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Input-to-state stability of infinite-dimensional control systems
Sergey Dashkovskiy, Andrii Mironchenko
TL;DR
The paper addresses limited attention to ISS for infinite-dimensional systems by developing Lyapunov-based tools, construction methods, a linearization principle, and an infinite-dimensional small-gain theorem. It establishes ISS implications under input assumptions and extends the framework across ODEs, time-delay systems, and many evolution PDEs.
Problem
ISS for infinite-dimensional systems has received relatively limited attention, motivating Lyapunov-based sufficient conditions and construction methods.
Method
The paper develops ISS-Lyapunov construction methods, a linearization method for local functions, and a small-gain design for interconnections.
Results
Under certain input-class assumptions, existence of a local or global ISS-Lyapunov function implies corresponding ISS, while the framework generalizes across several infinite-dimensional system classes.
Takeaways & Limitations
The framework provides ISS analysis and Lyapunov-function design tools for abstract systems, interconnections, and applications including evolution PDEs.
Takeaways & Limitations
Many results require piecewise-continuous or continuous inputs, which may be restrictive for applications such as PDEs and requires further research.
Abstract
from arXiv · showhide
We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.
1 Introduction
The paper develops Lyapunov-based tools for ISS of infinite-dimensional control systems, addressing a comparatively underdeveloped area. Its contributions include ISS-Lyapunov implications, linearization results, and an infinite-dimensional small-gain theorem.
- Motivation: ISS research for infinite-dimensional systems has received less attention than finite-dimensional ISS, apart from time-delay systems.
- Contributions: The paper develops Lyapunov-type sufficient conditions for ISS and methods for constructing ISS-Lyapunov functions in infinite-dimensional systems.
- Contributions: Under assumptions on admissible inputs, an ISS-Lyapunov function implies local or global ISS, consistently extending the finite-dimensional local definition.
- Contributions: For n ISS subsystems, the generalized infinite-dimensional small-gain theorem constructs an ISS-Lyapunov function for the interconnection when subsystem functions and the small-gain condition are available.
- Contributions: A linearization principle proves local ISS when the linearization is ISS; one proof applies on Banach spaces, while the Lyapunov-function construction requires Hilbert spaces.
2 Preliminaries
The preliminaries define abstract control systems, admissible inputs, and stability notions used throughout the paper. They also distinguish local, global, uniform, exponential, and input-to-state stability.
- Control-system framework: A control system is a triple of state space, admissible input-function space, and transition map satisfying existence, identity, causality, continuity, and semigroup properties.
- Control-system framework: The transition map φ(t,s,x,u) denotes the state at time t from state x at time s under input u.
- Standing assumptions: The paper assumes the BIC property: finite-time loss of existence can occur only when the solution norm becomes unbounded.
- Stability notions: ISS bounds state trajectories by a KL decay term and a K input term, while LISS restricts the state and input radii to finite values.
- Input spaces: For piecewise right-continuous inputs, the ISS input size can be expressed using the supremum norm over the entire input horizon.
3 Linear systems
The linear-systems section relates stability properties to semigroup behavior and illustrates infinite-dimensional distinctions with a counterexample and reaction-diffusion equations. In particular, exponential semigroup stability characterizes the relevant linear ISS behavior.
- Finite- versus infinite-dimensional behavior: For finite-dimensional linear systems, e0-GAS, eISS, 0-GAS, and ISS are equivalent, whereas infinite-dimensional systems do not generally preserve these equivalences.
- Semigroup stability: For the linear homogeneous system, 0-UGASx, uniform semigroup stability, uniform exponential stability, and exponential 0-UGASx are equivalent.
- Counterexample: The counterexample is 0-GAS but has unbounded trajectories for arbitrarily small inputs, showing that 0-GAS need not imply bounded-input bounded-state behavior.
- Reaction-diffusion example: For the linear parabolic Neumann system, eISS holds exactly when the reaction matrix R is Hurwitz.
- Reaction-diffusion example: The equal-diffusion assumption is essential for the stated reaction-diffusion criterion; unequal diffusion coefficients can produce Turing instability.
4 Lyapunov functions for nonlinear systems
This section establishes that ISS-Lyapunov functions imply local or global ISS under suitable input assumptions, and shows consistency with finite-dimensional definitions. It also applies the framework to semilinear parabolic systems and interconnections.
