Project

# Input-to-state stability and stabilization of distributed parameter systems

Goal: In this project, we are going to build a firm basis for the investigation
of input-to-state stability and stabilization of distributed parameter
systems. More specifically, our aims are:

1. To develop an ISS theory for linear and bilinear distributed parameter systems, including criteria for input-to-state stability and stabilizability of linear and bilinear DPS and sufficient conditions for robustness of ISS.

2. To obtain the infinite-dimensional counterparts of fundamental nonlinear results from ISS theory of finite-dimensional systems. In particular, Lyapunov characterizations of the ISS property, small-gain theorems for DPS and characterizations of ISS in terms of other stability properties.

3. To develop methods for robust stabilization of infinite-dimensional systems, namely, a robust version of continuum backstepping, finite-time robust stabilization of partial differential equations and design of ISS stabilizers for port-Hamiltonian systems.

Please check also our the YouTube ISS channel:

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## Project log

Dear Colleagues,
it is our pleasure to invite you to the next talk of the
Online Seminar on Input-to-State Stability and its Applications
Speaker:
Wilfrid Perruquetti (Ecole Centrale de Lille, CNRS, France)
Title:
Non-Asymptotic output feedback of a double integrator: a separation principle.
Date:
Thursday, 03 February 2022, at 5:00 pm (Time zone of Amsterdam, Berlin, Paris, Rome, Vienna)
To check the current time in this zone you may look to
Abstract: Usually, in control/estimation problems, one is looking at exponential decaying rates for many reasons: ease of understanding, many tools for tuning and getting a time response estimate. But nowadays, control theory has to meet more and more demanding performances in many areas such as aerospace, manufacturing, robotics and transportation to mention a few. A necessary property for these algorithms is stability. The convergence time for the system to reach the goal may be infinite (e.g., asymptotic or exponential convergence) or finite. Combining stability with these convergence types leads to asymptotic or non-asymptotic stability properties.
These concepts may help in obtaining a separation principle when designing output feedback as seen on an example for an double integrator system where ISS properties of homogeneous systems is applied without building a Lyapunov function for the closed-loop system.
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The talk will be given via Zoom.
The login data for the Zoom meeting is provided at the bottom of this email.
Please feel free to forward this invitation to anyone who might be interested. For being included in the mailing list, it is enough to write a short email to
If you are unable to take part in the seminar, you may check/subscribe to the ISS YouTube Channel
We plan to put the recorded talks there (after the confirmation of the speaker).
If you do not wish to get any further invitations to our seminar, please let us know by replying to this email.
With best regards,
Andrii Mironchenko & Daniel Liberzon
--- Invitation to Zoom meeting ---
Andrii Mironchenko is inviting you to a scheduled Zoom meeting.
Topic: ISS Seminar
Time: 03 February, 2022 05:00 PM Amsterdam, Berlin, Rome, Stockholm, Vienna
For upcoming talks see the schedule at
Join Zoom Meeting
Meeting ID: 967 9073 7954
Passcode: 031793
Join by SIP
Join by H.323
162.255.37.11 (US West)
162.255.36.11 (US East)
213.19.144.110 (Amsterdam Netherlands)
213.244.140.110 (Germany)
Meeting ID: 967 9073 7954
Passcode: 031793
Join by Skype for Business

