SU\G(𝔸)/K·U
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All published works (64)

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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Comparison Theorem for Infinite-Dimensional Linear Impulsive Systems

2023-12-13
Vladyslav Bivziuk , Sergey Dashkovskiy , В. И. Слынько
We consider a linear impulsive system in an infinite-dimensional Banach space. It is assumed that the moments of impulsive action satisfy the averaged dwell-time condition and the linear operator on … We consider a linear impulsive system in an infinite-dimensional Banach space. It is assumed that the moments of impulsive action satisfy the averaged dwell-time condition and the linear operator on the right side of the differential equation generates an analytic semigroup in the state space. Using commutator identities, we prove a comparison theorem that reduces the problem of asymptotic stability of the original system to the study of a simpler system with constant dwell-times. An illustrative example of a linear impulsive system of parabolic type in which the continuous and discrete dynamics are both unstable is given.
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Asymptotic analysis of optimal control problems on the semiaxes for Carathéodory differential inclusions with fast oscillating coefficients

2023-10-27
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Olena A. Kapustian +1 more
We consider an optimal control problem for a differential inclusion of the Carathéodory type affine with respect to the control with a coercive cost functional on a semiaxis and with … We consider an optimal control problem for a differential inclusion of the Carathéodory type affine with respect to the control with a coercive cost functional on a semiaxis and with fast oscillating time-dependent coefficients. We prove that, when the small parameter converges to zero, the solution to this problem tends to some solution of the optimal control problem with averaged coefficients, where the averaging we understand in the sense of the Kuratowski upper limit.
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Robust Stability of a Nonlinear ODE-PDE System

2023-06-22
Sergey Dashkovskiy , Oleksiy V. Kapustyan , В. И. Слынько
This work studies stability and robustness of a nonlinear system given as an interconnection of an ODE and a parabolic PDE subjected to external disturbances entering through the boundary conditions … This work studies stability and robustness of a nonlinear system given as an interconnection of an ODE and a parabolic PDE subjected to external disturbances entering through the boundary conditions of the parabolic equation. To this end we develop an approach for construction of a suitable coercive Lyapunov function as one of our main results. Based on this Lyapunov function, we establish well-posedness of the considered system and establish conditions that guarantee the input-to-state stable (ISS) property. ISS estimates are derived explicitly for the particular case of globally Lipschitz nonlinearities.

Lyapunov Functions for Linear Hyperbolic Systems

2023-02-22
Ivan Atamas , Sergey Dashkovskiy , В. И. Слынько
In this article, we develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for … In this article, we develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for which we give an example demonstrating that existing approaches cannot provide sufficient conditions for the asymptotic stability, but our approach does. Sufficient conditions for exponential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> -stability for a connected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2 \times 2$</tex-math></inline-formula> system of linear 1-D hyperbolic systems are obtained. By means of examples, we compare the capabilities of our approach with the existing ones.

Comparison theorem for infinite-dimensional linear impulsive systems

2023-01-01
Vladyslav Bivziuk , Sergey Dashkovskiy , В. И. Слынько
We consider a linear impulsive system in an infinite-dimensional Banach space. It is assumed that the moments of impulsive action satisfy the averaged dwell-time condition and the linear operator on … We consider a linear impulsive system in an infinite-dimensional Banach space. It is assumed that the moments of impulsive action satisfy the averaged dwell-time condition and the linear operator on the right side of the differential equation generates an analytic semigroup in the state space. Using commutator identities, we prove a comparison theorem that reduces the problem of asymptotic stability of the original system to the study of a simpler system with constant dwell-times. An illustrative example of a linear impulsive system of parabolic type in which the continuous and discrete dynamics are both unstable is given.
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Asymptotic stability conditions for linear coupled impulsive systems with time-invariant subsystems

2023-01-01
В. И. Слынько , Sergey Dashkovskiy , Ivan Atamas
This article proposes an approach to construct a Lyapunov function for a linear coupled impulsive system consisting of two time-invariant subsystems. In contrast to various variants of small-gain stability conditions … This article proposes an approach to construct a Lyapunov function for a linear coupled impulsive system consisting of two time-invariant subsystems. In contrast to various variants of small-gain stability conditions for coupled systems, the asymptotic stability property of independent subsystems is not assumed. To analyze the asymptotic stability of a coupled system, the direct Lyapunov method is used in combination with the discretization method. The periodic case and the case when the Floquet theory is not applicable are considered separately. The main results are illustrated with examples.
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Dwell-time stability conditions for infinite dimensional impulsive systems

2022-11-07
Sergey Dashkovskiy , В. И. Слынько
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Relative Stability in the Sup-Norm and Input-to-State Stability in the Spatial Sup-Norm for Parabolic PDEs

2022-07-19
Jun Zheng , Guchuan Zhu , Sergey Dashkovskiy
In this article, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relative <inline-formula><tex-math notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula>-equi-stability</i> (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic … In this article, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relative <inline-formula><tex-math notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula>-equi-stability</i> (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic partial differential equations (PDEs). Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a superlinear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this article.
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Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

2022-06-18
Sergey Dashkovskiy , V.A. Zaitsev

Robustness of global attractors: Abstract framework and application to dissipative wave equations

2021-10-14
Sergey Dashkovskiy , Oleksiy V. Kapustyan
&lt;p style='text-indent:20px;'&gt;We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply … &lt;p style='text-indent:20px;'&gt;We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of dissipative wave equations with nontrivial global attractors.&lt;/p&gt;
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Asymptotic gain results for attractors of semilinear systems

2021-09-08
Jochen Schmid , Oleksiy V. Kapustyan , Sergey Dashkovskiy
&lt;p style='text-indent:20px;'&gt;We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a … &lt;p style='text-indent:20px;'&gt;We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee–Infante equations, for example.&lt;/p&gt;
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Stability of uniform attractors of impulsive multi-valued semiflows

2021-02-27
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Yu. M. Perestyuk

Well-posedness and robust stability of a nonlinear ODE-PDE system

2021-01-01
Sergey Dashkovskiy , Oleksiy V. Kapustyan , В. И. Слынько
This work studies stability and robustness of a nonlinear system given as an interconnection of an ODE and a parabolic PDE subjected to external disturbances entering through the boundary conditions … This work studies stability and robustness of a nonlinear system given as an interconnection of an ODE and a parabolic PDE subjected to external disturbances entering through the boundary conditions of the parabolic equation. To this end we develop an approach for a construction of a suitable coercive Lyapunov function as one of the main results. Based on this Lyapunov function we establish the well-posedness of the considered system and establish conditions that guarantee the ISS property. ISS estimates are derived explicitly for the particular case of globally Lipschitz nonlinearities.
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Dwell-time stability conditions for infinite dimensional impulsive systems

2021-01-01
Sergey Dashkovskiy , В. И. Слынько
We consider nonlinear impulsive systems on Banach spaces subjected to disturbances and look for dwell-time conditions guaranteeing the the ISS property. In contrary to many existing results our conditions cover … We consider nonlinear impulsive systems on Banach spaces subjected to disturbances and look for dwell-time conditions guaranteeing the the ISS property. In contrary to many existing results our conditions cover the case where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov type methods are use for this purpose. The effectiveness of our approach is illustrated on a rather nontrivial example, which is feedback connection of an ODE and a PDE systems.
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Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input

2020-10-13
Ricardo Almeida , Snezhana Hristova , Sergey Dashkovskiy
Summary The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and … Summary The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.
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Uniform $\mathcal {K}$-continuity and ISS in $L^\infty$-norm of nonlinear parabolic PDEs

2020-08-05
Jun Zheng , Guchuan Zhu , Sergey Dashkovskiy
We introduce the notion \emph{uniform $\mathcal {K}$-continuity (UKC)} to describe uniformly continuous dependence of solutions on external disturbances for nonlinear parabolic PDEs. As an application of UKC, input-to-state stability (ISS) … We introduce the notion \emph{uniform $\mathcal {K}$-continuity (UKC)} to describe uniformly continuous dependence of solutions on external disturbances for nonlinear parabolic PDEs. As an application of UKC, input-to-state stability (ISS) estimates in $L^\infty$-norm are established for weak solutions of a class of nonlinear parabolic PDEs with mixed/ Dirichlet/ Robin boundary disturbances. The properties of UKC and ISS estimates are established by the De Giorgi iteration.
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A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations

2020-05-06
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Jochen Schmid
Abstract We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system. Abstract We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system.

Robustness properties of a large-amplitude, high-frequency extremum seeking control scheme

2020-01-01
Raik Suttner , Sergey Dashkovskiy
We analyze stability and robustness properties of an extremum seeking scheme that employs oscillatory dither signals with sufficiently large amplitudes and frequencies. Our study takes both input and output disturbances … We analyze stability and robustness properties of an extremum seeking scheme that employs oscillatory dither signals with sufficiently large amplitudes and frequencies. Our study takes both input and output disturbances into account. We consider general $L_\infty$-disturbances, which may resonate with the oscillatory dither signals. A suitable change of coordinates followed by an averaging procedure reveals that the closed-loop system approximates the behavior of an averaged system. This leads to the effect that stability and robustness properties carry over from the averaged system to the closed-loop system. In particular, we show that, if the averaged system is input-to-state stable (ISS), then the closed-loop system has ISS-like properties.
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Stability of Uniformly Attracting Sets for Impulsive-Perturbed Multi-Valued Semiflows

2020-01-01
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Yu. M. Perestyuk
In this paper we investigate stability of uniformly attracting sets for semiflows generated by impulsive infinite-dimensional dynamical systems without uniqueness. Obtained abstract results are applied to weakly nonlinear parabolic system, … In this paper we investigate stability of uniformly attracting sets for semiflows generated by impulsive infinite-dimensional dynamical systems without uniqueness. Obtained abstract results are applied to weakly nonlinear parabolic system, whose trajectories have jumps at moments of intersection with certain surface in the phase space.

The ISS property for a feedback connection of an ODE with a parabolic PDE

2020-01-01
Sergey Dashkovskiy , В. И. Слынько
We consider a linear non-autonomous ODE with an input signal, which is a solution of a linear non-autonomous parabolic PDE for which the solution of the ODE enters as an … We consider a linear non-autonomous ODE with an input signal, which is a solution of a linear non-autonomous parabolic PDE for which the solution of the ODE enters as an input. Moreover there are external disturbances entering through the boundary conditions of the parabolic equation. In this work we derive ISS estimates for this feedback connection under certain linear matrix inequality conditions. The derivation of results is based on the Lyapunov method.

Robust stability of a perturbed nonlinear wave equation

2020-01-01
Sergey Dashkovskiy , В. И. Слынько
In this work we consider a nonlinear wave equation subject to both distributed as well as boundary perturbations and derive several ISS-like estimates for solutions for such equations by means … In this work we consider a nonlinear wave equation subject to both distributed as well as boundary perturbations and derive several ISS-like estimates for solutions for such equations by means of Lyaponov and Faedo-Galerkin methods. Depending on the regularity of the boundary input signals different types of estimates are derived.

Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs

2020-01-01
Jun Zheng , Guchuan Zhu , Sergey Dashkovskiy
In this paper, we introduce the notion of relative $\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. Based on … In this paper, we introduce the notion of relative $\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a $1$-D parabolic PDE with a destabilizing term, are provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this paper.
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Approximation of Solutions to the Optimal Control Problems for Systems with Maximum

2019-10-19
Sergey Dashkovskiy , О. Д. Кичмаренко , Kateryna Yu. Sapozhnikova
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Well-posedness of non-autonomous semilinear systems

2019-01-01
Jochen Schmid , Sergey Dashkovskiy , Birgit Jacob +1 more
Abstract We present well-posedness - along with global stability - results for non-autonomous semilinear input-output systems, the central assumption being that the considered system is scattering-passive. We consider both systems … Abstract We present well-posedness - along with global stability - results for non-autonomous semilinear input-output systems, the central assumption being that the considered system is scattering-passive. We consider both systems with distributed control and observation and systems with boundary control and observation.

A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations

2019-01-01
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Jochen Schmid
We establish the local input-to-state stability of a large class of disturbed nonlinear reaction-diffusion equations w.r.t. the global attractor of the respective undisturbed system. We establish the local input-to-state stability of a large class of disturbed nonlinear reaction-diffusion equations w.r.t. the global attractor of the respective undisturbed system.
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Asymptotic gain results for attractors of semilinear systems

2019-01-01
Jochen Schmid , Oleksiy V. Kapustyan , Sergey Dashkovskiy
We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large … We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee--Infante equations, for example.
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Existence and Invariance of Global Attractors for Impulsive Parabolic System Without Uniqueness

2018-11-29
Sergey Dashkovskiy , Petro Feketa , Oleksiy V. Kapustyan +1 more
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Asymptotic properties of Zeno solutions

2018-07-06
Sergey Dashkovskiy , Petro Feketa

Invariance and stability of global attractors for multi-valued impulsive dynamical systems

