Andrii Mironchenko

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All published works (90)

Infinite-dimensional port-Hamiltonian systems with a stationary interface

2025-02-05

Alexander Kilian , Bernhard Maschke , Andrii Mironchenko +1 more

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Modeling and stability analysis of live systems with time-varying dimension

2025-01-27

Andrii Mironchenko

A major limitation of the classical control theory is the assumption that the state space and its dimension do not change with time. This prevents analyzing and even formalizing the … A major limitation of the classical control theory is the assumption that the state space and its dimension do not change with time. This prevents analyzing and even formalizing the stability and control problems for open multi-agent systems whose agents may enter or leave the network, industrial processes where the sensors or actuators may be exchanged frequently, smart grids, etc. In this work, we propose a framework of live systems that covers a rather general class of systems with a time-varying state space. We argue that input-to-state stability is a proper stability notion for this class of systems, and many of the classic tools and results, such as Lyapunov methods and superposition theorems, can be extended to this setting.

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Input-to-state stability in integral norms for linear infinite-dimensional systems

2025-01-13

Sahiba Arora , Andrii Mironchenko

We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. … We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. While input-to-state stability is typically characterized by exponential stability and finite-time admissibility, we show that this equivalence does not extend directly to integral norms. For analytic semigroups, we establish a precise characterization using maximal regularity theory. Additionally, we provide direct Lyapunov theorems and construct Lyapunov functions for $L^p$-$L^q$-ISS and demonstrate the results with examples, including diagonal systems and diffusion equations.

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Stability criteria for positive semigroups on ordered Banach spaces

2024-12-18

Jochen Glück , Andrii Mironchenko

Characterization of input-to-output stability for infinite dimensional systems

2024-10-08

Patrick Bachmann , Sergey Dashkovskiy , Andrii Mironchenko

We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special … We prove a superposition theorem for input-to-output stability (IOS) of a broad class of nonlinear infinite-dimensional systems with outputs including both continuous-time and discrete-time systems. It contains, as a special case, the superposition theorem for input-to-state stability (ISS) of infinite-dimensional systems from [1] and the IOS superposition theorem for systems of ordinary differential equations from [2]. To achieve this result, we introduce and examine several novel stability and attractivity concepts for infinite dimensional systems with outputs: We prove criteria for the uniform limit property for systems with outputs, several of which are new already for systems with full-state output, we provide superposition theorems for systems which satisfy both the output-Lagrange stability property (OL) and IOS, give a sufficient condition for OL and characterize ISS in terms of IOS and input/output-to-state stability. Finally, by means of counterexamples, we illustrate the challenges appearing on the way of extension of the superposition theorems from [1] and [2] to infinite-dimensional systems with outputs.

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Input-to-state stability meets small-gain theory

2024-06-25

Andrii Mironchenko

Input-to-state stability (ISS) unifies global asymptotic stability with respect to variations of initial conditions with robustness with respect to external disturbances. First, we present Lyapunov characterizations for input-to-state stability as … Input-to-state stability (ISS) unifies global asymptotic stability with respect to variations of initial conditions with robustness with respect to external disturbances. First, we present Lyapunov characterizations for input-to-state stability as well as ISS superpositions theorems showing relations of ISS to other robust stability properties. Next, we present one of the characteristic applications of the ISS framework - the design of event-based control schemes for the stabilization of nonlinear systems. In the second half of the paper, we focus on small-gain theorems for stability analysis of finite and infinite networks with input-to-state stable components. First, we present a classical small-gain theorem in terms of trajectories for the feedback interconnection of 2 nonlinear systems. Finally, a recent Lyapunov-based small-gain result for a network with infinitely many ISS components is shown.

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For Time-Invariant Delay Systems, Global Asymptotic Stability Does Not Imply Uniform Global Attractivity

2024-01-01

Antoine Chaillet , Fabian Wirth , Andrii Mironchenko +1 more

Adapting a counterexample recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero … Adapting a counterexample recently proposed by J.L. Mancilla-Aguilar and H. Haimovich, we show here that, for time-delay systems, global asymptotic stability does not ensure that solutions converge uniformly to zero over bounded sets of initial states. Hence, the convergence might be arbitrarily slow even if initial states are confined to a bounded set.

