currently available for the linearized system.
  • Burgers’ equation: Burgers’ equation is 0-UGAS in L2(0, 1), but the available ISS result applies only to sufficiently small inputs.Whether ISS holds without restrictions on input magnitude remains an open problem.

5.3. ISS Lyapunov methods for stabilization of stationary hyperbolic systems.

This section develops disturbance-to-state stabilization for boundary-controlled stationary hyperbolic PDEs using dynamic feedback and Lyapunov analysis. Dynamic controllers preserve H1-regularity under measurement disturbances and yield DSS estimates with exponential decay and disturbance-dependent bounds.

  • Boundary measurements are disturbed, motivating feedback laws that stabilize the hyperbolic PDE despite sensor, environmental, or communication errors.
  • DSS guarantees exponential decay without disturbances and disturbance-dependent bounds otherwise, but dynamic-controller DSS need not stabilize the controller’s internal state.
  • Static controllers may fail to produce the H1-regular solutions required for DSS because discontinuous disturbances destroy spatial differentiability.
  • Dynamic controllers smooth disturbance discontinuities while ensuring boundary values and H1 norms are bounded by disturbance magnitude plus exponential transients.
  • The closed-loop system has a unique mild solution for bounded disturbances and admissible initial states.
  • Lyapunov-based parameter conditions establish DSS, while vanishing disturbances imply uniform spatial convergence of the PDE state to zero.

5.4. ISS Lyapunov methods for time-varying hyperbolic systems.

For time-varying hyperbolic systems, the survey constructs periodic Lyapunov functions under matrix and transport assumptions. The resulting criterion yields an exponential ISS Lyapunov function and extends earlier stationary stability conditions.

  • The section studies linear hyperbolic PDEs with time- and space-dependent coefficients, periodic dynamics, disturbances, and boundary conditions.
  • Under the stated regularity and compatibility assumptions, the hyperbolic boundary-value problem has a unique classical solution defined for all t ≥ 0.
  • Assumption 8 requires nonnegative diagonal transport coefficients and supports a diagonal positive definite matrix Q with a periodic auxiliary function r.
  • Theorem 5.24 constructs a periodic Lyapunov function that is an exponential ISS Lyapunov function under the stated parameter condition on µ.
  • The criterion does not require each pointwise ordinary differential equation to be stable and generalizes sufficient conditions to time-varying and semilinear perturbed systems.
  • In the time-invariant case, the periodic Lyapunov construction reduces to a conventional time-invariant PDE Lyapunov candidate and has been applied to shallow-water boundary feedback.

6. Interconnected systems.

The section extends ISS small-gain methods to interconnected infinite-dimensional systems, including heterogeneous PDE, delay, and ODE components. Under suitable gain conditions, subsystem ISS properties yield ISS of the whole network and a composite Lyapunov function.

  • Small-gain theory reduces stability analysis of complex networks to subsystem stability properties and their interconnection structure.
  • The interconnection framework accommodates heterogeneous components and both in-domain and boundary couplings.
  • For forward-complete ISS subsystems with a well-defined interconnection, small-gain conditions imply ISS of the network through UGS and UAG arguments.
  • Subsystem ISS estimates are reformulated using separate internal gains so each subsystem’s response to particular neighboring states can be analyzed.
  • A Lyapunov-form small-gain theorem constructs an ISS Lyapunov function for the whole network from subsystem ISS Lyapunov functions and internal gains.
  • For the considered interconnections of integral ISS systems, the stated gain condition yields iISS, and additional K∞ assumptions strengthen the conclusion to ISS.

Infinite networks with linear gains.

For countably infinite networks with linear gains, spectral-radius conditions provide exponential ISS and a coercive Lyapunov function for the full interconnection. The approach avoids requiring quasi-compactness assumptions used by some infinite-dimensional Perron-Frobenius results.

  • The infinite-network result assumes exponentially ISS subsystems with exponential ISS Lyapunov functions in dissipative form and linear neighbor gains.
  • If r(Ψ) < 1, the whole infinite interconnection is exponentially ISS and admits a coercive exponential ISS Lyapunov function formed as a weighted sum.
  • The spectral approach addresses infinite couplings without relying on quasi-compactness assumptions required by available infinite-dimensional Perron-Frobenius or Krein-Rutman results.
  • The theorem has been extended to exponential ISS with respect to closed sets and applied to time-varying networks, infinite-agent consensus, and distributed observers.

Infinite networks with nonlinear gains.

For infinite networks with nonlinear gains, decisive small-gain results remain unavailable. Existing partial results impose conservative conditions, while relationships among alternative small-gain conditions remain unclear.

  • Decisive small-gain results are not yet available for infinite networks with nonlinear gains.
  • A countably infinite network of ISS systems is ISS when all neighboring-subsystem gain functions are less than identity.The survey characterizes this condition as rather conservative.
  • For infinite networks, the relationships among spectral, robust strong, Ω-path, and monotone-invertibility small-gain conditions remain unclear.

7. Input-to-state stability of time-delay systems.

The survey develops ISS theory for retarded time-delay systems as a special class of infinite-dimensional control systems. It establishes their control-system formulation, equivalent ISS characterization, and complementary Lyapunov-Krasovskii and Lyapunov-Razumikhin methods, while identifying unresolved equivalences.

  • Overview: Retarded time-delay systems connect delay-specific ISS theory with the general infinite-dimensional ISS framework.The section emphasizes this relationship and surveys time-invariant retarded differential equations.
  • Control-system formulation: The state space is C([−Td, 0], Rn), with histories represented by shifted state segments and inputs taken from globally essentially bounded measurable functions.
  • Control-system formulation: Under Assumption 9, the delay equation defines a control system with unique solutions for every initial state and admissible input.The argument uses Lipschitz, continuity, measurability, and growth conditions to obtain Carathéodory solutions.
  • ISS characterization: General ISS, Lyapunov, superposition, and trajectory-based small-gain results remain valid for delay equations.
  • Open problems: It remains unknown whether forward completeness and bounded reachability sets, or LIM and ULIM, are equivalent for time-delay systems.
  • Lyapunov methods: Lyapunov-Krasovskii and Lyapunov-Razumikhin methods provide sufficient ISS conditions, while Lyapunov-Krasovskii theory also gives necessary and sufficient conditions.The converse result yields a coercive functional with an exponential decay rate.

8. Applications.

ISS methods support control and stabilization applications across distributed-parameter systems, including moving-boundary heat processes, tokamak profiles, delay systems, and nonlinear flexible structures. The examples illustrate ISS’s role in handling disturbances, interconnections, and higher-order PDE dynamics.

  • Scope of applications: ISS applications in PDE control span stabilization, disturbance rejection, observer design, event-triggering, and large-scale interconnections.
  • Moving-boundary systems: The one-phase Stefan problem uses a 1-D heat equation on a time-varying domain to stabilize an interface position under a time-varying disturbance.
  • Tokamak systems: In tokamak control, ISS addresses external perturbations and coupling with dynamical actuators such as current control.The cited work includes Lyapunov-based designs and real experiments.
  • Delay systems: Delay-system applications combine logarithmic-norm and freezing techniques for slowly varying coefficients and nonlinear perturbations.
  • Nonlinear PDEs: A Lyapunov approach proves ISS of mild solutions for a semilinear railway-track PDE modeling flexible structures with Kelvin-Voigt damping.

9. Further topics.

The survey extends ISS analysis to strong, practical, Lur’e, hybrid, and computational settings beyond standard infinite-dimensional ISS. These extensions clarify weaker stability notions, accommodate implementation constraints, and combine continuous and discrete dynamics through conditions such as dwell time.

  • Strong ISS: Strong ISS is weaker than ISS for general infinite-dimensional systems, although the two notions coincide for ODEs.
  • Strong ISS: Strong ISS combines uniform global stability with a strong asymptotic gain property whose convergence time depends on the initial state and tolerance.
  • Practical ISS: Input-to-state practical stability adds a constant residual bound and is used for stochastic, quantized, sample-data, and interconnected control systems.
  • Practical ISS: For infinite-dimensional systems, superposition results show that a non-coercive ISS Lyapunov function with bounded reachability sets implies ISpS.
  • Lur’e systems: Lur’e-system ISS analysis uses transfer functions, sector bounds, circle criteria, and passivity-related tools for linear systems with static nonlinear feedback.
  • Computation and PDEs: SOS programming can numerically construct ISS Lyapunov functions for evolution equations with polynomial nonlinearities.
  • Boundary inputs: ISS analysis is harder for PDEs with boundary inputs than for PDEs with distributed inputs, though monotonicity can enable a transformation between them in suitable systems.
  • Impulsive systems: If both continuous and discrete dynamics are ISS, an impulsive system is uniformly ISS across impulse sequences; otherwise dwell-time conditions may be required.

10. Open problems.

The section identifies unresolved ISS problems for infinite-dimensional systems, especially fully nonlinear PDEs, output stability, and time-varying, discrete-time, and hybrid settings.

  • ISS and integral ISS: Relations between ISS and integral ISS for linear systems with unbounded operators remain incompletely understood, and corresponding superposition theorems are unavailable.The section also questions whether Lyapunov methods have limitations for iISS systems that are not ISS.
  • Systems with outputs: Infinite-dimensional IOS theory is largely undeveloped, despite its relevance when stability of the full state is unnecessary or only selected errors are measured.IOS includes outputs such as tracking, observer, and drifting errors.
  • Fully nonlinear PDEs: General ISS methods for fully nonlinear PDEs remain highly desirable because important examples such as porous medium and Navier–Stokes equations are not semilinear.These equations fall outside the class where the unbounded part is linear.
  • Other open directions: ISS theory for time-varying infinite-dimensional systems is in its infancy, while discrete-time and hybrid infinite-dimensional systems remain almost unexplored.These are identified as broader open areas beyond the other problems discussed.

11. Conclusion and discussion.

The survey consolidates the state of ISS theory for infinite-dimensional systems across linear and nonlinear models and multiple analytical approaches. It organizes criteria, PDE methods, network stability, delay systems, and applications into a unified review.

  • Scope: The survey covers ISS results for both linear and nonlinear infinite-dimensional systems using Lyapunov, dynamical-systems, semigroup, admissibility, and PDE methods.Its stated aim is to outline the field's state of the art.
  • Organization: It presents Lyapunov and superposition criteria, functional-analytic tests for linear and boundary-control systems, PDE analysis, infinite-dimensional networks, delay systems, and applications.The sections connect these methods to distinct classes of control systems.

12. Appendix.

The appendix collects inequalities, notation, function-space conventions, set definitions, and abbreviations used throughout the survey.

  • Inequalities: The appendix records Cauchy, Jensen, Poincare, and Agmon inequalities used throughout the paper.The listed propositions include references for the proofs of these inequalities.
  • Notation: It defines notation for operators, spectra, norms, kernels, images, function spaces, and common mathematical sets.The notation covers Banach-space operators, Sobolev spaces, measurable functions, and sequence spaces.
  • Abbreviations and sets: The abbreviation list includes stability properties such as AG, BRS, DSS, IOS-related terms, UAG, UGS, and 0-UGAS.It also defines standard number sets and the nonnegative real line.
  • Function and sequence spaces: It also specifies conventions for continuous and piecewise-continuous functions, comparison-function classes, balls, closures, derivatives, boundaries, and measures.These conventions support the ISS definitions and estimates used in the main text.
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