2017-09-06
Sergey Dashkovskiy , Petro Feketa , Oleksiy V. Kapustyan +1 more

Behavior of solutions to systems with maximum

2017-07-01
Sergey Dashkovskiy , Snezhana Hristova , О. Д. Кичмаренко +1 more
In this work we consider dynamics of systems given by differential equations in which the unknown function depends on its maximal value over a prehistory time interval and on some … In this work we consider dynamics of systems given by differential equations in which the unknown function depends on its maximal value over a prehistory time interval and on some input signal. Such systems, called systems with maximum are a special subclass of systems with time delays. In this work we are interested in stability properties of the solution with respect to the external signals. The input-to-state stability is used for this purpose.

Almost ISS property for feedback connected systems

2017-03-06
Petro Feketa , Humberto Stein Shiromoto , Sergey Dashkovskiy
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Global attractors of impulsive parabolic inclusions

2017-01-01
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Iryna V. Romaniuk
In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ and with impulsive multi-valued perturbations. Moments of impulses … In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter $\varepsilon>0$ this system has a global attractor.
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NUMERICAL SOLUTION TO INITIAL VALUE PROBLEM FOR ONE CLASS OF DIFFERENTIAL EQUATION WITH MAXIMUM

2016-10-09
Sergey Dashkovskiy , О. Д. Кичмаренко , Kateryna Yu. Sapozhnikova +1 more
This paper presents a numerical method for solution computation for one type of initial value problems given by differential equations with maximum of the unknown function over a prehistory.We prove … This paper presents a numerical method for solution computation for one type of initial value problems given by differential equations with maximum of the unknown function over a prehistory.We prove the convergence of the method and illustrate it by several examples.
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Almost ISS property for feedback connected systems

2016-10-07
Petro Feketa , Humberto Stein Shiromoto , Sergey Dashkovskiy
Small-gain conditions used in analysis of feedback interconnections are contraction conditions which imply certain stability properties. Such conditions are applied to a finite or infinite interval. In this paper we … Small-gain conditions used in analysis of feedback interconnections are contraction conditions which imply certain stability properties. Such conditions are applied to a finite or infinite interval. In this paper we consider the case, when a small-gain condition is applied to several disjunct intervals and use the density propagation condition in the gaps between these intervals to derive global stability properties for an interconnection. This extends and improves recent results from [1].
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Input-to-state Stability of Impulsive Systems with Different Jump Maps

2016-05-27
Sergey Dashkovskiy , Petro Feketa
The paper introduces sufficient conditions for input-to-state stability (ISS) of a class of impulsive systems with jump maps that depend on time. Such systems can naturally represent an interconnection of … The paper introduces sufficient conditions for input-to-state stability (ISS) of a class of impulsive systems with jump maps that depend on time. Such systems can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Using a concept of ISS-Lyapunov function for subsystems a small-gain type theorem equipped with a new dwell-time condition to verify ISS of an interconnection has been proven.
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Example Demonstrating the Application of Small-gain and Density Propagation Conditions for Interconnections

2016-01-01
Humberto Stein Shiromoto , Petro Feketa , Sergey Dashkovskiy
This work provides an example that motivates and illustrates theoretical results related to a combination of small-gain and density propagation conditions. Namely, in case the small-gain fails to hold at … This work provides an example that motivates and illustrates theoretical results related to a combination of small-gain and density propagation conditions. Namely, in case the small-gain fails to hold at certain points or intervals the density propagation condition can be applied to assure global stability properties. We repeat the theoretical results here and demonstrate how they can be applied in the proposed example.
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Input-to-state Stability of Impulsive Systems with Different Jump Maps

2016-01-01
Sergey Dashkovskiy , Petro Feketa
The paper introduces sufficient conditions for input-to-state stability (ISS) of a class of impulsive systems with jump maps that depend on time. Such systems can naturally represent an interconnection of … The paper introduces sufficient conditions for input-to-state stability (ISS) of a class of impulsive systems with jump maps that depend on time. Such systems can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Using a concept of ISS-Lyapunov function for subsystems a small-gain type theorem equipped with a new dwell-time condition to verify ISS of an interconnection has been proven.
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Almost ISS property for feedback connected systems

