Averaging of functional differential inclusions in Banach spaces

Donchev, Tzanko; Grammel, G.

doi:10.1016/j.jmaa.2005.02.050

Cited by 8 publications

(8 citation statements)

References 13 publications

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“…Averaging results for inclusions with upper semi-continuous right-hand side were obtained by Plotnikov [13] under conditions of Lipschitz continuity of the averaged inclusion and for inclusions with a piecewise Lipschitzian right-hand side. Some interesting results were proved by Donchev and Grammel [10] under a onesided Lipschitz condition. Klimov [11] obtained a version of Bogolyubov's first theorem for inclusions satisfying a unilateral continuity condition.…”

Section: Introductionmentioning

confidence: 87%

“…The use of this concept of solution makes it necessary to generalize the averaging method to differential inclusions. Many results extending Bogolyubov's first theorem to differential inclusions were obtained (see, e.g., [6][7][8][9][10][11]). In the case of Lipschitzian differential inclusions the problem was completely solved by Plotnikov [6] and for inclusions with continuous right-hand side by Plotnikova [8] and Lakrib [12].…”

Section: Introductionmentioning

confidence: 99%

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On the asymptotic stability of discontinuous systems analysed via the averaging method

Gama¹,

Guerman

Smirnov

2011

Nonlinear Analysis: Theory, Methods & Applications

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a b s t r a c tThe averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semicontinuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov's first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semicontinuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov's theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

On the asymptotic stability of discontinuous systems analysed via the averaging method

Gama¹,

Guerman

Smirnov

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…In this case, a rigorous justification of the averaging method was given by Sokolovskaya [117]. Under the same one-sided Lipschitz condition, other extensions of the method were obtained in the papers by Donchev et al [13], for perturbed one sided Lipschitz differential inclusions, by Donchev & Grammel [12], for functional differential inclusions in Banach spaces, and by Donchev [10] for evolution inclusions in Banach spaces. In the papers by Sokolovskaya [114][115][116] and Sokolovskaya & Filatov [119] some approximation theorems were proved using the averaging method for inclusions with r.h.s.…”

Section: Averaging On a Finite Time Intervalmentioning

confidence: 99%

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Gama

Smirnov

2013

Set-Valued Var. Anal

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The averaging method is one of the most used tools to study dynamical systems. With the development of the theory of differential inclusions, a respective generalization of the averaging method followed the steps outlined in the theory of differential equations. Presently, it has been successfully applied to a wide range of problems involving differential inclusions, simplifying the study of the systems under consideration. In this work, the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions are surveyed. A detailed list of references is given and some examples of applications are presented.

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“…Then, Assumption 3 holds for the function η wa (x, w, τ, τ 0 , 0). For PWM control system (24), noting the structure of f allows for strong average in (12), there exists a strong average if g i (x, w) and h i (x, w) are independent of w, i.e., g i (x, w) := g i (x) and h i (x, w) := h i (x). In this case, following the calculations used to establish the weak average, we get that f sa (x, w) :…”

Section: Averaging Analysismentioning

confidence: 99%

“…The averaging method was developed for continuous-time systems, discretetime systems, and differential inclusions [3,12,13,31], but there exist only a few results for special classes of hybrid systems. For example, results on averaging of switched linear systems and dither systems were considered in [17,18,30].…”

Section: Introductionmentioning

confidence: 99%

Input-to-state stability for a class of hybrid dynamical systems via averaging

Wang

Nešić

Teel

2011

Math. Control Signals Syst.

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Input-to-state stability (ISS) properties for a class of time-varying hybrid dynamical systems via averaging method are considered. Two definitions of averages, strong average and weak average, are used to approximate the time-varying hybrid systems with time-invariant hybrid systems. Closeness of solutions between the timevarying system and solutions of its weak or strong average on compact time domains is given under the assumption of forward completeness for the average system. We also show that ISS of the strong average implies semi-global practical (SGP)-ISS of the actual system. In a similar fashion, ISS of the weak average implies semi-global practical derivative ISS (SGP-DISS) of the actual system. Through a power converter example, we show that the main results can be used in a framework for a systematic design of hybrid feedbacks for pulse-width modulated control systems.

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Averaging of functional differential inclusions in Banach spaces

Cited by 8 publications

References 13 publications

On the asymptotic stability of discontinuous systems analysed via the averaging method

On the asymptotic stability of discontinuous systems analysed via the averaging method

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Input-to-state stability for a class of hybrid dynamical systems via averaging

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