Abstract
The method of directing function was extended to the nonsmooth case. It is used to solve the problem of periodic oscillations of the control plants obeying the functional-differential inclusions whose right side is not convex-valued and satisfies the condition for almost semicontinuity from below.
Similar content being viewed by others
References
Myshkis, A.D., General Theory of Differential Delay Equations, Usp. Mat. Nauk, 1949, vol. 4, no. 5, pp. 99–141.
Krasnoselskii, A.M., Krasnoselskii, M.A., Mawhin, J., and Pokrovskii, A., Generalized Guiding Functions in a Problem on High Frequency Forced Oscillations, Nonlin. Anal. Theory, Methods Appl., 1994, vol. 22, no. 11, pp. 1357–1371.
Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (Operator of Shift Along the Trajectories of Differential Equations) Moscow: Nauka, 1966.
Krasnosel’skii, M.A. and Perov, A.I., On One Principle of Existence of Bounded Periodic and Almostperiodic Solutions of Systems of Ordinary Differential Equations, Dokl. Akad. Nauk SSSR, 1958, vol. 123, no. 2, pp. 235–238.
Borisovich, Yu.G., Gel’man, B.D., Myshkis, A.D., and Obukhovskii, V.V., Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii (Introduction to the Theory of Multivalued Maps and Differential Inclusions), Moscow: “Librokom,” 2011.
Górniewicz, L., Topological Fixed Point Theory of Multivalued Mappings, Dordrecht: Kluwer, 1999.
Fonda, A., Guiding Functions and Periodic Solutions to Functional Differential Equations, Proc. Am. Math. Soc., 1987, vol. 99, no. 1, pp. 79–85.
Rachinskii, D.I., Forced Oscillations in Control Systems under Near-Resonance Conditions, Autom. Remote Control, 1995, vol. 56, no. 11, part 1, pp. 1575–1584.
Kornev, S.V. and Obukhovskii, V.V., On Integral Directing Functions for Functional-Differential Inclusions, in Topologicheskie metody nelineinogo analiza (Topological Methods of Nonlinear Analysis), Voronezh: Voronezh. Gos. Univ., 2000, pp. 87–107.
Kornev, S.V., On the Method of Multivalent Guiding Functions for Periodic Solutions of Differential Inclusions, Autom. Remote Control, 2003, vol. 64, no. 3, pp. 409–419.
De Blasi, F.S., Górniewicz, L., and Pianigiani, G., Topological Degree and Periodic Solutions of Differential Inclusions, Nonlin. Anal., 1999, vol. 37, pp. 217–245.
Kornev, S.V. and Obukhovskii, V.V., On Nonsmooth Multivalent Directing Functions, Differ. Uravn., 2003, vol. 39, no. 11, pp. 1497–1502.
Kornev, S. and Obukhovskii, V., On Some Developments of the Method of Integral Guiding Functions, Differ. Uravn., 2003, vol. 12, nos. 3–4, pp. 303–310.
Obukhovskii, V., Zecca, P., Loi, N.V., and Kornev, S., Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes Math., vol. 2076, Berlin: Springer, 2013.
Kamenskii, M., Obukhovskii, V., and Zecca, P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Berlin: Walter de Gruyter, 2001.
Bressan, A. and Colombo, G., Extensions and Selections of Maps with Decomposable Values, Studia Math., 1988, vol. 90, pp. 69–86.
Fryszkowski, A., Fixed Point Theory for Decomposable Sets, Dordrecht: Kluwer, 2004.
Emel’yanov, S.V., Korovin, S.K., Bobylev, N.A., and Bulatov, A.V., Gomotopii ekstremal’nykh zadach (Homotopies of Extremal Problems), Moscow: Nauka, 2001.
Clarke, F.H., Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988.
Bader, R., Gel’man, B.D., and Obukhovskii, V.V., On a Class of Multivalued Maps, Vestn. Voronezh. Gos. Univ., Ser. Fiz.-Mat., 2003, vol. 2, pp. 35–38.
Michael, E., Continuous Selections. I, Ann. Math., 1956, vol. 63, pp. 361–382.
Kornev, S.V. and Obukhovskii, V.V., On Some Variants of the Topological Degree Theory for the Nonconvexvalued Multimaps and Periodic Problems for Functional-Differential Inclusion, Tr. Mat. Fak., Voronezh. Gos. Univ., 2004, vol. 8, pp. 56–75.
Kornev, S. and Obukhovskii, V., On Asymptotics of Solutions for a Class of Functional Differential Inclusions, Discuss. Math. Differ. Incl., Control Optim., 2014, vol. 34, no. 2, pp. 219–227.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.V. Kornev, 2015, published in Avtomatika i Telemekhanika, 2015, No. 9, pp. 31–43.
Rights and permissions
About this article
Cite this article
Kornev, S.V. Nonsmooth integral directing functions in the problems of forced oscillations. Autom Remote Control 76, 1541–1550 (2015). https://doi.org/10.1134/S0005117915090027
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117915090027