2016-01-01
Petro Feketa , Humberto Stein Shiromoto , Sergey Dashkovskiy
Small-gain conditions used in analysis of feedback interconnections are contraction conditions which imply certain stability properties. Such conditions are applied to a finite or infinite interval. In this paper we … Small-gain conditions used in analysis of feedback interconnections are contraction conditions which imply certain stability properties. Such conditions are applied to a finite or infinite interval. In this paper we consider the case, when a small-gain condition is applied to several disjunct intervals and use the density propagation condition in the gaps between these intervals to derive global stability properties for an interconnection. This extends and improves recent results from [1].
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The life after Zeno

2015-07-06
Sergey Dashkovskiy , Petro Feketa
The paper proposes a unified framework for the construction of solutions to a hybrid dynamical system that exhibit Zeno behavior. A new approach that enables solution to be prolonged after … The paper proposes a unified framework for the construction of solutions to a hybrid dynamical system that exhibit Zeno behavior. A new approach that enables solution to be prolonged after reaching its Zeno time is developed. It allows a comprehensive stability analysis and long-term asymptotic behavior characterization of solutions. The results are applicable to a wide class of hybrid systems and match with practical experience of simulation of real-world phenomena.
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A combination of small-gain and density propagation inequalities for stability analysis of networked systems

2015-01-01
Humberto Stein Shiromoto , Petro Feketa , Sergey Dashkovskiy
In this paper, the problem of stability analysis of a large-scale interconnection of nonlinear systems for which the small-gain condition does not hold globally is considered. A combination of the … In this paper, the problem of stability analysis of a large-scale interconnection of nonlinear systems for which the small-gain condition does not hold globally is considered. A combination of the small-gain and density propagation inequalities is employed to prove almost input-to-state stability of the network.
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Prolongation and stability of Zeno solutions to hybrid dynamical systems

2015-01-01
Sergey Dashkovskiy , Petro Feketa
The paper proposes a framework for the construction of solutions to a hybrid dynamical system that exhibit Zeno behavior. A new approach that enables solution to be prolonged after reaching … The paper proposes a framework for the construction of solutions to a hybrid dynamical system that exhibit Zeno behavior. A new approach that enables solution to be prolonged after reaching its Zeno time is developed. It allows for a comprehensive stability analysis and asymptotic behavior characterization of such solutions. The results are applicable to a wide class of hybrid systems and match with practical experience of simulation of real-world phenomena. Moreover they are potentially useful for applications to interconnections of hybrid systems.
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Reduction of the small gain condition for large‐scale interconnections

2013-11-28
Sergey Dashkovskiy , Michael Kosmykov
Summary The small gain condition is sufficient for input‐to‐state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of … Summary The small gain condition is sufficient for input‐to‐state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of a large size of the system. To facilitate this procedure, we aggregate the subsystems and the gains between the subsystems that belong to certain interconnection patterns (motifs) by using three heuristic rules. These rules are based on three motifs: sequentially connected nodes, nodes connected in parallel, and almost disconnected subgraphs. Aggregation of these motifs keeps the structure of the mutual influences between the subsystems in the network. Furthermore, fulfillment of the reduced small gain condition implies ISS of the large network. Thus, such reduction allows to decrease the number of computations needed to verify the small gain condition. Finally, an ISS‐Lyapunov function for the large network can be constructed using the reduced small gain condition. Applications of these rules is illustrated on an example. Copyright © 2013 John Wiley &amp; Sons, Ltd.
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Input-to-state stability of interconnected hybrid systems

2013-02-26
Sergey Dashkovskiy , Michael Kosmykov
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Stabilization of generalized triangular form systems with dynamic uncertainties by means of small gain theorems

2013-01-01
Sergey Dashkovskiy , Svyatoslav Pavlichkov , Zhong‐Ping Jiang

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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Input-to-state stability of nonlinear impulsive systems

2012-12-21
Sergey Dashkovskiy , Andrii Mironchenko
We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, … We prove that impulsive systems, which possess an 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 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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Input-to-state stability of infinite-dimensional control systems

2012-08-27
Sergey Dashkovskiy , Andrii Mironchenko
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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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Dwell-time conditions for robust stability of impulsive systems

2012-02-15
Sergey Dashkovskiy , Andrii Mironchenko
We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for impulse time sequences, which satisfy the fixed dwell-time condition. If the ISS Lyapunov function is the … We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for impulse time sequences, which satisfy the fixed dwell-time condition. If the ISS Lyapunov function is the exponential one, we provide stronger result, which guarantees uniform ISS of the whole system over sequences of impulse times, which satisfy the generalized average dwell-time condition.
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Common Coauthors

Coauthor Papers Together
Oleksiy V. Kapustyan 13
Petro Feketa 12
Michael Kosmykov 12
В. И. Слынько 10
Andrii Mironchenko 10
Fabian Wirth 6
Humberto Stein Shiromoto 5
Iryna V. Romaniuk 3
О. Д. Кичмаренко 3
Jochen Schmid 3
Kateryna Yu. Sapozhnikova 3
Lars Naujok 3
Guchuan Zhu 3
Svyatoslav Pavlichkov 2
Yu. M. Perestyuk 2
Snezhana Hristova 2
Björn S. Rüffer 2
Vladyslav Bivziuk 2
Ivan Atamas 2
Jochen Schmid 2
Jun Zheng 2
Tetyana Zhuk 1
V.A. Zaitsev 1
Jun Zheng 1
Patrick Bachmann 1
A. N. Vityuk 1
Ricardo Almeida 1
Birgit Jacob 1
Raik Suttner 1
Hafida Laasri 1
Olena A. Kapustian 1
Zhong‐Ping Jiang 1

Commonly Cited References

An ISS small gain theorem for general networks

2007-05-01
Sergey Dashkovskiy , Björn S. Rüffer , Fabian Wirth
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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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Attractors for Equations of Mathematical Physics

2001-11-13
Vladimir V. Chepyzhov , Mark Vishik
Introduction Attractors of autonomous equations: Attractors of autonomous ordinary differential equations Attractors of autonomous partial differential equations Dimension of attractors Attractors of non-autonomous equations: Processes and attractors Translation compact functions … Introduction Attractors of autonomous equations: Attractors of autonomous ordinary differential equations Attractors of autonomous partial differential equations Dimension of attractors Attractors of non-autonomous equations: Processes and attractors Translation compact functions Attractors of non-autonomous partial differential equations Semiprocesses and attractors Kernels of processes Kolmogorov $\varepsilon$-entropy of attractors Trajectory attractors: Trajectory attractors of autonomous ordinary differential equations Attractors in Hausdorff spaces Trajectory attractors of autonomous equations Trajectory attractors of autonomous partial differential equations Trajectory attractors of non-autonomous equations Trajectory attractors of non-autonomous partial differential equations Approximation of trajectory attractors Perturbation of trajectory attractors Averaging of attractors of evolution equations with rapidly oscillating terms Proofs of Theorems II.1.4 and II.1.5 Lattices and coverings Bibliography Index.

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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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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Global attractors of impulsive parabolic inclusions

2017-01-01
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Iryna V. Romaniuk
In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ and with impulsive multi-valued perturbations. Moments of impulses … In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter $\varepsilon>0$ this system has a global attractor.
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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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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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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

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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Flows of characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:math> in impulsive semidynamical systems

2006-11-08
E.M. Bonotto

Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization

2002-01-01
Lars Grüne
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On Stability in Impulsive Dynamical Systems

2004-01-01
Krzysztof Ciesielski
Several results on stability in impulsive dynamical systems are proved. The first main result gives equivalent conditions for stability of a compact set. In particular, a generalization of Ura's theorem … Several results on stability in impulsive dynamical systems are proved. The first main result gives equivalent conditions for stability of a compact set. In particular, a generalization of Ura's theorem to the case of impulsive systems is shown. The secon
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Input-to-state stability of interconnected hybrid systems

2013-02-26
Sergey Dashkovskiy , Michael Kosmykov
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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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On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity

2013-11-25
Nataliia V. Gorban , Oleksiy V. Kapustyan , Pavlo O. Kasyanov +1 more

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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Stability and asymptotic stability in impulsive semidynamical systems

1994-01-01
Saroop K. Kaul
In this paper we generalize two results of Lasalle′s, the invariance theorem and asymptotic stability theorem of discrete and continuous semidynamical systems, to impulsive semidynamical systems. In this paper we generalize two results of Lasalle′s, the invariance theorem and asymptotic stability theorem of discrete and continuous semidynamical systems, to impulsive semidynamical systems.
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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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Attractors of impulsive dissipative semidynamical systems

2012-12-19
E.M. Bonotto , Daniela Paula Demuner

Global attractors for impulsive dynamical systems – a precompact approach

2015-04-15
E.M. Bonotto , Matheus C. Bortolan , Alexandre N. Carvalho +1 more

Partial Differential Equations

1988-01-01
Shiing-Shen Chern
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A local input-to-state stability result w.r.t. attractors of nonlinear reaction–diffusion equations

2020-05-06
Sergey Dashkovskiy , Oleksiy V. Kapustyan , Jochen Schmid
Abstract We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system. Abstract We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system.

A Vector Small-Gain Theorem for General Nonlinear Control Systems

2009-01-01
Iasson Karafyllis , Zhong‐Ping Jiang
A new Small-Gain Theorem is presented for general nonlinear control systems. The novelty of this research work is that vector Lyapunov functions and functionals are utilized to derive various input-to-output … A new Small-Gain Theorem is presented for general nonlinear control systems. 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 recovers several recent results as special instances and is extendible to several important classes of control systems such as large-scale complex systems, nonlinear sampled-data systems and nonlinear 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 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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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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A region-dependent gain condition for asymptotic stability

2015-01-08
Humberto Stein Shiromoto , Vincent Andrieu , Christophe Prieur
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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.

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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Global Attractors in Impulsive Infinite-Dimensional Systems

2016-09-01
Oleksiy V. Kapustyan , М. О. Perestyuk

Semicontinuity of attractors for impulsive dynamical systems

2016-07-03
E.M. Bonotto , Matheus C. Bortolan , Rodolfo Collegari +1 more

Impulsive Differential Equations

1995-08-01
А. М. Самойленко , N. A. Perestyuk
General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of … General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of oscillations in impulsive systems a periodic and almost periodic impulsive system.

Solutions to hybrid inclusions via set and graphical convergence with stability theory applications

2006-02-21
Rafal Goebel , Andrew R. Teel

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

2012-01-18
Christophe Prieur , Frédéric Mazenc
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Erratum to: Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations

1998-04-01
J. M. Ball
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Input-to-state stability for parabolic boundary control:linear and semilinear systems

2020-01-01
Felix L. Schwenninger
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Is there life after Zeno? Taking executions past the breaking (Zeno) point

2006-01-01
Aaron D. Ames , Haiyang Zheng , Robert D. Gregg +1 more
Understanding Zeno phenomena plays an important role in understanding hybrid systems. A natural - and intriguing - question to ask is: what happens after a Zeno point? Inspired by the … Understanding Zeno phenomena plays an important role in understanding hybrid systems. A natural - and intriguing - question to ask is: what happens after a Zeno point? Inspired by the construction of Filippov (1988), we propose a method for extending Zeno executions past a Zeno point for a class of hybrid systems: Lagrangian hybrid systems. We argue that after the Zeno point is reached, the hybrid system should switch to a holonomically constrained dynamical system, where the holonomic constraints are based on the unilateral constraints on the configuration space that originally defined the hybrid system. These principles are substantiated with a series of examples.
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Weak input-to-state stability: characterizations and counterexamples

2019-10-11
Jochen Schmid
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Asymptotic properties of Zeno solutions

2018-07-06
Sergey Dashkovskiy , Petro Feketa

Semigroups of Linear Operators and Applications to Partial Differential Equations

1983-01-01
A. Pazy

Input-to-state stability of unbounded bilinear control systems

2018-11-20
René Hosfeld , Birgit Jacob , Felix L. Schwenninger
We study input-to-state stability of bilinear control systems with possibly unbounded control operators. Natural sufficient conditions for integral input-to-state stability are given. The obtained results are applied to a bilinearly … We study input-to-state stability of bilinear control systems with possibly unbounded control operators. Natural sufficient conditions for integral input-to-state stability are given. The obtained results are applied to a bilinearly controlled Fokker-Planck equation.
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On the exponential solution of differential equations for a linear operator

1954-11-01
Wilhelm Magnus

Geometric Theory of Semilinear Parabolic Equations

1981-01-01
Daniel Henry
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Numerical construction of LISS Lyapunov functions under a small-gain condition

2012-03-12
Roman Geiselhart , Fabian Wirth
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Pullback Attractors for a Class of Extremal Solutions of the 3D Navier–Stokes System

2012-01-01
Michael Z. Zgurovsky , Pavlo O. Kasyanov , Oleksiy V. Kapustyan +2 more

Asymptotic behaviour of reaction–diffusion equations with non-damped impulsive effects

2007-02-12
Gerardo Iovane , Oleksiy V. Kapustyan , José Valero

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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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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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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