Partial averaging of discrete-time set-valued systems
DOI:
https://doi.org/10.24193/subbmath.2018.4.09Keywords:
Averaging method, discrete-time system, set-valued mapping, Hukuhara difference.Abstract
In the introduction of the article we given an overview of the results for set-valued equations. Further we considered the set-valued discrete-time dynamical systems and substantiates the averaging method for nonlinear set-valued discrete-time systems with a small parameter. Mathematics Subject Classification (2010): 49M25, 34C29, 49J53.References
Baier, R., Donchev, T., Discrete approximation of impulsive differential inclusions, Numer. Funct. Anal. Optim., 31(2010), no. 6, 653–678.
Belan, E.P., Averaging in the theory of finite-difference equations, Ukr. Math. J., 19(1967), no. 3, 319–323.
Burd, V., Method of Averaging for Differential Equations on an Infinite Interval. Theory and Applications, Lect. Pure Appl. Math., 255, Chapman & Hall/CRC, Boca Ra- ton, FL, 2007.
Chahma, I.A., Set-valued discrete approximation of state-constrained differential inclusions, Bayreuth. Math. Schr., 67(2003), 3–162.
de Blasi, F.S., Iervolino, F., Equazioni differentiali con soluzioni a valore compatto convesso, Boll. Unione Mat. Ital., 2(4–5)(1969), 491–501.
de Blasi, F., Iervolino, F., Euler method for differential equations with set-valued solutions, Boll. Unione Mat. Ital., 4(1971), no. 4, 941–949.
Deimling, K., Multivalued Differential Equations, De Gruyter series in nonlinear analysis and applications, 1, Walter de Gruyter, 1992.
Elaydi, S.N., Discrete Chaos: With Applications in Science and Engineering, Second Edition, Chapman and Hall/CRC, Boca Raton, FL, 2008.
Gu, R.B., Guo, W.J., On mixing properties in set valued discrete system, Chaos Solitons Fractals, 28(2006), no. 3, 747–754.
Hukuhara, M., Integration des applications mesurables dont la valeur est un compact convexe, Funkc. Ekvacioj, Ser. Int., 10(1967), 205–223.
Janiak, T., Luczak -Kumorek, E., Method on partial averaging for functional-differential equations with Hukuhara’s derivative, Stud. Univ. Babe¸s-Bolyai, Math., 48(2003), no. 2, 65–72.
Janiak, T., Luczak-Kumorek, E., Bogolubov’s type theorem for functional-differential inclusions with Hukuhara’s derivative, Stud. Univ. Babe¸s-Bolyai, Math., 36(1991), no. 1, 41–55.
Khan, A., Kumar, P., Chaotic properties on time varying map and its set valued extension, Adv. Pure Math., 3(2013), 359–364.
Kichmarenko, O.D., Averaging of differential equations with Hukuhara derivative with maxima, Int. J. Pure Appl. Math., 57(2009), no. 3, 447–457.
Kisielewicz, M., Method of averaging for differential equations with compact convex valued solutions, Rend. Mat., VI, Ser., 9(1976), no. 3, 397–408.
Komleva, T.A., Plotnikova L.I., Plotnikov A.V., Skripnik, N.V., Averaging in fuzzy controlled systems, Nonlinear Oscil., 14(2012), no. 3, 342–349.
Krylov, N.M., Bogoliubov, N.N., Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1947.
Kulenovic, M.R.s., Merino, O., Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall / CRC, 2002.
Lakshmikantham, V., Granna Bhaskar, T., Vasundhara Devi, J., Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.
Lakshmikantham, V., Mohapatra, R.N., Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 2003.
Martynyuk, D.I., Danilov, Ya., Pan’kov, G., Second Bogolyubov theorem for systems of difference equations, Ukr. Math. J., 48(1996), no. 4, 516–529.
Perestyuk, N.A., Plotnikov, V.A., Samoilenko, A.M., Skripnik, N.V., Differential equations with impulse effects: multivalued right-hand sides with discontinuities, de Gruyter Stud. Math. 40, Berlin/Boston, Walter De Gruyter GmbH & Co, 2011.
Perestyuk, N.A., Skripnik, N.V., Averaging of set-valued impulsive systems, Ukr. Math. J., 65(2013), no. 1, 140–157.
Petersen, I.R., Savkin, A.V., Discrete-Time Set-Valued State Estimation, In: Robust Kalman Filtering for Signals and Systems with Large Uncertainties. Control Engineering. Birkhauser, Boston, MA, 1999.
Plotnikov, A.V., Averaging differential embeddings with Hukuhara derivative, Ukr. Math. J., 41(1989), no. 1, 112–115.
Plotnikov, V.A., Kichmarenko, O.D., Averaging of controlled equations with the Hukuhara derivative, Nonlinear Oscil., 9(2006), no. 3, 365–374.
Plotnikov, V.A., Plotnikov, A.V., Vityuk, A.N., Differential equations with a multivalued right-hand side. Asymptotic methods, AstroPrint, Odessa, 1999.
Plotnikov, V.A., Plotnikova, L.I., Yarovoi, A.T., Averaging method for discrete systems and its application to control problems, Nonlinear Oscil., 7(2004), no. 2, 240–253.
Plotnikov, V.A., Rashkov, P.I., Averaging in differential equations with hukuhara derivative and delay, Funct. Differ. Equ., 8(2001), no. 3-4, 371–381.
Plotnikov, A.V., Skripnik, N.V. Differential equations with ”clear” and fuzzy multivalued right-hand side. Asymptotics methods, AstroPrint, Odessa, 2009.
Plotnikov, A.V., Skripnik, N.V., An existence and uniqueness theorem to the cauchy problem for generalised set differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20(2013), no. 4, 433–445.
Polovinkin, E.S., Multivalued analysis and differential inclusions, FIZMATLIT, Moscow, 2014.
Polovinkin, E.S., Strongly convex analysis, Sb. Math., 187(1996), 259–286.
Robinson, R.C., An Introduction to Dynamical Systems: Continuous and Discrete, Pearson Education, Inc., 2004.
Roman-Flores, H., A Note on Transitivity in Set-Valued Discrete Systems, Chaos Solution Fractals, 17(2003), no. 1, 99–104.
Roman-Flores, H., Chalco-Cano, Y., Robinsons Chaos in Set-Valued Discrete Systems, Chaos Solitons Fractals, 25(2005), no. 1, 33–42.
Sanders, J.A., Verhulst, F., Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, 59, Springer-Verlag, New York, 1985.
Shi, Y.M., Chen, G.R., Chaos of time-varying discrete dynamical systems, J. Difference Equ. Appl., 15(2009), no. 5, 429–449.
Skripnik, N.V., Averaging of impulsive differential inclusions with Hukuhara derivative, Nonlinear Oscil., 10(2007), no. 3, 422–438.
Tolstonogov, A., Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, 2000.
Ungureanu, V., Lozan, V., Linear discrete-time set-valued Pareto-Nash-Stackelberg control processes and their principles, ROMAI J., 9(2013), no. 1, 185–198.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
