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K. Deimling (1992)
Multivalued Differential Equations
N.N. Bogolyubov, Yu.A. Mitropol’skii (1963)
Asimptoticheskie metody v teorii nelineinykh kolebanii
Елена Соколовская, E. Sokolovskaya, Олег Филатов, Oleg Filatov (2005)
Аппроксимация сверху систем дифференциальных включений с нелипшицевой правой частью@@@Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side, 78
T. Donchev, E. Farkhi (1998)
Stability and Euler Approximation of One-sided Lipschitz Differential InclusionsSiam Journal on Control and Optimization, 36
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.
Differential Equations – Springer Journals
Published: Jul 2, 2008
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