On averaging of differential inclusions in the case where the average of the right-hand side does not exist

Плотников, В. А.; Savchenko, Volodymyr

doi:10.1007/bf02529499

Cited by 6 publications

(4 citation statements)

References 1 publication

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“…2For non-periodic G(·, α), one still can consider an averaged version of (1)s, α) ds does not exist. Theorem 4 extends the main result of[21] and partially[11, Theorem 1].…”

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confidence: 69%

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Averaging of functional differential inclusions in Banach spaces

Donchev

Grammel

2005

Journal of Mathematical Analysis and Applications

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Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of nonexistence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved. Under stronger regularity conditions, approximation orders for Krylov-Bogoliubov-Mitropolskii type right-hand sides are derived.  2005 Elsevier Inc. All rights reserved.

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“…2For non-periodic G(·, α), one still can consider an averaged version of (1)s, α) ds does not exist. Theorem 4 extends the main result of[21] and partially[11, Theorem 1].…”

supporting

confidence: 69%

“…Theorem 4 of the present work deals with the averaging problem when the average does not necessarily exist. It extends earlier results of [21] (ordinary differential inclusions) and [11] (functional differential inclusions). We mention also [20], where the averaging method is applied to differential equation with Hukuhara derivative and infinite delay.…”

supporting

confidence: 67%

Averaging of functional differential inclusions in Banach spaces

Donchev

Grammel

2005

Journal of Mathematical Analysis and Applications

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“…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%

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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“…To overcome this difficulty, Plotnikov & Savchenko [100] suggested an extension of Theorem 1. Under the following conditions:…”

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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On averaging of differential inclusions in the case where the average of the right-hand side does not exist

Cited by 6 publications

References 1 publication

Averaging of functional differential inclusions in Banach spaces

Averaging of functional differential inclusions in Banach spaces

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

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

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