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 Mat. Zametki, 2005, Volume 78, Issue 5, Pages 763–772 (Mi mz2632)

Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

E. V. Sokolovskaya, O. P. Filatov

Samara State University

Abstract: Suppose that $\mathbb R^n$ is the $p$-dimensional space with Euclidean norm ${\|\cdot\|}$, $K(\mathbb R^p)$ is the set of nonempty compact sets in $\mathbb R^p$, $\mathbb R_+=0,+\infty)$, $D=\mathbb R_+\times\mathbb R^m\times\mathbb R^n\times[0,a]$, $D_0=\mathbb R_+\times\mathbb R^m$, $F_0\colon D_0\to K(\mathbb R^m)$, and $\operatorname{co}F_0$ is the convex cover of the mapping $F_0$. We consider the Cauchy problem for the system of differential inclusions
$$\dot x\in\mu F(t,x,y,\mu),\quad \dot y\in G(t,x,y,\mu),\qquad x(0)=x_0,\quad y(0)=y_0$$
with slow $x$ and fast $y$ variables; here $F\colon D\to K(\mathbb R^m)$, $G\colon D\to K(\mathbb R^n)$, and $\mu\in[0,a]$ is a small parameter. It is assumed that this problem has at least one solution on $[0,1/\mu]$ for all sufficiently small $\mu\in[0,a]$. Under certain conditions on $F$, $G$, and $F_0$, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any $\varepsilon>0$, there is a $\mu_0>0$ such that for any $\mu\in(0,\mu_0]$ and any solution $(x_\mu(t),y_\mu(t))$ of the problem under consideration, there exists a solution $u_\mu(t)$ of the problem $\dot u\in\mu\operatorname{co}F_0(t,u)$, $u(0)=x_0$ for which the inequality $\|x_\mu(t)-u_\mu(t)\|<\varepsilon$ holds for each $t\in[0,1/\mu]$.

DOI: https://doi.org/10.4213/mzm2632

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English version:
Mathematical Notes, 2005, 78:5, 709–718

Bibliographic databases:

UDC: 517.928

Citation: E. V. Sokolovskaya, O. P. Filatov, “Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side”, Mat. Zametki, 78:5 (2005), 763–772; Math. Notes, 78:5 (2005), 709–718

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2632
• https://doi.org/10.4213/mzm2632
• http://mi.mathnet.ru/eng/mz/v78/i5/p763

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This publication is cited in the following articles:
1. O. P. Filatov, “Kvaziresheniya differentsialnykh vklyuchenii i teorema raznostnoi approksimatsii”, Trudy Tretei Vserossiiskoi nauchnoi konferentsii (29–31 maya 2006 g.). Chast 3, Differentsialnye uravneniya i kraevye zadachi, Matem. modelirovanie i kraev. zadachi, SamGTU, Samara, 2006, 214–217
2. Klimov V.S., “Averaging of differential inclusions”, Differ. Equ., 44:12 (2008), 1673–1681
3. Filatov O.P., “Averaging of systems of differential inclusions with slow and fast variables”, Differ. Equ., 44:3 (2008), 349–363
4. Donchev T., Farkhi E., “On the theorem of Filippov-Pliś and some applications”, Control Cybernet., 38:4, Part A Sp. Iss. SI (2009), 1251–1271
5. Donchev T., “Singularly perturbed evolution inclusions”, SIAM J. Control Optim., 48:7 (2010), 4572–4590
6. V. S. Klimov, “The Bohl index of a homogeneous parabolic inclusion”, Izv. Math., 75:2 (2011), 347–370
7. Gama R. Smirnov G., “Stability and Optimality of Solutions To Differential Inclusions Via Averaging Method”, Set-Valued Var. Anal., 22:2 (2014), 349–374
8. Klimov V.S., “Strength and Stability of the Bohl Index”, Differ. Equ., 51:5 (2015), 592–604
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