Previous |  Up |  Next

Article

Title: Relative controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control (English)
Author: Joice Nirmala, Rajagopal
Author: Balachandran, Krishnan
Language: English
Journal: Kybernetika
ISSN: 0023-5954 (print)
ISSN: 1805-949X (online)
Volume: 53
Issue: 1
Year: 2017
Pages: 161-178
Summary lang: English
.
Category: math
.
Summary: This paper describes the controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control. Necessary and sufficient conditions for the controllability criteria for linear fractional delay system are established. Further sufficient conditions for the controllability of nonlinear fractional delay integrodifferential system are obtained by using fixed point arguments. