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Title: Cone-type constrained relative controllability of semilinear fractional systems with delays (English)
Author: Sikora, Beata
Author: Klamka, Jerzy
Language: English
Journal: Kybernetika
ISSN: 0023-5954 (print)
ISSN: 1805-949X (online)
Volume: 53
Issue: 2
Year: 2017
Pages: 370-381
Summary lang: English
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Category: math
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Summary: The paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function $f$. The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered. The results are illustrated by an example. (English)
Keyword: the Caputo derivative
Keyword: semilinear fractional systems
Keyword: relative controllability
Keyword: delays in control
Keyword: constraints
MSC: 34G20
MSC: 93B05
MSC: 93C05
MSC: 93C10
idZBL: Zbl 06770173
idMR: MR3661357
DOI: 10.14736/kyb-2017-2-0370
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Date available: 2017-06-25T18:05:48Z
Last updated: 2018-01-10
Stable URL: http://hdl.handle.net/10338.dmlcz/146810
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Reference: [1] Ahmed, E., Hashis, A. H., Rihan, F. A.: On fractional order cancer model..J. Fractional Calculus Appl. 3 (2012), 1-6. MR 1330571
Reference: [2] Babiarz, A., Niezabitowski, M.: Controllability Problem of Fractional Neutral Systems: A Survey..Math. Problems Engrg., ID 4715861 (2017), 15 pages. MR 3603402, 10.1155/2017/4715861
Reference: [3] Balachandran, K., Kokila, J., Trujillo, J. J.: Relative controllability of fractional dynamical systems with multiple delays in control..Comp. Math. Apll. 64 (2012), 3037-3045. Zbl 1268.93021, MR 2989332, 10.1016/j.camwa.2012.01.071
Reference: [4] Balachandran, K., Zhou, Y., Kokila, J.: Relative controllability of fractional dynamical systems with delays in control..Commun. Nonlinear. Sci. Numer. Simulat. 17 (2012), 3508-3520. Zbl 1248.93022, MR 2913988, 10.1016/j.cnsns.2011.12.018
Reference: [5] Balachandran, K., Zhou, Y., Kokila, J.: Relative controllability of fractional dynamical systems with discributed delays in control..Comp. Math. Apll. 64 (2012), 3201-3206. MR 2989348, 10.1016/j.camwa.2011.11.061
Reference: [6] Balachandran, K.: Controllability of Nonlinear Fractional Delay Dynamical Systems with Multiple Delays in Control.Lecture Notes in Electrical Engineering. Theory and Applications of Non-integer Order Systems 407 (2016), 321-332. 10.1007/978-3-319-45474-0_29
Reference: [7] Bodnar, M., Piotrowska, J.: Delay differential equations: theory and applications..Matematyka Stosowana 11 (2011), 17-56 (in Polish). MR 2755711
Reference: [8] Haque, M. A.: A predator-prey model with discrete time delay considering different growth function of prey..Adv. Apll. Math. Biosciences 2 (2011), 1-16. 10.1016/j.mbs.2011.07.003
Reference: [9] He, X.: Stability and delays in a predator-prey system..J. Math. Anal. Appl. 198 (1996), 355-370. Zbl 0952.34061, MR 1376269, 10.1006/jmaa.1996.0087
Reference: [10] Kaczorek, T.: Selected Problems of Fractional Systems Theory..Lect. Notes Control Inform. Sci. 411 2011. Zbl 1221.93002, MR 2798773, 10.1007/978-3-642-20502-6
Reference: [11] Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits..Studies in Systems, Decision and Control 13 2015. Zbl 1354.93001, MR 3497539, 10.1007/978-3-319-11361-6
Reference: [12] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations..North-Holland Mathematics Studies 204 2006. Zbl 1092.45003, MR 2218073
Reference: [13] Klamka, J.: Controllability of Dynamical Systems..Kluwer Academic Publishers, 1991. Zbl 0818.93002, MR 1134783
Reference: [14] Klamka, J.: Constrained controllability of semilinear systems with delayed controls..Bull. Polish Academy of Sciences: Technical Sciences 56 (2008), 333-337.