- Lyapunov implications: The theorem requires admissible input classes to remain in the class under time shifts without increasing their norms.This assumption is stated for many standard classes, including piecewise right-continuous, Lp, L∞, and Sobolev inputs.
- Lyapunov implications: An ISS-Lyapunov function implies ISS, while a local ISS-Lyapunov function implies local ISS under the theorem’s input-class assumptions.The result applies globally when the state and input radii are infinite.
- Finite-dimensional consistency: For piecewise right-continuous inputs, the paper proves an equivalent local ISS-Lyapunov characterization using pointwise input values and the Lie derivative.The authors state that this formulation recovers the standard finite-dimensional definition for ODE systems.
- Extension by density: A density argument transfers ISS from a restricted system on dense state and input subspaces to the full system with the same β and γ estimates.The transfer assumes continuous dependence of the transition map on initial states and inputs.
- Parabolic PDE example: For a semilinear parabolic reaction-diffusion example, an ISS-Lyapunov function is established on smooth states and inputs, then ISS is extended to the H1-based system.The extension uses density and the preceding transfer lemma.
5 Linearization
The section proves that ISS of the linearization implies local ISS for the nonlinear system, and develops two corresponding construction routes for LISS-Lyapunov functions. One route applies on Banach spaces, while the Lyapunov-function construction additionally assumes a Hilbert state space and a coercive operator.
- The first proof requires only a Banach state space but does not provide a LISS-Lyapunov function.
- The second proof assumes a Hilbert state space and provides a LISS-Lyapunov function for the nonlinear system.
- The nonlinear model is decomposed around a linear approximation whose generator is R = A + B, with B bounded.
- The paper proves that if the linearized system is ISS, then the nonlinear system is LISS.
- ISS of the linearized system yields exponential stability of its semigroup, supporting fading-memory estimates that establish nonlinear LISS.
- Under the Hilbert-space assumptions, a coercive positive operator P satisfying the stated inequality yields a quadratic LISS-Lyapunov function.
6 Interconnections of input-to-state stable systems
The section generalizes Lyapunov small-gain analysis to interconnections of infinite-dimensional ISS systems and constructs an ISS-Lyapunov function for the whole network under a small-gain condition.
- The paper generalizes the Lyapunov small-gain theorem to interconnections of n infinite-dimensional ISS subsystems.
- The interconnected system is formulated over a Banach product space with subsystem generators and piecewise-continuous inputs.
- Subsystem ISS-Lyapunov functions are characterized using bounds, gain functions, and an implication condition for trajectories driven by admissible inputs.
- The derivation depends on the selected norm for the interconnection input space, and this norm dependence cannot generally be removed in infinite dimensions.
- An Ω-path and the gain operator encode interconnection effects and enable construction of a composite Lyapunov function.
- If the gain operator satisfies the small-gain condition, the whole interconnection is ISS and possesses an ISS-Lyapunov function.
- The construction extends from finite-dimensional linear systems to linear systems over Banach spaces and is applied to systems whose stability conditions yield 0-UGASx.
- For the nonlinear example, the small-gain condition produces a parameter condition guaranteeing 0-UGASx, while existence and uniqueness for all times follow when ISS excludes blow-up.
7 Conclusion
The paper extends ISS theory to broad classes of infinite-dimensional systems through Lyapunov implications, linearization, and small-gain constructions, while identifying input regularity and converse characterizations as open issues.
- The framework encompasses ODEs, time-delay systems, and many evolution PDEs while remaining consistent with established ISS definitions for ODEs and time-delay systems.
- For general control systems, an ISS-Lyapunov function implies ISS under specified input assumptions, and the finite-dimensional local definition is recovered.
- For Banach-space differential systems, the small-gain theorem constructs an ISS-Lyapunov function for an interconnection from subsystem functions under the small-gain condition.
- Linearization provides an alternative for constructing local ISS-Lyapunov functions when the system is linearizable.
- Most results assume piecewise-continuous or continuous inputs, which may be restrictive for applications such as PDEs and requires further research.
- Characterizations of ISS analogous to finite-dimensional results and a converse Lyapunov theorem for infinite-dimensional systems remain beyond the paper’s scope.