Dear Colleagues,
it is our pleasure to invite you to the next talk of the
Online Seminar on Input-to-State Stability and its Applications
Speaker:
Pierdomenico Pepe (University of L'Aquila, Italy)
Title:
Nonlinear Halanay's Inequalities for ISS of Retarded Systems: the Continuous and the Discrete Time Case
Date:
Thursday, 20 January 2022, at 5:00 pm (Time zone of Amsterdam, Berlin, Paris, Rome, Vienna)
To check the current time in this zone you may look to
Abstract: Nonlinear versions of continuous-time and discrete-time Halanay's inequalities are presented as sufficient conditions for the convergence of involved functions to the origin, uniformly with respect to bounded sets of initial values. The same results are shown in the case forcing terms are also present, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations and by discrete-time equations with delays are shown.
References: [1] Pierdomenico Pepe, A Nonlinear Version of Halanay’s Inequality for the Uniform Convergence to the Origin, Mathematical Control and Related Fields, 2021, doi: 10.3934/mcrf.2021045
[2] Maria Teresa Grifa, Pierdomenico Pepe, On Stability Analysis of Discrete-Time Systems With Constrained Time-Delays via Nonlinear Halanay-Type Inequality, IEEE Control Systems Letters, Volume 5, Issue 3, July 2021, doi: 10.1109/LCSYS.2020.3007096
*************************
The talk will be given via Zoom.
The login data for the Zoom meeting is provided at the bottom of this email.
Please feel free to forward this invitation to anyone who might be interested. For being included in the mailing list, it is enough to write a short email to
If you are unable to take part in the seminar, you may check/subscribe to the ISS YouTube Channel
We plan to put the recorded talks there (after the confirmation of the speaker).
If you do not wish to get any further invitations to our seminar, please let us know by replying to this email.
With best regards,
Andrii Mironchenko & Daniel Liberzon
--- Invitation to Zoom meeting ---
Andrii Mironchenko is inviting you to a scheduled Zoom meeting.
Topic: ISS Seminar
Time: 20 January, 2022 05:00 PM Amsterdam, Berlin, Rome, Stockholm, Vienna
For upcoming talks see the schedule at
Join Zoom Meeting
Meeting ID: 967 9073 7954
Passcode: 031793
Join by SIP
Join by H.323
162.255.37.11 (US West)
162.255.36.11 (US East)
213.19.144.110 (Amsterdam Netherlands)
213.244.140.110 (Germany)
Meeting ID: 967 9073 7954
Passcode: 031793
Join by Skype for Business

Dear Colleagues,
the interest in infinite-dimensional ISS theory is growing.
In particular, recently a Topical Issue "Input-to-state stability for infinite-dimensional systems"
has appeared in the MCSS Journal
This topical collection presents recent progress in input-to-state stability for infinite-dimensional systems and provides an overview of various techniques employed in this field. These encompass methods from nonlinear control, operator and semigroup theory, Lyapunov theory, nonlinear networks, and partial differential equations (PDEs).
Guest Editors of this Special Issue:
Birgit Jacob, Andrii Mironchenko, Felix Schwenninger

Dear Colleagues!
We would like to invite you to take part in the 3rd Workshop on Stability and Control of Infinite-Dimensional Systems (SCINDIS)
• The Workshop will take place on September 27 – 29, 2021
• Fully online
• Scheduled from 13:00 – 21:30 (Amsterdam-Berlin-Paris-Vienna time)
• No registration fees
• Organized by: Sergey Dashkovskiy, René Hosfeld, Birgit Jacob, Andrii Mironchenko, and Fabian Wirth.
The registration site is open right now. As of 03.09.2021, there are more than 90 registered participants in the workshop. https://www.fan.uni-wuppertal.de/de/scindis-2020/list-of-participants.html
We encourage you to contribute to the interactive sessions of the Workshop.
In a 1-hour interactive session, the speakers will be able to present their work by means of a short presentation, that will engage the participants in a lively discussion. An interactive session can be understood as an online version of a poster session. Each speaker of an interactive session will be a host of an individual Zoom meeting, and she/he can fully use her/his imagination to create a way of presenting the material.
We are looking forward to seeing you online at SCINDIS! All the best,
Organizers
Sergey Dashkovskiy, René Hosfeld, Birgit Jacob, Andrii Mironchenko, Fabian Wirth

