The partial averaging scheme for impulsive differential inclusions with fuzzy right-hand side

Skripnik, Natalia V.

doi:10.15330/ms.43.2.129-139

Cited by 3 publications

(3 citation statements)

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“…In the proof of the obtained theorems the proximity of the α-solutions is shown and therefore the proximity of solutions sets of the initial and averaged inclusions is proved. The similar results for impulsive differential inclusions with fuzzy right-hand side are obtained in [25,26,27,28].…”

Section: Definition 2 ([9]supporting

confidence: 80%

Averaging method for impulsive differential inclusions with fuzzy right-hand side

Skripnik

2021

Mat. Stud.

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In this paper the substantiation of the partial scheme of the averaging method for impulsive differential inclusions with fuzzy right-hand side in terms of R - solutions on the finite interval is considered.Consider the impulsive differential inclusion with the fuzzy right-hand side $$\dot x \in \varepsilon F(t,x) ,\ t \not= t_i,\ x(0)\in X_0,\quad\Delta x \mid _{t=t_i} \in \varepsilon I_i (x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$ where $t\in \mathbb{R}_+ $ is time, $x \in \mathbb{R}^n $ is a phase variable, $\varepsilon > 0 $ is a small parameter,$ F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n,$ $I_i \colon \mathbb{R}^n \to \mathbb{E}^n $ are fuzzy mappings, moments $t_i$ are enumerated in the increasing order.Associate with inclusion (1) the following partial averaged differential inclusion $$\dot\xi \in \varepsilon \widetilde F (t, \xi ),\ t \not= s_j ,\ \xi (0) \in X_0,\quad \Delta \xi \vert _{t=s_j} \in \varepsilon K_j (\xi ),\qquad\qquad\qquad\qquad\qquad\qquad\quad (2),$$ where the fuzzy mappings $ \widetilde F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n ; \quad K_j \colon \mathbb{R} \to \mathbb{E}^n $ satisfy the condition $$\lim _{T \to \infty } \frac 1T D \Big( \int\limits_t^{t+T} F(t,x) dt + \sum_{t \leq t_i < t+T} I_i (x),\int\limits_t^{t+T} \widetilde F(t,x)dt +\sum_{t \leq s_j < t+T} K_j (x) \Big) = 0,\quad\quad (3)$$ moments $s_j$ are enumerated in the increasing order. In the paper is proved the following main theorem:{\sl Let in the domain $ Q = \lbrace t \geq 0 , x \in G\subset \mathbb{R}^n \rbrace $ the following conditions fulfill:$1)$ fuzzy mappings $ F (t,x), \widetilde F(t,x), I_i(x),K_j(x) $are continuous, uniformly bounded with constant $M$, concave in $x,$ satisfy Lipschitz condition in $x$ with constant $ \lambda ;$$2)$ uniformly with respect to $t, x$ limit (3) exists and $\frac 1T i(t,t+T) \leq d < \infty ,\ \frac 1T j(t,t+T) \leq d < \infty,$where $i(t,t+T)$ and $j(t,t+T)$ are the quantities of impulse moments $t_i$ and $s_j$ on the interval$ [ t, t+T ] $;$3)$ {\rm R}-solutions of inclusion (2) for all $ X_0 \subset G^{\prime} \subset G $for $ t \in [0,L^{\ast} \varepsilon ^{-1} ] $ belong to the domain $G$ with a $ \rho $- neighborhood.Then for any $\eta > 0 $ and $L \in (0,L^{\ast}]$ there exists $\varepsilon _0 (\eta,L) \in (0,\sigma ] $ such that for all $\varepsilon \in (0, \varepsilon _0 ]$ and $t \in [0,L \varepsilon ^{-1}] $ the inequality holds:$D(R(t, \varepsilon ), \widetilde R (t, \varepsilon)) < \eta,$ where $R(t, \varepsilon), \widetilde R(t, \varepsilon ) $ are the {\rm R-} solutions of inclusions (1) and (2), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$

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Section: Definition 2 ([9]supporting

confidence: 80%

Averaging method for impulsive differential inclusions with fuzzy right-hand side

Skripnik

2021

Mat. Stud.

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“…Recently there have been new advances in the theory of fuzzy differential equations [12][13][14][15][16], fuzzy integrodifferential equations [17][18][19][20], differential inclusions with fuzzy right-hand side [21][22][23][24] and fuzzy differential inclusions [25][26][27] as well as in the theory of control fuzzy d i f f e r e n t i a l e q u a t i o n s [ 2 8 -3 0 ] , c o n t r o l f u z z y integrodifferential equations [31][32][33][34][35], controlfuzzy differential inclusions [36][37][38][39], and control fuzzy integrodifferential inclusions [40]. In works [41][42][43][44][45][46][47][48] various schemes of an average for the fuzzy differential equations and inclusions have been considered. In this article we prove the substantiation of the method of full averaging for the integrodifferential inclusions with small parameter on the metric space ( )…”

Section: Introductionmentioning

confidence: 99%

Averaging of Fuzzy Integrodifferential Inclusions

Plotnikov¹

2012

CONTROL

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“…The applications of fuzzy set theory can be found in many branches of science as physical, mathematical, differential equations and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [16,25,26,35,36,44], fuzzy integral equations [12,13,15,31,32,37,46,47,54], fuzzy integro-differential equations [1,6,7,8], differential inclusions with fuzzy right-hand side [2,4,5,14] and fuzzy differential inclusions [43,48,49] as well as in the theory of control fuzzy differential equations [39,50,24], control fuzzy integrodifferential equations [21,22,23,33,34,41], control fuzzy differential inclusions [29,30,40,42], and control fuzzy integro-differential inclusions [53]. Almost in all papers mentioned above the authors also consider equivalent fuzzy integral equations.…”

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confidence: 99%

Averaging of fuzzy integral equations

Skripnik¹

2017

Discrete &Amp; Continuous Dynamical Systems - B

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The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

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The partial averaging scheme for impulsive differential inclusions with fuzzy right-hand side

Cited by 3 publications

References 7 publications

Averaging method for impulsive differential inclusions with fuzzy right-hand side

Averaging method for impulsive differential inclusions with fuzzy right-hand side

Averaging of Fuzzy Integrodifferential Inclusions

Averaging of fuzzy integral equations

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