Reference: [15] Klamka, J., Sikora, B.: New controllability Criteria for Fractional Systems with Varying Delays..Lect. Notes Electr. Engrg. Theory and Applications of Non-integer Order Systems 407 (2017), 333-344. 10.1007/978-3-319-45474-0_30
Reference: [16] Krishnaveni, K., Kannan, K., Balachandar, S. R.: Approximate analytical solution for fractional population growth model..Int. J. Engrg. Technol. 5 (2013), 2832-2836.
Reference: [17] Machado, J. T., Costa, A. C., Quelhas, M. D.: Fractional dynamics in DNA..Comm. Nonlinear Sciences and Numerical Simulation 16 (2011), 2963-2969. Zbl 1218.92038, 10.1016/j.cnsns.2010.11.007
Reference: [18] Malinowska, A. B., Odziejewicz, T., Schmeidel, E.: On the existence of optimal control for the fractional continuous-time Cucker-Smale model..Lect. Notes Electr. Engrg., Theory and Applications of Non-integer Order Systems 407 (2016), 227-240. 10.1007/978-3-319-45474-0_21
Reference: [19] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Calculus..Villey 1993. MR 1219954
Reference: [20] Monje, A., Chen, Y., Viagre, B. M., Xue, D., Feliu, V.: Fractional-order Systems and Controls. Fundamentals and Applications..Springer-Verlag 2010. MR 3012798, 10.1007/978-1-84996-335-0
Reference: [21] Nirmala, R. J., Balachandran, K., Rodriguez-Germa, L., Trujillo, J. J.: Controllability of nonlinear fractional delay dynamical systems..Rep. Math. Physics 77 (2016), 87-104. MR 3461800, 10.1016/s0034-4877(16)30007-6
Reference: [22] Oldham, K. B., Spanier, J.: The Fractional Calculus..Academic Press 1974. Zbl 0292.26011, MR 0361633
Reference: [23] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications..In: Mathematics in Science and Engineering, Academic Press 1999. Zbl 0924.34008, MR 1658022, 10.1016/s0034-4877(16)30007-6
Reference: [24] Robinson, S. M.: Stability theory for systems of inequalities. Part II. Differentiable nonlinear systems..SIAM J. Numerical Analysis 13 (1976), 497-513. MR 0410522, 10.1137/0713043
Reference: [25] Sabatier, J., Agrawal, O. P., Machado, J. A. Tenreiro: Advances in Fractional Calculus..In: Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag 2007. MR 3184154, 10.1007/978-1-4020-6042-7
Reference: [26] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications..Gordan and Breach Science Publishers 1993. Zbl 0818.26003, MR 1347689
Reference: [27] Sikora, B.: Controllability of time-delay fractional systems with and without constraints..IET Control Theory Appl. 10 (2016), 320-327. MR 3468656, 10.1049/iet-cta.2015.0935
Reference: [28] Sikora, B.: Controllability criteria for time-delay fractional systems with a retarded state..Int. J. Applied Math. Computer Sci. 26 (2016), 521-531. Zbl 1347.93057, MR 3560625, 10.1515/amcs-2016-0036
Reference: [29] Srivastava, V. K., Kumar, S., Awasthi, M., Singh, B. K.: Two-dimensional time fractional-order biological population model and its analytical solution..Egyptian J. Basic Appl. Sci. 1 (2014), 71-76. 10.1016/j.ejbas.2014.03.001
Reference: [30] Wei, J.: The controllability of fractional control systems with control delay..Comput. Math. Appl. 64 (2012), 3153-3159. Zbl 1268.93027, MR 2989343, 10.1016/j.camwa.2012.02.065
Reference: [31] Zduniak, B., Bodnar, M., Foryś, U.: A modified Van der Pol equation with delay in a description of the heart action..Int. J. Appl. Math. Computer Sci. 24 (2014), 853-863. Zbl 1309.93076, MR 3309453, 10.2478/amcs-2014-0063
Reference: [32] Zhang, H., Cao, J., Jiang, W.: Controllability criteria for linear fractional differential systems with state delay and impulses..J. Appl. Math., ID146010 (2013) 9 pages. Zbl 1271.93028, MR 3064923, 10.1155/2013/146010
.

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