Dear Colleagues,
it is our pleasure to invite you to the next talk of the
Online Seminar on Input-to-State Stability and its Applications
Speaker:
Romain Postoyan (CNRS, Université de Lorraine, France)
Title:
Event-Triggered Control Through the Eyes of Hybrid Small-Gain Theorem
Date:
Thursday, 29 July 2021, at 5:00 pm (Time zone of Amsterdam, Berlin, Paris, Rome, Vienna)
To check the current time in this zone you may look to
Abstract: A common approach to design event-triggered controllers is emulation. The idea is to first construct a feedback law in continuous-time, which ensures the desired closed-loop properties. Then, the communication constraints between the plant and the controller are taken into account and a triggering rule is synthesized to generate the transmission instants in such a way that the properties of the continuous-time closed-loop system are preserved, and a strictly positive minimum inter-event time exists, which is essential in practice.
Various triggering rules have been proposed in this context in the literature, including relative threshold, fixed threshold, dynamic triggering law to mention a few. We will show in this talk that these seemingly unrelated techniques can all be interpreted in a unified manner. Indeed, it appears that all them guarantee the satisfaction of the conditions of a hybrid small-gain theorem. This unifying perspective provides clear viewpoints on the essential differences and similarities of existing event-triggering policies. Interestingly, for all the considered laws, the small-gain condition vacuously holds in the sense that one of the interconnection gains is zero. We then exploit this fact to modify the original triggering law in such a way that the small-gain condition is no longer trivially satisfied. By doing so, we obtain redesigned strategies, which may reduce the number of transmissions as illustrated by an example.
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The talk will be given via Zoom.
The login data for the Zoom meeting is provided at the bottom of this email.
Please feel free to forward this invitation to anyone who might be interested. For being included in the mailing list, it is enough to write a short email to
If you are unable to take part in the seminar, you may check/subscribe to the ISS YouTube Channel
We plan to put the recorded talks there (after the confirmation of the speaker).
If you do not wish to get any further invitations to our seminar, please let us know by replying to this email.
With best regards,
Andrii Mironchenko & Daniel Liberzon
--- Invitation to Zoom meeting ---
Andrii Mironchenko is inviting you to a scheduled Zoom meeting.
Topic: ISS Seminar
Time: 29 July, 2021 05:00 PM Amsterdam, Berlin, Rome, Stockholm, Vienna
For upcoming talks see the schedule at
Join Zoom Meeting
Meeting ID: 967 9073 7954
Passcode: 031793
Join by SIP
Join by H.323
162.255.37.11 (US West)
162.255.36.11 (US East)
213.19.144.110 (Amsterdam Netherlands)
213.244.140.110 (Germany)
Meeting ID: 967 9073 7954
Passcode: 031793
Join by Skype for Business

Dear Colleagues,
it is my pleasure to invite you to the next talk of the
Online Seminar on Input-to-State Stability and its Applications
Speaker:
Miroslav Krstic (University of California, San Diego, USA)
Title:
Fixed-Time ISS and Prescribed-Time Stabilization
Date:
Thursday, 22 July 2021, at 5:00 pm (Central European Time)
Abstract: In prescribed-time stabilization the task is to design a feedback law that guarantees completion of the convergence to a set point no later than a time that is prescribed by the user and independent of the initial condition of the plant. When the plant model is known perfectly and the full state is measured, ISS issues do not arise. However, in the presence of disturbances or under observer-based feedback, ISS with respect to various inputs becomes of interest. Perhaps unexpectedly, once prescribed-time stabilization is achieved, an ISS-like property stronger than the conventional ISS is obtained as a bonus. Specifically, the origin, which is not necessarily the system’s equilibrium, is made attractive in prescribed time even in the presence of non-vanishing disturbances. Or, in simpler language, the ISS gain is a function of time and decays to zero at the terminal time. I will discuss the ISS issues associated with prescribed-time feedback design for general linear ODEs, some nonlinear ODEs with a disturbance matched by control, and briefly for parabolic PDEs (in hyperbolic PDEs, finite-time stabilization, when possible, is obtained as easily as exponential stabilization).
*************************
The talk will be given via Zoom.
The login data for the Zoom meeting is provided at the bottom of this email.
Please feel free to forward this invitation to anyone who might be interested. For being included in the mailing list, it is enough to write a short email to
If you are unable to take part in the seminar, you may check/subscribe to the ISS YouTube Channel
We plan to put the recorded talks there (after the confirmation of the speaker).
If you do not wish to get any further invitations to our seminar, please let us know by replying to this email.
With best regards,
Andrii Mironchenko & Daniel Liberzon
--- Invitation to Zoom meeting ---
Andrii Mironchenko is inviting you to a scheduled Zoom meeting.
Topic: ISS Seminar
Time: 22 July, 2021 05:00 PM Amsterdam, Berlin, Rome, Stockholm, Vienna
(Every week on Thu, 05:00 PM CET)
Join Zoom Meeting
Meeting ID: 967 9073 7954
Passcode: 031793
Join by SIP
Join by H.323
162.255.37.11 (US West)
162.255.36.11 (US East)
213.19.144.110 (Amsterdam Netherlands)
213.244.140.110 (Germany)
Meeting ID: 967 9073 7954
Passcode: 031793
Join by Skype for Business

