Previous |  Up |  Next


the Caputo derivative; semilinear fractional systems; relative controllability; delays in control; constraints
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.
[1] Ahmed, E., Hashis, A. H., Rihan, F. A.: On fractional order cancer model. J. Fractional Calculus Appl. 3 (2012), 1-6. MR 1330571
[2] Babiarz, A., Niezabitowski, M.: Controllability Problem of Fractional Neutral Systems: A Survey. Math. Problems Engrg., ID 4715861 (2017), 15 pages. DOI 10.1155/2017/4715861 | MR 3603402
[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. DOI 10.1016/j.camwa.2012.01.071 | MR 2989332 | Zbl 1268.93021
[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. DOI 10.1016/j.cnsns.2011.12.018 | MR 2913988 | Zbl 1248.93022
[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. DOI 10.1016/j.camwa.2011.11.061 | MR 2989348
[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. DOI 10.1007/978-3-319-45474-0_29
[7] Bodnar, M., Piotrowska, J.: Delay differential equations: theory and applications. Matematyka Stosowana 11 (2011), 17-56 (in Polish). MR 2755711
[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. DOI 10.1016/j.mbs.2011.07.003
[9] He, X.: Stability and delays in a predator-prey system. J. Math. Anal. Appl. 198 (1996), 355-370. DOI 10.1006/jmaa.1996.0087 | MR 1376269 | Zbl 0952.34061
[10] Kaczorek, T.: Selected Problems of Fractional Systems Theory. Lect. Notes Control Inform. Sci. 411 2011. DOI 10.1007/978-3-642-20502-6 | MR 2798773 | Zbl 1221.93002
[11] Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Studies in Systems, Decision and Control 13 2015. DOI 10.1007/978-3-319-11361-6 | MR 3497539 | Zbl 1354.93001
[12] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204 2006. MR 2218073 | Zbl 1092.45003
[13] Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Publishers, 1991. MR 1134783 | Zbl 0818.93002
[14] Klamka, J.: Constrained controllability of semilinear systems with delayed controls. Bull. Polish Academy of Sciences: Technical Sciences 56 (2008), 333-337.
[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. DOI 10.1007/978-3-319-45474-0_30
[16] Krishnaveni, K., Kannan, K., Balachandar, S. R.: Approximate analytical solution for fractional population growth model. Int. J. Engrg. Technol. 5 (2013), 2832-2836.
[17] Machado, J. T., Costa, A. C., Quelhas, M. D.: Fractional dynamics in DNA. Comm. Nonlinear Sciences and Numerical Simulation 16 (2011), 2963-2969. DOI 10.1016/j.cnsns.2010.11.007 | Zbl 1218.92038
[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. DOI 10.1007/978-3-319-45474-0_21
[19] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Calculus. Villey 1993. MR 1219954
[20] Monje, A., Chen, Y., Viagre, B. M., Xue, D., Feliu, V.: Fractional-order Systems and Controls. Fundamentals and Applications. Springer-Verlag 2010. DOI 10.1007/978-1-84996-335-0 | MR 3012798
[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. DOI 10.1016/s0034-4877(16)30007-6 | MR 3461800
[22] Oldham, K. B., Spanier, J.: The Fractional Calculus. Academic Press 1974. MR 0361633 | Zbl 0292.26011
[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. DOI 10.1016/s0034-4877(16)30007-6 | MR 1658022 | Zbl 0924.34008
[24] Robinson, S. M.: Stability theory for systems of inequalities. Part II. Differentiable nonlinear systems. SIAM J. Numerical Analysis 13 (1976), 497-513. DOI 10.1137/0713043 | MR 0410522
[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. DOI 10.1007/978-1-4020-6042-7 | MR 3184154
[26] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Gordan and Breach Science Publishers 1993. MR 1347689 | Zbl 0818.26003
[27] Sikora, B.: Controllability of time-delay fractional systems with and without constraints. IET Control Theory Appl. 10 (2016), 320-327. DOI 10.1049/iet-cta.2015.0935 | MR 3468656
[28] Sikora, B.: Controllability criteria for time-delay fractional systems with a retarded state. Int. J. Applied Math. Computer Sci. 26 (2016), 521-531. DOI 10.1515/amcs-2016-0036 | MR 3560625 | Zbl 1347.93057
[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. DOI 10.1016/j.ejbas.2014.03.001
[30] Wei, J.: The controllability of fractional control systems with control delay. Comput. Math. Appl. 64 (2012), 3153-3159. DOI 10.1016/j.camwa.2012.02.065 | MR 2989343 | Zbl 1268.93027
[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. DOI 10.2478/amcs-2014-0063 | MR 3309453 | Zbl 1309.93076
[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. DOI 10.1155/2013/146010 | MR 3064923 | Zbl 1271.93028
Partner of
EuDML logo