# Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions

@article{Mironchenko2020InputtoStateSO, title={Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions}, author={Andrii Mironchenko and Christophe Prieur}, journal={SIAM Rev.}, year={2020}, volume={62}, pages={529-614} }

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…

## 68 Citations

Input-to-state stability and integral input-to-state stability of non-autonomous infinite-dimensional systems

- MathematicsInt. J. Syst. Sci.
- 2021

It is proved that for a class of admissible inputs the existence of an ISS Lyapunov function implies the ISS of a system in Banach spaces and it is shown that uniform global asymptotic stability is equivalent to their integral input-to-state stability for non-autonomous generalised bilinear systems overBanach spaces.

ISS small-gain criteria for infinite networks with linear gain functions

- MathematicsSyst. Control. Lett.
- 2021

Characterization of integral input-to-state stability for nonlinear time-varying systems of infinite dimension

- MathematicsArXiv
- 2022

. For large classes of inﬁnite-dimensional time-varying control systems, the equivalence between integral input-to-state stability (iISS) and the combination of global uniform asymptotic stability…

Lyapunov functions for input-to-state stability of infinite-dimensional systems with integrable inputs

- Mathematics
- 2020

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order N ∈ ℕ on a bounded 1-dimensional spatial domain (a, b). In order to achieve stabilization,…

Non-coercive Lyapunov functions for input-to-state stability of infinite-dimensional systems

- Mathematics
- 2019

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of…

Growth Conditions for Global Exponential Stability and Exp-Iss of Time-Delay Systems Under Point-Wise Dissipation

- MathematicsSSRN Electronic Journal
- 2022

For time-delay systems, it is known that global asymptotic stability is guaranteed by the existence of a Lyapunov-Krasovskii functional that dissipates in a point-wise manner along solutions, namely…

Robust Stability of Global Attractors for Reaction-Diffusion System w.r.t. Disturbances

- Mathematics
- 2021

Asymptotic stability of an equilibrium is a fundamental property of evolutionary processes and plays important role for many applications. It is well-known that a globally asymptotically stable…

Lyapunov Characterization of Uniform Exponential Stability for Nonlinear Infinite-Dimensional Systems

- MathematicsIEEE Transactions on Automatic Control
- 2022

In this article, we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to uncertainties. We first extend the well-known Datko lemma to the framework of the…

N ov 2 01 9 NON-COERCIVE LYAPUNOV FUNCTIONS FOR INPUT-TO-STATE STABILITY OF INFINITE-DIMENSIONAL SYSTEMS

- Mathematics
- 2019

We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of…

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It is shown that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system, and it is proved a linearization principle that allows a construction of a local ISS- Lyap unov function for a system.

Characterizations of Input-to-State Stability for Infinite-Dimensional Systems

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The new notion of strong ISS (sISS) is introduced that is equivalent toISS in the ODE case, but is strictly weaker than ISS in the infinite-dimensional setting and several criteria for the sISS property are proved.

Input-to-State Stability of Nonlinear Impulsive Systems

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It is proved that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition and two small-gain theorems are proved that provide a construction of an ISS Lyap unov function for an interconnection of impulsive Systems if the ISS LyAPunov functions for subsystems are known.

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Construction of Lyapunov Functions for Interconnected Parabolic Systems: An iISS Approach

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Stability of two highly nonlinear reaction-diffusion systems is established by the the proposed small-gain criterion, and for interconnections of partial differential equations, the choice of a right state and input spaces is crucial.

Lyapunov functions for input-to-state stability of infinite-dimensional systems with integrable inputs

- Mathematics
- 2020

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order N ∈ ℕ on a bounded 1-dimensional spatial domain (a, b). In order to achieve stabilization,…

ISS implies iISS even for switched and time-varying systems (if you are careful enough)

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Characterizations of input-to-state stability for hybrid systems

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