Dear Colleagues,
We would like to invite you to take part in the Tutorial Session
"Stability and Robust Control of PDEs and Large Scale Networks"
organised by
Andrii Mironchenko and Christophe Prieur
at ECC 2021:
Date: Thursday, July 1, 15:30-17:30 (CET)
Abstract:
In this tutorial we introduce to a broad audience key concepts, results and applications of the infinite-dimensional stability theory, with a particular focus on input-to-state stability and robustness analysis. The scope of techniques which we discuss includes Lyapunov functions, nonlinear systems theory, semigroup theory, spectral decompositions and boundary control. We discuss the applications of these methods to robust stability of boundary control systems, robust control of partial differential equations and to stability of large-scale and infinite networks.
Please feel free to forward this invitation to anyone who might be interested.
With best regards,
Andrii Mironchenko

added a research item
This paper provides a Lyapunov-based small-gain theorem for input-to-state stability (ISS) of networks composed of infinitely many finite-dimensional systems. We model these networks on infinite-dimensional $\ell_{\infty}$-type spaces. A crucial assumption in our results is that the internal Lyapunov gains, modeling the influence of the subsystems on each other, are linear functions. Moreover, the gain operator built from the internal gains is assumed to be subadditive and homogeneous, which covers both max-type and sum-type formulations for the ISS Lyapunov functions of the subsystems. As a consequence, the small-gain condition can be formulated in terms of a generalized spectral radius of the gain operator. Through an example, we show that the small-gain condition can easily be checked if the interconnection topology of the network has some kind of symmetry. While our main result provides an ISS Lyapunov function in implication form for the overall network, an ISS Lyapunov function in a dissipative form is constructed under mild extra assumptions.
Please subscribe also to our YouTube channel on ISS theory:
This is going to become a hub for recent talks on ISS theory at various conferences and seminars.

added a research item
We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results apply to the large class of ordered Banach spaces that have normal and generating cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.
added 2 research items
We introduce the concept of non-uniform input-to-state stability for networks, which combines the uniform global stability together with the uniform attractivity of any finite number of modes of the system, but which does not guarantee the uniform convergence of all modes. We show that given an infinite network of ISS subsystems, which do not have a uniform $\mathcal{KL}$-bound on the transient behavior, and if the gain operator satisfies the bounded monotone invertibility property, then the whole network is non-uniformly ISS and its any finite subnetwork is uniformly ISS.
We prove nonlinear small-gain theorems for input-to-state stability of infinite heterogeneous networks, consisting of input-to-stable subsystems of possibly infinite dimension. Furthermore, we prove small-gain results for the uniform global stability of infinite networks. Our results extend available theorems for finite networks of finite- or infinite-dimensional systems. These results are shown either under the so-called monotone limit property or under the monotone bounded invertibility property, which is equivalent to a uniform small-gain condition. We show that for finite networks of nonlinear systems these properties are equivalent to the so-called strong small-gain condition of the gain operator, and for infinite networks with linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.
Dear Colleagues,
we would like to invite you to participate in the
Pre-Conference Workshop of the IFAC 2020 World Congress on
"Input-to-state stability and control of infinite-dimensional systems",
which will be held virtually on Saturday, 11 July 2020 (full-day workshop).
Speakers at this Workshop are:
• Iasson Karafyllis, National Technical University of Athens, Greece
• Miroslav Krstic, University of California, San Diego, California, USA
• Hugo Lhachemi, University College Dublin, Ireland
• Andrii Mironchenko, University of Passau, Germany
• Pierdomenico Pepe, University of l'Aquila, Italy
• Christophe Prieur, CNRS, Université Grenoble Alpes, France
• Fabian Wirth, University of Passau, Germany
More information about the topic of the workshop can be found at:
Participation fee for this workshop is
• 70 Euro for the standard participation
• 40 Euro for the students
Please register to the Workshop until the 8th of July, at
Participation at the Workshop is decoupled from the participation at the
IFAC World Congress itself, that is one can register for the workshop even
if you do not plan to participate in IFAC World Congress.
Please free to distribute this invitation to anyone who might be interested.
Looking forward to seeing You virtually at the Workshop.
Best regards,
Christophe Prieur and Andrii Mironchenko,
The organizers of the IFAC WC Pre-Conference Workshop
"Input-to-state stability and control of infinite-dimensional systems"

The speaker of the next Zoom Seminar on "Dynamical systems" will be
Dr. Andrii Mironchenko
Title: Small-gain theorems for infinite networks of input-to-state stable systems
Time: June 2, 2020 at 2pm (local time in Amsterdam, Berlin, Rome, Stockholm, Vienna)
Abstract:
In this talk we discuss sufficient conditions for stability of infinite networks of heterogeneous control systems, whose subsystems can be itself infinite-dimensional systems of different classes.
We show that a well-posed interconnection of an infinite number of input-to-state stable systems is again input-to-state stable provided
the discrete-time system, induced by the gain operator is stable.
Small-gain theorems for input-to-state stability of finite networks are a special case of our general result.
For infinite networks with linear gains our small-gain condition is equivalent to the fact that the spectral radius of the gain operator is less than 1, which is a tight condition, which cannot be improved.
Additionally, we show the small-gain theorem for infinite networks of systems which are merely uniform globally stable as well as a nonuniform ISS small-gain theorem, which gives a property which is somewhat weaker than ISS, but under weaker assumptions for subsystems.
We show our main results both in summation and in semimaximum formulation of the ISS properties for subsystems, which further increases the applicability of the results.
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Below, you will find the details for all further meetings as well as the login-data.
Please free to distribute this invitation to anyone who might be interested.
Zoom-Meeting:
Meeting-ID: 930 4949 8885

Dear all,
let us announce the following talk at Zoom-Seminar "Dynamical Systems"
Title: Conditions for uniform stability of positive linear operators in infinite dimensions
Speaker: Jochen Glück, University of Passau
Abstract:
Motivated by the study of small gain theorems for input-to-state
stability of infinite networks we discuss under which conditions
a positive linear operator T on an ordered Banach space X is uniformly
stable in the sense that T^n → 0 with respect to the operator norm as n → ∞.
In the ﬁnite-dimensional case, a necessary and suﬃcient condition for uni-
form stability is that $T \not\geq x$ whenever x ≥ 0. We present a “uniform version”
of this condition which can be used to obtain a similar characterisation of
uniform stability in inﬁnite dimensions.
Then, we use a large part of the talk to discuss a general method that
can serve as a guideline to derive uniform inﬁnite dimensional results if their
ﬁnite-dimensional counterparts are already known: namely, one can construct
so-called ﬁlter powers of a Banach space X; a ﬁlter power is a certain larger
Banach space X^F that contains X, and X^F is very useful to encode important
properties of operators on X.
Time: 25.Mai.2020 02:00 PM Amsterdam, Berlin, Rom, Stockholm, Wien
Below, you will find the log-in data for this (and all subsequent) zoom meetings.
Looking forward to seeing you!
Best regards,
Andrii Mironchenko
********************************************************************
Invitation for Zoom-Meeting:
Topic: Oberseminar (Lehrstuhl für Dynamische Systeme)
Time: 25.Mai.2020 02:00 PM Amsterdam, Berlin, Rom, Stockholm, Wien
Import the ics-files to your schedule (.ics):
Weekly:
Zoom-Meeting:
Meeting-ID: 930 4949 8885
SIP:
H.323:
162.255.37.11 (US West)
162.255.36.11 (US East)
213.19.144.110 (EMEA)
Passwort: 823588
Meeting-ID: 930 4949 8885

Dear Colleagues,
It is a pleasure for us to inform you about the Pre-Conference Workshop
“Input-to-state stability and control of infinite-dimensional systems“
which be offered at the 21rst IFAC World Congress to be held in Berlin, Germany, 12-17 July 2020.
The Workshop will take place on Sunday, July 12th, 2020.
Summary:
In this workshop, we provide to a broad audience an overview of key concepts, results, and applications of the infinite-dimensional input-to-state stability theory.
The scope of techniques that we discuss includes Lyapunov functions, semigroup theory, spectral methods, boundary control and nonlinear systems theory. We discuss the applications of these methods to robust stability of boundary control systems, robust control of partial differential equations and to the stability of networks with infinite-dimensional components.
Homepage:
Speakers:
• Fabian Wirth, University of Passau, Germany
• Christophe Prieur, CNRS, Université Grenoble Alpes, France
• Andrii Mironchenko, University of Passau, Germany
• Hugo Lhachemi, University College Dublin, Ireland
• Pierdomenico Pepe, University of l’Aquila, Italy
• Miroslav Krstic, University of California, San Diego, California, USA
Organisers:
Each participant of the IFAC World Congress may apply for participation at this Workshop (for the additional fee).
Within this Workshop, there will be a poster session. All participants may present posters related to the workshop subject. To register a poster please send a short email to the organisers with the title of the poster.
Looking forward to seeing you in Berlin!
All the best,
Christophe P. and Andrii M.

A survey on ISS of infinite-dimensional systems
A. Mironchenko, Ch. Prieur. Input-to-state stability of infinite-dimensional systems: recent results and open questions
has been accepted to SIAM Review. The final preprint is attached.

added 3 research items
In this paper we consider countable couplings of finite-dimensional input-to-state stable systems. We consider the whole interconnection as an infinite-dimensional system on the ∞ state space. We develop stability conditions of the small-gain type to guarantee that the whole system remains ISS and highlight the differences between finite and infinite couplings by means of examples. We show that using our methodology it is possible to study uniform global asymptotic stability of nonlinear spatially invariant systems by solving a finite number of nonlinear algebraic inequalities.
This paper presents a small-gain theorem for networks composed of a countably infinite number of finite-dimensional subsystems. Assuming that each subsystem is exponentially input-to-state stable, we show that if the gain operator, collecting all the information about the internal Lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially input-to-state stable. The effectiveness of our result is illustrated through several examples including nonlinear spatially invariant systems with sector nonlinearities and a road traffic network.
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 such Lyapunov functions implies integral-to-integral input-to-state stability. Assuming further regularity it is possible to conclude input-to-state stability. For a particular class of linear systems with unbounded admissible input operators, explicit constructions of noncoercive Lyapunov functions are provided. The theory is applied to a heat equation with Dirichlet boundary conditions.
added a research item
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 stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. 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 a special attention devoted to ISS theory of boundary control systems. 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 ISS framework to study 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. Furthermore, this survey suggests many open problems throughout the paper.
added 3 research items
We show by means of counterexamples that many characterizations of input-to-state stability (ISS) known for ODE systems are not valid for general differential equations in Banach spaces. Moreover, these notions or combinations of notions are not equivalent to each other, and can be classified into several groups according to the type and grade of nonuniformity. 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. We characterize strong ISS as a strong asymptotic gain property plus global stability.
In this paper a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite-and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.
We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.
added 2 research items
We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the same class of control systems we derive small-gain theorems for asymptotic gain, uniform global stability and weak input-to-state stability properties. We show that our technique is applicable for different formulations of ISS property (summation, maximum, semimaximum) and discuss tightness of achieved small-gain theorems. Finally, we introduce variations of asymptotic gain and limit properties, which are particularly useful for small-gain arguments and characterize ISS in terms of these notions.
We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.
Dear all!
An updated full list of publications of this project you may find here:
Meanwhile ISS of infinite-dimensional systems is getting more and more attention within the control theoretic community.
There were/will be invited sessions on ISS of infinite-dimensional systems at
• 23rd International Symposium on the Mathematical Theory of Networks and Systems (MTNS ), Hong Kong, 2018
• 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Vienna, 2019
Furthermore, it is one of the key topics of
• 2nd Workshop on Stability and Control of Infinite-Dimensional Systems (SCINDIS-2018)
Looking forward to meeting you at these events.

added a research item
In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.
Dear all!
We would like to invite you to take part in the 2nd Workshop on Stability and Control of Infinite-Dimensional Systems (SCINDIS-2018)
The workshop will take place in
Würzburg, Germany, October 10 – 12, 2018
and is organized by Sergey Dashkovskiy, Birgit Jacob, Fabian Wirth and Andrii Mironchenko.
The first workshop has attracted leading researchers and talented young scientists from 11 countries and was a great success. This time we expect even more participants and excellent talks.
you can find the list of the invited speakers and the scope of the workshop.
The registration site will be open soon.
We are looking forward to see you in Würzburg!
All the best,
Organizers

