zbMATH — the first resource for mathematics

The Krasnosel’skii-Krein theorem for differential inclusions. (English. Russian original) Zbl 1109.34306
Differ. Equ. 41, No. 7, 1049-1053 (2005); translation from Differ. Uravn. 41, No. 7, 997-1000 (2005).
A family of differential inclusions in a finite-dimensional space \[ \dot{x} \in F(t,x,\lambda) \qquad x(0,\lambda) = x_0, \] is considered under the assumption that the convex-valued right-hand side is integrally continuous with respect to \(\lambda\) at the point \(\lambda_0\). Extending the result of M. A. Krasnoselskii and S. G. Krein [Usp. Mat. Nauk 10, No. 3(65), 147–152 (1955; Zbl 0064.33901)], the author proves that under some additional conditions, for each \(\eta >0\), there exists a neighborhood \(U(\lambda_0)\) of \(\lambda_0\) such that for each solution \(x(t,\lambda),\) \(\lambda \in U(\lambda_0)\), of the above problem defined for \(0 \leq t \leq T\) there exists a solution \(x(t,\lambda_0)\) such that \(\| x(t,\lambda) - x(t,\lambda_0)\| < \eta,\) \(0 \leq t \leq T.\) These results are applied to obtain the analogs of the Bogolyubov average theorem for differential inclusions.

34A60 Ordinary differential inclusions
34C29 Averaging method for ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
Full Text: DOI
[1] Vasil’ev, A.B., Ukr. Mat. Zh., 1983, vol. 35, no.5, pp. 607–611.
[2] Plotnikov, V.A., Plotnikov, A.V., and Vityuk, A.N., Differentsial’nye uravneniya s mnogoznachnoi pravoi chast’yu. Asimptoticheskie metody (Differential Equations with Multivalued Right-Hand Side. Asymptotic Methods), Odessa, 1999.
[3] Filatov, O.P. and Khapaev, M.M., Usrednenie sistem differentsial’nykh vklyuchenii (Averaging of Systems of Differential Inclusions), Moscow, 1988.
[4] Janiak, T. and Luczak-Kumorek, E., Discuss. Math., 1991, no. 11, pp. 63–73.
[5] Krasnosel’skii, M.A. and Krein, S.G., Uspekhi Mat. Nauk, 1955, vol. 10, no.3(65), pp. 147–152.
[6] Dawidowski, M., Funct. et Approxim. (PRL), 1979, no. 7, pp. 55–70.
[7] Panasyuk, A.I. and Panasyuk, V.I., Asimptoticheskaya magistral’naya optimizatsiya upravlyaemykh sistem (Asymptotic Turnpike Optimization of Control Systems), Minsk, 1986. · Zbl 0613.49003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.