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

• N. V. Skripnik Odessa I.I. Mechnikov National University
Keywords: fuzzy system; differential inclusion; impulse; averaging method; R-solution

### Abstract

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), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$

### Author Biography

N. V. Skripnik, Odessa I.I. Mechnikov National University

Odessa I.I. Mechnikov National University

Odessa, Ukraine

### References

Aubin J.-P. Fuzzy differential inclusions// Problems of Control and Information Theory. – 1990. – V.19, №1. – P. 55–67.

Baidosov V.A. Differential inclusions with fuzzy right-hand side// Soviet Mathematics. – 1990. – V.40, №3. – P. 567–569.

Baidosov V.A. Fuzzy differential inclusions// Journal of Applied Mathematics and Mechanics. – 1990. – V.54, №1. – P. 8–13.

Bogoliubov N.N., Mitropolsky Yu.A. Asymptotic methods in the theory of non-linear oscillations. – New York: Gordon and Breach, 1961.

Burd V. Method of averaging for differential equations on an infinite interval. Theory and applications (Lecture Notes in Pure and Applied Mathematics, vol. 255). – Boca Raton, FL: Chapman and Hall/CRC, 2007.

Gama R., Smirnov G. Stability and optimality of solutions to differential inclusions via averaging method// Set-Valued and Variational Analysis. – 2014. – V.22, №2. – P. 349–374.

Guo M., Xue X., Li R. Impulsive functional differential inclusions and fuzzy population models// Fuzzy Sets and Systems. – 2003. – V.138. – P. 601–615.

Hale J.K. Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, 3. – New York-Heidelberg: Springer-Verlag, 1977.

H¨ullermeier E. An approach to modelling and simulation of uncertain dynamical system// International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. – 1997. – V.5, №2. – P. 117–137.

Klimchuk S., Plotnikov A., Skripnik N. Overview of V.A. Plotnikov ’s research on averaging of differential inclusions// Physica D. – 2012. – V.241, №22. – P. 1932–1947.

Krylov N.M., Bogoliubov N.N. Introduction to nonlinear mechanics. – Princeton: Princeton University Press, 1947.

Lochak P., Meunier C. Multiphase averaging for classical systems, Appl. Math. Sci., 72. – New York: Springer-Verlag, 1988.

Mitropolskiy Yu.A. Lectures of averaging in nonlinear mechanics. – Kiev: Naukova Dumka, 1986.

Park J.Y., Han H.K. Existence and uniqueness theorem for a solution of fuzzy differential equations// International Journal of Mathematics and Mathematical Sciences. – 1999. – V.22, №2. – P. 271–279.

Perestyuk N.A., Plotnikov V.A., Samoilenko A.M., Skripnik N.V. Differential equations with impulse effects: multivalued right-hand sides with discontinuities (De Gruyter Studies in Mathematics: 40). – Berlin/Boston: Walter De Gruyter GmbHCo., 2011.

Plotnikov A.V. A Procedure of complete averaging for fuzzy differential inclusions on a finite segment// Ukrainian Mathematical Journal. – 2015. – V.67, №3. – P. 421–430.

Plotnikov A.V., Komleva T.A. The averaging of fuzzy linear differential inclusions on finite interval// Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms. – 2016. – V.23, №1. – P. 1–9.

Plotnikov A.V., Komleva T.A., Plotnikova L.I. On the averaging of differential inclusions with fuzzy right-hand side when the average of the right-hand side is absent// Iranian journal of optimization. – 2010. – V.2, №3. – P. 506–517.

Plotnikov A.V., Komleva T.A., Plotnikova L.I. The partial averaging of differential inclusions with fuzzy right-hand side// Journal of Advanced Research in Dynamical and Control Systems. – 2010. – V.2, №2. – P. 26–34.

Plotnikov A.V., Skripnik N.V. Differential equations with “clear” and fuzzy set valued right–hand side. Asymptotical methods. – Odessa: Astroprint, 2009.

Plotnikov V.A., Plotnikov A.V., Vityuk A.N. Differential equations with a multivalued right-hand side. Asymptotic methods. – Odessa: AstroPrint, 1999.

Samoilenko A.M., Perestyuk N.A. Impulsive differential equations. – Singapore: World Scientific, 1995.

Sanders J.A., Verhulst F. Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences 59. – New York: Springer-Verlag, 1985.

Sanders J.A., Verhulst F., Murdock J. Averaging methods in nonlinear dynamical systems, 2nd edition, Appl. Math. Sci., 59. – New York: Springer-Verlag, 2007.

Skripnik N.V. Averaging of impulsive differential inclusions with fuzzy right-hand side// Ukrainian Mathematical Journal. – 2015. – V.66, №11. – P. 1756–1772.

Skripnik N.V. The scheme of partial averaging for impulsive differential inclusions with fuzzy right-hand side// Mat. Stud. – 2015. – V.43, №2. – P. 129–139.

Skripnik N.V. Averaging of impulsive differential inclusions with fuzzy right-hand side when the average is absent// Asian-European Journal of Mathematics. – 2015. – V.12, №4. – P. 1550086-1–1550086-12.

Skripnik N.V. Step scheme of averaging method for impulsive differential inclusions with fuzzy right-hand side// Contemporary Methods in Mathematical Physics and Gravitation. – 2015. – V.1, №1. – P. 9–26.

Skripnik N.V. The full averaging scheme for impulsive differential inclusions with fuzzy right-hand side in terms of R–solutions// Researches in mathematics and mechanics. – 2018. – V.23, №2. – P. 86–100.

Zadeh L. Fuzzy sets// Information and Control. – 1965. – №8. – P. 338–353.

Published
2021-03-04
How to Cite
1.
Skripnik NV. Averaging method for impulsive differential inclusions with fuzzy right-hand side. Mat. Stud. [Internet]. 2021Mar.4 [cited 2021Oct.16];55(1):76-4. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/88
Issue
Section
Articles