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Averaging of systems of differential inclusions with slow and fast variables

Averaging of systems of differential inclusions with slow and fast variables We consider the system of differential inclusions $$ \dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$ , where F,G: D → Kυ ( $$ R^{m_1 } $$ ), Kυ ( $$ R^{m_2 } $$ ) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces $$ R^{m_1 } $$ and $$ R^{m_2 } $$ , respectively, D = R + × $$ R^{m_1 } $$ × $$ R^{m_2 } $$ × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem $$ \dot u $$ ∈ µF 0(u), u(0) = x 0, F 0(u) ∈ Kυ ( $$ R^{m_1 } $$ ), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ɛ > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ0] and any solution x µ(·), y µ(·) of the original problem, there exists a solution u µ(·) of the averaged problem such that ∥x µ(t) − y µ(t) ∥ ≤ ɛ for t ∈ [0, 1/µ]. Furthermore, for each solution u µ(·)of the averaged problem, there exists a solution x µ(·), y µ(·) of the original problem with the same estimate. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Averaging of systems of differential inclusions with slow and fast variables

Differential Equations , Volume 44 (3) – Jul 2, 2008

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References (4)

Publisher
Springer Journals
Copyright
Copyright © 2008 by MAIK Nauka
Subject
Mathematics; Difference and Functional Equations; Partial Differential Equations; Ordinary Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266108030063
Publisher site
See Article on Publisher Site

Abstract

We consider the system of differential inclusions $$ \dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$ , where F,G: D → Kυ ( $$ R^{m_1 } $$ ), Kυ ( $$ R^{m_2 } $$ ) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces $$ R^{m_1 } $$ and $$ R^{m_2 } $$ , respectively, D = R + × $$ R^{m_1 } $$ × $$ R^{m_2 } $$ × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem $$ \dot u $$ ∈ µF 0(u), u(0) = x 0, F 0(u) ∈ Kυ ( $$ R^{m_1 } $$ ), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ɛ > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ0] and any solution x µ(·), y µ(·) of the original problem, there exists a solution u µ(·) of the averaged problem such that ∥x µ(t) − y µ(t) ∥ ≤ ɛ for t ∈ [0, 1/µ]. Furthermore, for each solution u µ(·)of the averaged problem, there exists a solution x µ(·), y µ(·) of the original problem with the same estimate.

Journal

Differential EquationsSpringer Journals

Published: Jul 2, 2008

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