References
S. C. Zaremba, “Sure les equations an paratingent,”Bull. Sci. Math.,60, No. 5, 139–160 (1936).
A. Marchaud, “Sur les champs de demicones et equations differentielles du premier order,”Bull. Soc. Math.,62, 1–38 (1934).
T. Wazewski, “Systemes de commande et equations an contingent,”Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,9, No. 3, 151–155 (1961).
T. Wazewski, “Sur une condition equvalente a l'equation an contingent,”Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron. Phys.,9, No. 12, 865–867 (1961).
A. F. Filippov, “Classical solutions of differential equations with a multi-valued right-hand side,”Vestn. Mosk. Gos. Univ., No. 3, 16–26 (1967).
A. F. Filippov,Differential Equations with a Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).
V. I. Blagodatskikh, “Some results in the theory of differential inclusions (review),” in:Summer School on Ord. Diff. Equat., Czechoslovakia, Part XI (1974), pp. 26–40.
V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control,”Trudy Mat. Inst. Akad. Nauk SSSR,169, 194–252 (1985).
Yu. G. Borisovich, B. D. Gel'man, A. D. Myskiš, and V. V. Obukhovsky, “Multi-valued mappings,” in:Itogi Nauki i Tekhn., Mat. Analiz, Vol. 19, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1982), pp. 127–231.
V. A. Plotnikov, “Method of homogenization for differential inclusions and its applications to optimal control problems,”Differents. Uravneniya, No. 8, 1427–1433 (1979).
V. A. Plotnikov, “Homogenization of differential inclusions,”Ukr. Mat. Zh., 573–576 (1979).
V. A. Plotnikov, “Partial homogenization of differential inclusions,”Mat. Zametki,27, No. 6, 947–952 (1980).
V. A. Plotnikov,Method of Homogenization in Control Problems [in Russian], Lybid', Odessa (1992).
A. N. Vityuk and S. S. Klimenko, “The Bogolyubov theorem for hyperbolic differential inclusions,”Ukr. Mat. Zh., No. 5, 641–645 (1987).
O. P. Filatov and M. M. Khapaev, “On the mutual ε-approximation of solutions of a system of differential inclusions with fast and slow variables,”Mat. Zametki,47, No. 5, 127–134 (1990).
O. P. Filatov and M. M. Khapaev, “Homogenization of differential inclusions with fast and slow variables,”Mat. Zametki,47, No. 6, 102–109 (1990).
M. A. Aizerman and F. R. Gantmakher, “Stability with respect to linear approximation of a periodic solution of a system of differential equations with discontinuous right-hand sides,”Prikl. Mat. Mekh.,21, No. 5, 658–669 (1957).
A. F. Filippov, “Differential equations with a discontinuous right-hand side,”Mat. Sb.,51, No. 1, 99–128 (1960).
A. M. Samoilenko, “Substantiation of the homogenization method for differential equations with a discontinuous right-hand side,” in:Approxim. Methods of Solving Diff. Equat. [in Russian], Akad. Nauk Ukr. SSR, Kiev (1963), pp. 90–95.
E. F. Mishchenko and N. Kh. Rozov,Differential Equations with a Small Parameter and Relaxation Oscillations [in Russian], Nauka, Moscow (1975).
V. I. Utkin,Sliding Conditions in Optimization and Control Problems [in Russian], Nauka, Moscow (1981).
V. A. Plotnikov and T. S. Zverkova, “Method of homogenization for systems of standard type with discontinuous right-hand sides,”Differents. Uravneniya, No. 6, 1091–1093 (1982).
L. S. Pontryagin and E. F. Mishchenko, “Some problems in the theory of differential equations with a small parameter,”Trudy Mat. Inst. Akad. Nauk SSSR,169, 99–118 (1985).
L. T. Ashchepkov,Optimal Control of Discontinuous Systems [in Russian], Nauka, Novosibirsk (1987).
A. M. Samoilenko and N. A. Perestyuk,Differential Equations with Pulsed Forcing [in Russian], Vishcha Shkola, Kiev (1987).
V. A. Plotnikov and S. S. Klimchuk, “Homogenization of equations of the sliding regime in control problems,” in:Dynamics of Systems [in Russian] (1989), pp. 89–92.
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 17, Dinamicheskie Sistemy-2, 1994.
Rights and permissions
About this article
Cite this article
Plotnikov, V.A. Asymptotic methods in the theory of differential equations with discontinuous and multi-valued right-hand sides. J Math Sci 80, 1605–1616 (1996). https://doi.org/10.1007/BF02363930
Issue Date:
DOI: https://doi.org/10.1007/BF02363930