added a research item
For a broad class of infinite-dimensional systems, we characterize input-to-state practical stability (ISpS) using the uniform limit property and in terms of input-to-state stability. We specialize our results to the systems with Lipschitz continuous flows and evolution equations in Banach spaces. Even for the special case of ordinary differential equations our results are novel and improve existing criteria for ISpS.
Dear all!
Now the full list of publications of this project you may find here:
During the last months several new papers have been submitted, in particular:
1. Nabiullin, R. and Schwenninger, F.. Strong input-to-state stability for infinite dimensional linear systems.  Submitted, 2017.
2. Jacob, B.; Schwenninger, F. and Zwart, H. L_infty-admissibility for analytic semigroups and applications to input-to-state stability. Submitted, 2017.
3. Mironchenko, A. and Wirth, F. Lyapunov characterization of input-to-state stability fo semilinear control systems over Banach spaces. Submitted to Systems & Control Letters, 2017.
Also the final versions of the following manuscripts have been submitted.
1. Mironchenko, A. and Wirth, F. Characterizations of input-to-state stability for infinite-dimensional systems. Accepted to IEEE Transactions on Automatic Control, 2017.
2. Mironchenko, A. and Wirth, F. A non-coercive Lyapunov framework for stability of distributed parameter systems. Accepted to the 56th IEEE Conference on Decision and Control, 2017.
3. Mironchenko, A. and Wirth, F. Input-to-state stability of time-delay systems: criteria and open problems. Accepted to the 56th IEEE Conference on Decision and Control, 2017.
To be continued...

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We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.
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This paper deals with strong versions of input-to-state stability and integral input-to-state stability of infinite-dimensional linear systems with an unbounded input operator. We show that infinite-time admissibility with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable but, unlike in the case of exponentially stable systems, not a necessary one.
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We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant \textit{boundary disturbances} is equivalent to ISS of a closely related nonlinear parabolic PDE with constant \textit{distributed disturbances} and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques. As an application of our results, we show that the PDE backstepping controller which stabilizes linear reaction-diffusion equations from the boundary is robust with respect to additive actuator disturbances.
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We show that practical uniform global asymptotic stability (pUGAS) is equivalent to existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of robustly forward complete distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in Banach spaces. We apply this criterion to show that existence of a non-coercive Lyapunov function ensures pUGAS of robustly forward complete systems. For ordinary differential equations with uniformly bounded disturbances the concept of uniform weak attractivity is equivalent to the usual weak attractivity. It is however essentially stronger than weak attractivity for infinite-dimensional systems, even for linear ones.
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In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions with respect to $L^\infty$ are equivalent.
This work contributes to the recently intensified study of input-to-state stability for infinite-dimensional systems. The focus is laid on the relation between input-to-state stability and integral input-to-state stability for linear systems with a possibly unbounded control operator. The main result is that for parabolic diagonal systems both notions coincide, even in the setting of inputs in L ∞ , and a simple criterion is derived.
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We prove characterizations of ISS for a large class of infinite-dimensional control systems, including differential equations in Banach spaces, time-delay systems, ordinary differential equations, switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang in \cite{SoW96} for ODE systems. For the special case of differential equations in Banach spaces we prove even broader criteria for ISS property. Using an important technical result from \cite{SoW96} we show that the characterizations obtained in \cite{SoW96} are a special case of our results. 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 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.
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We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel ‘integral’ construction of non-coercive Lipschitz continuous Lyapunov functions. Finally, converse Lyapunov theorems are used in order to prove Lyapunov characterizations of input-to-state stability of infinite-dimensional systems.
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We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted.
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We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Some examples show the necessity of the assumptions which are made.
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We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel 'integral' construction of non-coercive Lipschitz continuous Lyapunov functions. Finally, converse Lyapunov theorems are used in order to prove Lyapunov characterizations of input-to-state stability of infinite-dimensional systems.
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In this project, we are going to build a firm basis for the investigation
of input-to-state stability and stabilization of distributed parameter
systems. More specifically, our aims are:
1. To develop an ISS theory for linear and bilinear distributed parameter systems, including criteria for input-to-state stability and stabilizability of linear and bilinear DPS and sufficient conditions for robustness of ISS.
2. To obtain the infinite-dimensional counterparts of fundamental nonlinear results from ISS theory of finite-dimensional systems. In particular, Lyapunov characterizations of the ISS property, small-gain theorems for DPS and characterizations of ISS in terms of other stability properties.
3. To develop methods for robust stabilization of infinite-dimensional systems, namely, a robust version of continuum backstepping, finite-time robust stabilization of partial differential equations and design of ISS stabilizers for port-Hamiltonian systems.
Please check also our the YouTube ISS channel: