The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear …
The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear systems theory, such as robust stabilization of nonlinear systems, design of nonlinear observers, analysis of large-scale networks, etc. The success of the ISS theory of ODEs and the need for robust stability analysis of partial differential equations (PDEs) motivated the development of ISS theory in the infinite-dimensional setting. For instance, the Lyapunov method for analysis of iISS of nonlinear parabolic equations. For an overview of the ISS theory for distributed parameter systems. ISS of control systems with application to robust global stabilization of the chemostat has been studied with the help of vector Lyapunov functions.
In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class …
In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite-dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS-LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS-LF for a class of abstract infinite-dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in-domain disturbances. Since the nonlinear terms of the corresponding abstract system are not Lipschitz continuous, the well-posedness is proved based on the application of compact semigroup theory and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.
ABSTRACT In this paper, we extend the notion of finite‐time input‐to‐state stability (FTISS) for finite‐dimensional systems to infinite‐dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a …
ABSTRACT In this paper, we extend the notion of finite‐time input‐to‐state stability (FTISS) for finite‐dimensional systems to infinite‐dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite‐dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS‐LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS‐LF for a class of abstract infinite‐dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in‐domain disturbances. Since the non‐linear terms of the corresponding abstract system are not Lipschitz continuous, the well‐posedness is proved based on the application of compact semigroup theory, and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.
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), 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 non-coercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for 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.
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.
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), 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 non-coercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for 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.
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.
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.
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.
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.
We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as …
We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral input-to-state stable under a certain smoothness assumption on nonlinear 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. …
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.
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.
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.
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
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the …
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "norm-estimators," and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the …
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of ``norm-estimators'', and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.
This paper introduces the extension of Finite-Time Stability (FTS) to the input-output case, namely the Input-Output FTS (IO-FTS). The main differences between classic IO stability and IO-FTS are that the …
This paper introduces the extension of Finite-Time Stability (FTS) to the input-output case, namely the Input-Output FTS (IO-FTS). The main differences between classic IO stability and IO-FTS are that the latter involves signals defined over a finite time interval, does not necessarily require the inputs and outputs to belong to the same class of signals, and that quantitative bounds on both inputs and outputs must be specified. This paper revises some recent results on IO-FTS, both in the context of linear systems and in the context of switching systems. In the final example the proposed methodology is used to minimize the maximum displacement and velocity of a building subject to an earthquake of given magnitude.
У цій статті ми розглядаємо стійкість граничних режимів для загального класу нелінійних розподілених математичних моделей, які називаються моделями реакції-дифузії. Системи реакції-дифузії природно виникають у багатьох застосуваннях. Наприклад, при математичному моделюванні …
У цій статті ми розглядаємо стійкість граничних режимів для загального класу нелінійних розподілених математичних моделей, які називаються моделями реакції-дифузії. Системи реакції-дифузії природно виникають у багатьох застосуваннях. Наприклад, при математичному моделюванні в біології та у теорії передачі сигналів широко використовується модель ФітцХью–Нагумо (FitzHugh–Nagumo model), розподілений варіант якої є окремим випадком загальної системи реакції-дифузії. Досліджено проблему стійкості притягуючих множин для нескінченновимірної системи реакції-дифузії відносно обмежених зовнішніх сигналів (збурень). Функції взаємодії, а також нелінійні збурення не вважаються неперервними за Ліпшицем. Отже, ми не можемо очікувати єдиності розв’язку для відповідної початкової задачі, і ми повинні використовувати багатозначний напівгруповий підхід. Вважається, що незбурена система має глобальний атрактор, тобто мінімальну компактну рівномірно притягаючу множину. Основною метою дослідження є оцінка відхилення траєкторії збуреної системи від глобального атрактора незбуреної як функції величини зовнішніх сигналів. Таку оцінку можна отримати в рамках теорії стійкості входу до стану (ISS). У статті запропоновано новий підхід до отримання оцінок робастної стійкості атрактора у випадку багатозначного еволюційного оператора. Зокрема, доведено, що багатозначна напівгрупа, породжена слабкими розв’язками нелінійної системи типу реакції-дифузії, має властивість локальної ISS відносно атрактора незбуреної системи.
In this article, we examine a particular family of infinite-dimensional discrete autonomous systems given by a first-order state-space equation; the state transition matrix for this family is a Laurent polynomial …
In this article, we examine a particular family of infinite-dimensional discrete autonomous systems given by a first-order state-space equation; the state transition matrix for this family is a Laurent polynomial matrix A(σ, σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ), where σ is the shift operator on R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -valued sequences. We term this family of systems as Laurent systems. We give necessary and sufficient conditions for the exponential ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -stability and the exponential ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> -stability of Laurent systems. We also compare the following four different notions of stability for Laurent systems: the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -stability, the exponential ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -stability, the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> -stability, and the exponential ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> -stability; furthermore, we conclude that the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -stability is an outlier.
Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic …
Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic scheduling. In this paper, we study the problem of estimating their dynamically varying probability density, given the agents' individual dynamics (which can be nonlinear and time-varying) and their states observed in real-time. The density evolution is shown to satisfy a linear partial differential equation uniquely determined by the agents' dynamics. We present a density filter which takes advantage of the system dynamics to gradually improve its estimation and is scalable to the agents' population. Specifically, we use kernel density estimators (KDE) to construct a noisy measurement and show that, when the agents' population is large, the measurement noise is approximately ``Gaussian''. With this important property, infinite-dimensional Kalman filters are used to design density filters. It turns out that the covariance of measurement noise depends on the true density. This state-dependence makes it necessary to approximate the covariance in the associated operator Riccati equation, rendering the density filter suboptimal. The notion of input-to-state stability is used to prove that the performance of the suboptimal density filter remains close to the optimal one. Simulation results suggest that the proposed density filter is able to quickly recognize the underlying modes of the unknown density and automatically ignore outliers, and is robust to different choices of kernel bandwidth of KDE.
In this chapter, dynamic systemsDynamic system and solutionSpace- Hilbert concepts are reviewed in the infinite-dimensional setting. After brief introduction of linear PDEs of parabolic, elliptic, and hyperbolic types, strongSolution- strong, …
In this chapter, dynamic systemsDynamic system and solutionSpace- Hilbert concepts are reviewed in the infinite-dimensional setting. After brief introduction of linear PDEs of parabolic, elliptic, and hyperbolic types, strongSolution- strong, mildSolution- mild, and weak solutions are addressed for these equations. Mathematical tools, presented for their analysis, involve Sturm–Liouville operators and their properties as well as the powerful method of separation of variables. ViscositySolution- viscosity and proximal solutionsSolution- proximal are additionally revisited for nonlinear first-order partial differential inequalities with discontinuous terms. Finally, modern stability paradigms such as ISSStability- ISS and finite timeStability- finite time stability among others are recalled with special attention to sliding mode dynamics in Hilbert spaceSpace- Hilbert and to homogeneous differential inclusions.
Railway tracks rest on a foundation known for exhibiting nonlinear viscoelastic behavior. Railway track deflections are modeled by a semilinear partial differential equation. This paper studies the stability of solutions …
Railway tracks rest on a foundation known for exhibiting nonlinear viscoelastic behavior. Railway track deflections are modeled by a semilinear partial differential equation. This paper studies the stability of solutions to this equation in presence of an input. With the aid of a suitable Lyapunov function, existence and exponential stability of classical solutions is established for certain inputs. The Lyapunov function is further used to find an a-priori estimate of the solutions, and also to study the input-to-state stability (ISS) of mild solutions.
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.
<p style='text-indent:20px;'>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 …
<p style='text-indent:20px;'>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.</p>
This paper investigates input-output properties of systems described by partial differential equations (PDEs). Analogous to systems described by ordinary differential equations (ODEs), dissipation inequalities are used to establish input-output properties …
This paper investigates input-output properties of systems described by partial differential equations (PDEs). Analogous to systems described by ordinary differential equations (ODEs), dissipation inequalities are used to establish input-output properties for PDE systems. Dissipation inequalities pertaining to passivity, induced L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -norm, reachability, and input-to-state stability (ISS) are formulated. For PDE systems with polynomial data, the dissipation inequalities are solved via polynomial optimization. The results are illustrated with an example.
For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary …
For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide 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.
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.
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.
This work studies how to estimate the mean-field density of large-scale systems in a distributed manner. Such problems are motivated by the recent swarm control technique that uses mean-field approximations …
This work studies how to estimate the mean-field density of large-scale systems in a distributed manner. Such problems are motivated by the recent swarm control technique that uses mean-field approximations to represent the collective effect of the swarm, wherein the mean-field density (especially its gradient) is usually used in feedback control design. In the first part, we formulate the density estimation problem as a filtering problem of the associated mean-field partial differential equation (PDE), for which we employ kernel density estimation (KDE) to construct noisy observations and use filtering theory of PDE systems to design an optimal (centralized) density filter. It turns out that the covariance operator of observation noise depends on the unknown density. Hence, we use approximations for the covariance operator to obtain a suboptimal density filter, and prove that both the density estimates and their gradient are convergent and remain close to the optimal one using the notion of input-to-state stability (ISS). In the second part, we continue to study how to decentralize the density filter such that each agent can estimate the mean-field density based on only its own position and local information exchange with neighbors. We prove that the local density filter is also convergent and remains close to the centralized one in the sense of ISS. Simulation results suggest that the centralized suboptimal density filter is able to generate convergent density estimates, and the local density filter is able to converge and remain close to the centralized filter.
— This paper considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic Partial Differential Equations (PDEs) of de Saint-Venant. The nonlinear system …
— This paper considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic Partial Differential Equations (PDEs) of de Saint-Venant. The nonlinear system dynamic is formulated by a Multi-Models approach over a wide operating range, where each local model is defined around a set of operating regimes. A Proportional Integral (PI) feedback was designed and performed through Bilinear Operator Inequality (BOI) and Linear Operator Inequality (LOI) techniques for infinite dimensional systems. The authors propose in this paper to improve the numerical part by introducing weight µi not only equal to {0,1}, but µi ∈ [0, 1].
In this work the 1-D heat equation with Dirichlet boundary conditions and disturbances at both boundaries is studied. It is shown that it is possible to obtain ISS estimates in …
In this work the 1-D heat equation with Dirichlet boundary conditions and disturbances at both boundaries is studied. It is shown that it is possible to obtain 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 obtained estimates of the gains of the disturbances are exact, in the sense that constant disturbances lead to equilibrium profiles for which the norms are exactly equal to the estimated gains. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation.
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.
In this work, decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the …
In this work, decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the L 2 and H 1 norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm–Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.
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.
We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive systems with inputs the well-established philosophy of …
We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive systems with inputs the well-established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the Lyapunov function on the system trajectories. This reduction is performed in such a way so that the resulting scalar system has no inputs. Novel sufficient conditions for ISS are provided, which generalize existing results for time-invariant and time-varying, switched and nonswitched, impulsive and nonimpulsive systems in several directions.
The paper investigates the issue of stability with respect to external disturbances for the global attractor of the wave equation under conditions that do not ensure the uniqueness of the …
The paper investigates the issue of stability with respect to external disturbances for the global attractor of the wave equation under conditions that do not ensure the uniqueness of the solution to the initial problem. Under general conditions for nonlinear terms, it is proved that the global attractor of the undisturbed problem is locally stable in the sense of ISS and has the AG property with respect to disturbances.
This note establishes the Exponential Input-to-State Stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed …
This note establishes the Exponential Input-to-State Stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.
We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption …
We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditions in light of existing results in the literature.
This work studies distributed (probability) density estimation of large-scale systems. Such problems are motivated by many density-based distributed control tasks in which the real-time density of the swarm is used …
This work studies distributed (probability) density estimation of large-scale systems. Such problems are motivated by many density-based distributed control tasks in which the real-time density of the swarm is used as feedback information, such as sensor deployment and city traffic scheduling. This work is built upon our previous work [1] which presented a (centralized) density filter to estimate the dynamic density of large-scale systems through a novel integration of mean-field models, kernel density estimation (KDE), and infinite-dimensional Kalman filters. In this work, we further study how to decentralize the density filter such that each agent can estimate the global density only based on its local observation and communication with neighbors. This is achieved by noting that the global observation constructed by KDE is an average of the local kernels. Hence, dynamic average consensus algorithms are used for each agent to track the global observation in a distributed way. We present a distributed density filter which requires very little information exchange, and study its stability and optimality using the notion of input-to-state stability. Simulation results suggest that the distributed filter is able to converge to the centralized filter and remain close to it.
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.
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.
This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates …
This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed by using an ISS Lyapunov Functional for the sup norm. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.
Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: …
Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.
Abstract Looking for the best possible smoothness (in terms of the upper index of the Besov spaces) for the solution of some semi–linear equations we consider a model case of …
Abstract Looking for the best possible smoothness (in terms of the upper index of the Besov spaces) for the solution of some semi–linear equations we consider a model case of a hypoelliptic operator, which acts between anisotropic Besov spaces. To obtain the best regularity we need some properties for the corresponding spaces, which we prove here. In particular we prove Fatou, Fubini and truncation properties. We give also some characterisations of the Besov and Triebel–Lizorkin spaces.
New trajectory-based small-gain results are obtained for nonlinear feedback systems under relaxed assumptions. Specifically, during a transient period, the solutions of the feedback system may not satisfy some key inequalities …
New trajectory-based small-gain results are obtained for nonlinear feedback systems under relaxed assumptions. Specifically, during a transient period, the solutions of the feedback system may not satisfy some key inequalities that previous small-gain results usually utilize to prove stability properties. The results allow the application of the small-gain perspective to various systems which satisfy less demanding stability notions than the Input-to-Output Stability property. The robust global feedback stabilization problem of an uncertain time-delayed chemostat model is solved by means of the trajectory-based small-gain results.
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.
This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also …
This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also established through the use of Lyapunov functionals.
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.
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.
This paper extends the nonlinear ISS small-gain theorem to a large-scale time delay system composed of three or more subsystems. En route to proving this small-gain theorem for systems of …
This paper extends the nonlinear ISS small-gain theorem to a large-scale time delay system composed of three or more subsystems. En route to proving this small-gain theorem for systems of differential equations with delays, a small-gain theorem for operators is examined. The result developed for operators allows applications to a wide class of systems, including state space systems with delays.
Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to …
Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.
In this work, small-gain theorems for large-scale time-delay systems are developed using a Razumikhin-type approach, both in terms of ISS and IOS. In the spirit of the Razumikhin theorem, the …
In this work, small-gain theorems for large-scale time-delay systems are developed using a Razumikhin-type approach, both in terms of ISS and IOS. In the spirit of the Razumikhin theorem, the state variables with delays are treated as disturbances to the system. This approach converts the problem of stability analysis for functional differential equations to the more tractable problem of stability analysis for delay-free systems with disturbances.
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.