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Forward Completeness Implies Bounded Reachable Sets for Time-Delay Systems on the State Space of Essentially Bounded Measurable Functions

2024-01-01

Lucas Brivadis , Antoine Chaillet , Andrii Mironchenko +1 more

We consider time-delay systems with a finite number of delays in the state space $L^\infty\times\mathbb{R}^n$. In this framework, we show that forward completeness implies the bounded reachability sets property, while … We consider time-delay systems with a finite number of delays in the state space $L^\infty\times\mathbb{R}^n$. In this framework, we show that forward completeness implies the bounded reachability sets property, while this implication was recently shown by J.L. Mancilla-Aguilar and H. Haimovich to fail in the state space of continuous functions. As a consequence, we show that global asymptotic stability is always uniform in the state space $L^\infty\times\mathbb{R}^n$.

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Input-to-state stability of infinite-dimensional systems: Foundations and present-day developments

2024-01-01

Andrii Mironchenko , Christophe Prieur

Input-to-state stability (ISS) unifies the stability and robustness in one notion, and serves as a basis for broad areas of nonlinear control theory. In this contribution, we covered the most … Input-to-state stability (ISS) unifies the stability and robustness in one notion, and serves as a basis for broad areas of nonlinear control theory. In this contribution, we covered the most fundamental facts in the infinite-dimensional ISS theory with a stress on Lyapunov methods. We consider various applications given by different classes of infinite-dimensional systems. Finally, we discuss a Lyapunov-based small-gain theorem for stability analysis of an interconnection of two ISS systems.

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Well-posedness and properties of the flow for semilinear evolution equations

2023-12-07

Andrii Mironchenko

Abstract We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz … Abstract We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general $$C_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.

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Lyapunov criteria for robust forward completeness of distributed parameter systems

2023-09-04

Andrii Mironchenko

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Infinite-dimensional port-Hamiltonian systems with a stationary interface

2023-01-01

Alexander Kilian , Bernhard Maschke , Andrii Mironchenko +1 more

We consider two systems of two conservation laws that are defined on complementary spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed … We consider two systems of two conservation laws that are defined on complementary spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed interface position, we characterize the boundary and interface conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. Furthermore, we present sufficient conditions for the exponential stability of the generated $C_0$-semigroup. The results are illustrated by the example of two acoustic waveguides coupled by an interface consisting of some membrane.

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Input-to-state stability of distributed parameter systems

2023-01-01

Andrii Mironchenko

Input-to-state stability (ISS) 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 … Input-to-state stability (ISS) 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 habilitation thesis, we provide a broad picture of infinite-dimensional input-to-state stability theory.

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A case study of port-Hamiltonian systems with a moving interface

2023-01-01

Alexander Kilian , Bernhard Maschke , Andrii Mironchenko +1 more

We model two systems of two conservation laws defined on complementary spatial intervals and coupled by a moving interface as a single non-autonomous port-Hamiltonian system, and provide sufficient conditions for … We model two systems of two conservation laws defined on complementary spatial intervals and coupled by a moving interface as a single non-autonomous port-Hamiltonian system, and provide sufficient conditions for its Kato-stability. An example shows that these conditions are quite restrictive. The more general question under which conditions an evolution family is generated remains open.

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Coercive quadratic converse ISS Lyapunov theorems for linear analytic systems

2023-01-01

Andrii Mironchenko , Felix L. Schwenninger

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive … We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigroup on a Hilbert space, then a quadratic ISS Lyapunov function always exists for any input operator that is bounded, or more generally, $p$-admissible with $p<2$. The constructions are semi-explicit and, in the case of self-adjoint generators, coincide with the canonical Lyapunov function being the norm squared. Finally, we construct a family of non-coercive ISS Lyapunov functions for analytic ISS systems under weaker assumptions on $B$.

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Ordinary Differential Equations with Measurable Inputs

2023-01-01

Andrii Mironchenko

In this chapter, we recall basic concepts and results of the theory of ordinary differential equations (ODEs) with measurable inputs (“Carathéodory theory”). In this chapter, we recall basic concepts and results of the theory of ordinary differential equations (ODEs) with measurable inputs (“Carathéodory theory”).

A Case Study of Port-Hamiltonian Systems With a Moving Interface

2023-01-01

Alexander Kilian , Bernhard Maschke , Andrii Mironchenko +1 more

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A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

2022-07-01

Christoph Kawan , Andrii Mironchenko , Majid Zamani

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the … In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

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Lyapunov criteria for robust forward completeness of distributed parameter systems

2022-01-01

Andrii Mironchenko

We show that the robust forward completeness for distributed parameter systems is equivalent to the existence of a corresponding Lyapunov function that increases at most exponentially along the trajectories. We show that the robust forward completeness for distributed parameter systems is equivalent to the existence of a corresponding Lyapunov function that increases at most exponentially along the trajectories.

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Stability criteria for positive $C_0$-semigroups on ordered Banach spaces

2022-01-01

Jochen Glück , Andrii Mironchenko

We consider resolvent positive operators $A$ on ordered Banach spaces and seek for conditions that ensure that their spectral bound $s(A)$ is negative. Our main result characterizes $s(A) < 0$ … We consider resolvent positive operators $A$ on ordered Banach spaces and seek for conditions that ensure that their spectral bound $s(A)$ is negative. Our main result characterizes $s(A) < 0$ in terms of so-called small-gain conditions that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property $s(A) < 0$ if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of $A$ is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. Our results can, in particular, be applied to generators $A$ of positive $C_0$-semigroups. Thus, they can be interpreted as stability results for this class of semigroups, and as such, they complement similar results recently proved for the discrete-time case. In the same vein we prove a Collatz-Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.

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Well-posedness and properties of the flow for semilinear evolution equations

2022-01-01

Andrii Mironchenko

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, … We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general $C_0$-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers' equation with nonlinear local dynamics and both distributed and boundary disturbances.

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

2021-10-25

Andrii Mironchenko , Navid Noroozi , Christoph Kawan +1 more

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Stability criteria for positive linear discrete-time systems

2021-08-22

Jochen Glück , Andrii Mironchenko

Abstract 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 … Abstract 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 are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive 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.

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Non-uniform ISS small-gain theorem for infinite networks

2021-08-09

Andrii Mironchenko

Abstract We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork while it allows for non-uniform convergence … Abstract We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork while it allows for non-uniform convergence of all components. For an infinite network consisting of input-to-state stable subsystems, which do not necessarily have a uniform $\mathscr{K}\mathscr{L}$-bound on the transient behaviour, we show the following: if the gain operator satisfies the uniform small-gain condition, then the whole network is non-uniformly input-to-state stable and all its finite subnetworks are input-to-state stable.

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Small Gain Theorems for General Networks of Heterogeneous Infinite-Dimensional Systems

2021-01-01

Andrii Mironchenko

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 14 January 2019Accepted: 15 January 2021Published online: 13 April 2021Keywordsinfinite-dimensional systems, input-to-state stability, … Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 14 January 2019Accepted: 15 January 2021Published online: 13 April 2021Keywordsinfinite-dimensional systems, input-to-state stability, interconnected systems, nonlinear systems, small-gain theoremsAMS Subject Headings93C25, 37C75, 93A15, 93C10Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

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A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems

2021-01-01

Christoph Kawan , Andrii Mironchenko , Majid Zamani

In this paper, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the … In this paper, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

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

2021-01-01

Andrii Mironchenko , Navid Noroozi , Christoph Kawan +1 more

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 … 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.

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A Lyapunov-Based Small-Gain Theorem for Infinite Networks

2020-12-03

Christoph Kawan , Andrii Mironchenko , Abdalla Swikir +2 more

This article 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 … This article 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.

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A relaxed small-gain theorem for infinite networks

2020-11-21

Navid Noroozi , Andrii Mironchenko , Fabian Wirth

Motivated by the scalability problem in large networks, we study stability of a network of infinitely many finite-dimensional subsystems. We develop a so-called relaxed small-gain theorem for input-to-state stability (ISS) … Motivated by the scalability problem in large networks, we study stability of a network of infinitely many finite-dimensional subsystems. We develop a so-called relaxed small-gain theorem for input-to-state stability (ISS) with respect to a closed set and show that every exponentially input-to-state stable system necessarily satisfies the proposed small-gain condition. Following our bottom-up approach, we study the well-posedness of the interconnection based on the behavior of the individual subsystems. Finally, we over-approximate large-but-finite networks by infinite networks and show that all the stability properties and the performance indices obtained for the infinite system can be transferred to the original finite one if each subsystem of the infinite network is individually ISS. Interestingly, the size of the truncated network does not need to be known. The effectiveness of our small-gain theorem is verified by application to an urban traffic network.

Stability criteria for positive linear discrete-time systems

2020-11-04

Jochen Glück , Andrii Mironchenko

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 … 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 are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive 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.

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Non-uniform ISS small-gain theorem for infinite networks

2020-07-14

Andrii Mironchenko

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, … 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.

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Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

2020-07-11

Andrii Mironchenko , Christoph Kawan , Jochen Glück

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 … 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.

Local Stabilization of an Unstable Parabolic Equation via Saturated Controls

2020-07-07

Andrii Mironchenko , Christophe Prieur , Fabian Wirth

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of … We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of the uncontrolled system and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop system, given in terms of linear and bilinear matrix inequalities. We show that our results can be used with distributed as well as scalar boundary control, and with different types of saturations. The efficiency of the proposed method is demonstrated by means of numerical simulations.

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Small-gain theorem for stability, cooperative control and distributed observation of infinite networks

2020-01-01

Navid Noroozi , Andrii Mironchenko , Christoph Kawan +1 more

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain … Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficient way. To demonstrate broad applicability of our small-gain theorem, we apply it to the stability analysis of infinite time-varying networks, to consensus in infinite-agent systems, as well as to the design of distributed observers for infinite networks.

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

2020-01-01

Andrii Mironchenko , Christophe Prieur

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 for the estimation of the impact of inputs … 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 for the estimation of 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 the ISS property for distributed parameter systems. Then fundamental properties are given, such 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 in studying the stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows for the reduction of 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 that are particularly suited for this class of systems. Finally, numerous applications are considered for which ISS properties play a crucial role in their study. The survey contains recent as well as classical results on systems theory and suggests many open problems.

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Noncoercive Lyapunov Functions for Input-to-State Stability of Infinite-Dimensional Systems

2020-01-01

Birgit Jacob , Andrii Mironchenko , J. R. Partington +1 more

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 4 November 2019Accepted: 06 July 2020Published online: 08 October 2020Keywordsinfinite-dimensional systems, input-to-state stability, … Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 4 November 2019Accepted: 06 July 2020Published online: 08 October 2020Keywordsinfinite-dimensional systems, input-to-state stability, Lyapunov functions, nonlinear systems, linear systemsAMS Subject Headings35Q93, 37B25, 37L15, 93C10, 93C25, 93D05, 93D09Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

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A relaxed small-gain theorem for infinite networks

2020-01-01

Navid Noroozi , Andrii Mironchenko , Fabian Wirth

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Stability criteria for positive linear discrete-time systems

2020-01-01

Jochen Glück , Andrii Mironchenko

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Non-uniform ISS small-gain theorem for infinite networks

2020-01-01

Andrii Mironchenko

We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork, while it allows for non-uniform convergence of … We introduce the concept of non-uniform input-to-state stability for networks. It combines the uniform global stability with the uniform attractivity of any subnetwork, while it allows for non-uniform convergence of all components. For an infinite network consisting of input-to-state stable subsystems, that do not necessarily have a uniform KL-bound on the transient behaviour, we show: If the gain operator satisfies the uniform small-gain condition, then the whole network is non-uniformly input-to-state stable and all its finite subnetworks are input-to-state stable.

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Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

2020-01-01

Andrii Mironchenko , Christoph Kawan , Jochen Glück

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain … We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a 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.

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A Lyapunov-based small-gain theorem for infinite networks

2019-10-28

Christoph Kawan , Andrii Mironchenko , Abdalla Swikir +2 more

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 … 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.

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Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

2019-03-14

Andrii Mironchenko , Fabian Wirth

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Small gain theorems for general networks of heterogeneous infinite-dimensional systems

2019-01-11

Andrii Mironchenko

We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable (ISS) control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the … We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable (ISS) 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 uniform asymptotic gain and uniform limit properties, which are particularly useful for small-gain arguments and characterize ISS in terms of these notions.

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Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances

2019-01-01

Andrii Mironchenko , Iasson Karafyllis , Miroslav Krstić

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 … 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 over a multidimensional spatial domain with Dirichlet boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and homogeneous Dirichlet 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 w.r.t. additive actuator disturbances.

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

2019-01-01

Birgit Jacob , Andrii Mironchenko , J. R. Partington +1 more

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 … 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 norm-to-integral input-to-state stability. This property in turn is equivalent to input-to-state stability, if the system satisfies certain mild regularity assumptions. 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.

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Design of saturated controls for an unstable parabolic PDE

2019-01-01

Andrii Mironchenko , Christophe Prieur , Fabian Wirth

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume Lyapunov stability of the … We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume Lyapunov stability of the uncontrolled system, and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop system, given in terms of linear and bilinear matrix inequalities. We show that our results can be used with distributed as well as scalar boundary control. The efficiency of the proposed method is demonstrated by means of a numerical simulation.

A Lyapunov-based small-gain theorem for infinite networks

2019-01-01

Christoph Kawan , Andrii Mironchenko , Abdalla Swikir +2 more

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

2019-01-01

Andrii Mironchenko , Christophe Prieur

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 … 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.

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Small gain theorems for general networks of heterogeneous infinite-dimensional systems

2019-01-01

Andrii Mironchenko

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Non-coercive Lyapunov functions for infinite-dimensional systems

2018-11-28

Andrii Mironchenko , Fabian Wirth

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Common Coauthors

Coauthor	Papers Together
Fabian Wirth	26
Sergey Dashkovskiy	10
Christoph Kawan	10
Majid Zamani	8
Navid Noroozi	8
Jochen Glück	7
Christophe Prieur	5
Alexander Kilian	4
Iasson Karafyllis	4
Miroslav Krstić	4
Hiroshi Ito	4
Bernhard Maschke	4
Abdalla Swikir	3
Lars Naujok	3
Guosong Yang	3
Michael Kosmykov	3
Daniel Liberzon	3
Jan Kozłowski	2
J. R. Partington	2
Lucas Brivadis	2
Birgit Jacob	2
Antoine Chaillet	2
Sahiba Arora	1
Felix L. Schwenninger	1
Kai Wulff	1
J. Kozłowski	1
Patrick Bachmann	1

Commonly Cited References

Small Gain Theorems for Large Scale Systems and Construction of ISS Lyapunov Functions

2010-01-01

Sergey Dashkovskiy , Björn S. Rüffer , Fabian Wirth

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. … We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS; the cases of summation, maximization, and separation with respect to external gains are obtained as corollaries.

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An ISS small gain theorem for general networks

2007-05-01

Sergey Dashkovskiy , Björn S. Rüffer , Fabian Wirth

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

2017-09-25

Andrii Mironchenko , Fabian Wirth

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), … 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), and 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 noncoercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS (sISS) that is equivalent to ISS in the ODE case, but 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.

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A vector small-gain theorem for general non-linear control systems

2011-06-02

Iasson Karafyllis , Zhong‐Ping Jiang

A new small-gain theorem is presented for general non-linear control systems and can be viewed as unification of previously developed non-linear small-gain theorems for systems described by ordinary differential equations, … A new small-gain theorem is presented for general non-linear control systems and can be viewed as unification of previously developed non-linear small-gain theorems for systems described by ordinary differential equations, retarded functional differential equations and hybrid models. The novelty of this research work is that vector Lyapunov functions and functionals are utilized to derive various input-to-output stability and input-to-state stability results. It is shown that the proposed approach is extendible to several important classes of control systems such as large-scale complex systems, non-linear sampled-data systems and non-linear time-delay systems. An application to a biochemical circuit model illustrates the generality and power of the proposed vector small-gain theorem.

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ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws

2012-01-18

Christophe Prieur , Frédéric Mazenc

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

2015-01-01

Andrii Mironchenko , Hiroshi Ito

This paper is devoted to two issues. The first one is to provide Lyapunov-based tools to establish integral input-to-state stability (iISS) and input-to-state stability (ISS) for some classes of nonlinear … This paper is devoted to two issues. The first one is to provide Lyapunov-based tools to establish integral input-to-state stability (iISS) and input-to-state stability (ISS) for some classes of nonlinear parabolic equations. The other one is to provide a stability criterion for interconnections of iISS parabolic systems. The results addressing the former problem allow us to overcome obstacles arising in tackling the latter one. The results for the latter problem are a small-gain condition and a formula of Lyapunov functions which can be constructed for interconnections whenever the small-gain condition holds. It is demonstrated that for interconnections of partial differential equations, the choice of a right state and input spaces is crucial, in particular for iISS subsystems which are not ISS. As illustrative examples, stability of two highly nonlinear reaction-diffusion systems is established by the proposed small-gain criterion.

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ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

2016-01-19

Iasson Karafyllis , Miroslav Krstić

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established … Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.

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An Introduction to Semilinear Evolution Equations

1998-10-29

Thierry Cazenave , Alain Haraux , Yvan Martel

Abstract This book presents in a self-contained form the typical basic properties of solutions to semilinear evolutionary partial differential equations, with special emphasis on global properties. It considers important examples, … Abstract This book presents in a self-contained form the typical basic properties of solutions to semilinear evolutionary partial differential equations, with special emphasis on global properties. It considers important examples, including the heat, Klein-Gordon, and Schro"odinger equations, placing each in the analytical framework which allows the most striking statement of the key properties. With the exceptions of the treatment of the Schro"odinger equation, the book employs the most standard methods, each developed in enough generality to cover other cases. This new edition includes a chapter on stability, which contains partial answers to recent questions about the global behavior of solutions. The self-contained treatment and emphasis on central concepts make this text useful to a wide range of applied mathematicians and theoretical researchers.

Geometric Theory of Semilinear Parabolic Equations

1981-01-01

Daniel Henry

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Input-to-state stability of infinite-dimensional control systems

2012-08-27

Sergey Dashkovskiy , Andrii Mironchenko

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Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space

2009-05-06

Björn S. Rüffer

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

2020-01-01

Andrii Mironchenko , Christophe Prieur

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Semigroups of Linear Operators and Applications to Partial Differential Equations

1983-01-01

A. Pazy

On continuity of solutions for parabolic control systems and input-to-state stability

2018-11-08

Birgit Jacob , Felix L. Schwenninger , Hans Zwart

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Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations

2018-08-27

Jun Zheng , Guchuan Zhu

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An Introduction to Infinite-Dimensional Linear Systems Theory

1995-01-01

Ruth F. Curtain , Hans Zwart

Non-coercive Lyapunov functions for infinite-dimensional systems

2018-11-28

Andrii Mironchenko , Fabian Wirth

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Input-to-State Stability of Nonlinear Impulsive Systems

2013-01-01

Sergey Dashkovskiy , Andrii Mironchenko

We prove that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the … We prove that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems if the ISS Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via the linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.

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Distributed control of spatially invariant systems

2002-07-01

Bassam Bamieh , Fernando Paganini , Munther A. Dahleh

We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the … We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), H/sub 2/ and H/sub /spl infin//, optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed.

A Small-Gain Theorem for a Wide Class of Feedback Systems with Control Applications

2007-01-01

Iasson Karafyllis , Zhong‐Ping Jiang

A small-gain theorem, which can be applied to a wide class of systems that includes systems satisfying the weak semigroup property, is presented in the present work. The result generalizes … A small-gain theorem, which can be applied to a wide class of systems that includes systems satisfying the weak semigroup property, is presented in the present work. The result generalizes all existing results in the literature and exploits notions of weighted, uniform, and nonuniform input-to-output stability properties. Applications to partial state feedback stabilization problems with sampled-data feedback applied with zero order hold and positive sampling rate are also presented.

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Admissibility of Unbounded Control Operators

1989-05-01

George H. Weiss

For linear systems described by $\dot x(t) = Ax(t) + Bu(t)$, where A generates a semigroup on the state space X and B is an unbounded operator, some necessary as … For linear systems described by $\dot x(t) = Ax(t) + Bu(t)$, where A generates a semigroup on the state space X and B is an unbounded operator, some necessary as well as some sufficient conditions are given for B to be admissible, i.e., for any t, the state $x(t)$ should be in X and should depend continuously on the input $u \in L^p $. This approach begins with an axiomatic description of such a system in terms of a functional equation. The results are applied to the wave equation on a bounded interval.

Uniform weak attractivity and criteria for practical global asymptotic stability

2017-06-12

Andrii Mironchenko

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String Stability and a Delay-Based Spacing Policy for Vehicle Platoons Subject to Disturbances

2017-03-16

Bart Besselink , Karl Henrik Johansson

A novel delay-based spacing policy for the control of vehicle platoons is introduced together with a notion of disturbance string stability. The delay-based spacing policy specifies the desired intervehicular distance … A novel delay-based spacing policy for the control of vehicle platoons is introduced together with a notion of disturbance string stability. The delay-based spacing policy specifies the desired intervehicular distance between vehicles and guarantees that all vehicles track the same spatially varying reference velocity profile, as is for example required for heavy-duty vehicles driving over hilly terrain. Disturbance string stability is a notion of string stability of vehicle platoons subject to external disturbances on all vehicles that guarantees that perturbations do not grow unbounded as they propagate through the platoon. Specifically, a control design approach in the spatial domain is presented that achieves tracking of the desired spacing policy and guarantees disturbance string stability with respect to a spatially varying reference velocity. The results are illustrated by means of simulations.

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One-parameter semigroups for linear evolution equations

2001-06-01

Klaus‐Jochen Engel , Rainer Nagel

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Input-to-state stability of interconnected hybrid systems

2013-02-26

Sergey Dashkovskiy , Michael Kosmykov

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Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization

2002-01-01

Lars Grüne

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Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances

2019-01-01

Andrii Mironchenko , Iasson Karafyllis , Miroslav Krstić

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Small-gain stability analysis of certain hyperbolic–parabolic PDE loops

2018-06-14

Iasson Karafyllis , Miroslav Krstić

Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

2019-03-14

Andrii Mironchenko , Fabian Wirth

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Input-to-State Stability for PDEs

2018-06-07

Iasson Karafyllis , Miroslav Krstić

This book studies input-to-state stability of parabolic and hyperbolic partial differential equations and equips the reader for many applications. In addition to stability results, the book develops existence and uniqueness … This book studies input-to-state stability of parabolic and hyperbolic partial differential equations and equips the reader for many applications. In addition to stability results, the book develops existence and uniqueness theory for all systems that are considered.

Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods

2012-03-29

Sergey Dashkovskiy , Michael Kosmykov , Andrii Mironchenko +1 more

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A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances

2018-11-16

Jun Zheng , Guchuan Zhu

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration … This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)

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Criteria for Input-to-State Practical Stability

2018-04-09

Andrii Mironchenko

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 systems … 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 systems with Lipschitz continuous flows and evolution equations in Banach spaces. The characterization of ISpS in terms of the limit property is novel already in the special case of ordinary differential equations.

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Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems

2017-01-01

Aneel Tanwani , Christophe Prieur , Sophie Tarbouriech

We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances … We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances in the measurement, the problem of designing dynamic controllers is considered so that the closed-loop system is robust with respect to measurement errors. Assuming that the disturbance is a locally essentially bounded measurable function of time, we derive a disturbance-to-state estimate which provides an upper bound on the maximum norm of the state (with respect to the spatial variable) at each time in terms of $\mathcal{L}^\infty$-norm of the disturbance up to that time. The analysis is based on constructing a Lyapunov function for the closed-loop system, which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived.

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Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One-Dimensional Semilinear Parabolic PDEs

2019-01-01

Iasson Karafyllis , Miroslav Krstić

This paper presents a novel methodology for the design of boundary feedback stabilizers for one-dimensional, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can … This paper presents a novel methodology for the design of boundary feedback stabilizers for one-dimensional, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.

Weak input-to-state stability: characterizations and counterexamples

2019-10-11

Jochen Schmid

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Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions

2016-08-01

Andrii Mironchenko , Hiroshi Ito

For bilinear infinite-dimensional dynamical systems,we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over … For bilinear infinite-dimensional dynamical systems,we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.

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Recovering a Function from a Dini Derivative

2006-01-01

John W. Hagood , Brian S. Thomson

provides a clear answer if we can assume that F' is Riemann integrable. Students of analysis will learn that if F' is Lebesgue integrable the same formula can be used, … provides a clear answer if we can assume that F' is Riemann integrable. Students of analysis will learn that if F' is Lebesgue integrable the same formula can be used, interpreting the integral in this more general sense. A full resolution of the problem requires a more general integral still, that of Denjoy and Perron (known frequently now as the Henstock-Kurzweil integral). The main question of this paper is, as it was for Lebesgue, whether a function can be recovered as an indefinite integral of one of its Dini derivatives?that is, when does the formula

ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances

2017-01-01

Iasson Karafyllis , Miroslav Krstić

For one-dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input-to-state stability (ISS) estimates in various norms. Due to the lack of an ISS … For one-dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input-to-state stability (ISS) estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.

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Contributions to Stability Theory

1956-07-01

J. L. Massera

A Lyapunov-Based Small-Gain Theorem for Infinite Networks

2020-12-03

Christoph Kawan , Andrii Mironchenko , Abdalla Swikir +2 more

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Noncoercive Lyapunov Functions for Input-to-State Stability of Infinite-Dimensional Systems

2020-01-01

Birgit Jacob , Andrii Mironchenko , J. R. Partington +1 more

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Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

2018-01-01

Birgit Jacob , Robert Nabiullin , J. R. Partington +1 more

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 … 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.

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A generalization of the nonlinear small-gain theorem for large-scale complex systems

2008-01-01

Zhong‐Ping Jiang , Yuan Wang

This paper presents a generalization of the nonlinear small-gain theorem for large-scale complex systems consisting of multiple input-to-output stable systems. It includes as a special case the previous nonlinear small-gain … This paper presents a generalization of the nonlinear small-gain theorem for large-scale complex systems consisting of multiple input-to-output stable systems. It includes as a special case the previous nonlinear small-gain theorems with two interconnected systems, and recent small-gain theorems for networks of input-to-state stable subsystems. It is expected that this new small-gain theorem will find wide applications in the analysis and control synthesis of large-scale complex systems.

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Classical converse theorems in Lyapunov's second method

2015-08-01

Christopher M. Kellett

Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a … Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.

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Nonconservative Discrete-Time ISS Small-Gain Conditions for Closed Sets

2017-08-02

Navid Noroozi , Roman Geiselhart , Lars Grüne +2 more

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and … This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of ωISS via finite-step ωISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee ωISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.

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Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces

2018-08-02

Andrii Mironchenko , Fabian Wirth

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Wave Equation With Cone-Bounded Control Laws

2016-01-19

Christophe Prieur , Sophie Tarbouriech , J.M. Gomes da Silva

This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection.Two kinds of control are considered: a distributed action and a boundary … This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection.Two kinds of control are considered: a distributed action and a boundary control.It is supposed that the control signal is subject to a cone-bounded nonlinearity.This kind of feedback laws includes (but is not restricted to) saturating inputs.By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation.The well-posedness is proven by using nonlinear semigroups techniques.Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closedloop system is proven by Lyapunov techniques.Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations.

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Vector-valued laplace transforms and cauchy problems

1987-10-01

Wolfgang Arendt

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Cones and Duality

2007-06-12

Charalambos D. Aliprantis , Rabee Tourky

Cones Cones in topological vector spaces Yudin and pull-back cones Krein operators $\mathcal{K}$-lattices The order extension of $L'$ Piecewise affine functions Appendix: Linear topologies Bibliography Index. Cones Cones in topological vector spaces Yudin and pull-back cones Krein operators $\mathcal{K}$-lattices The order extension of $L'$ Piecewise affine functions Appendix: Linear topologies Bibliography Index.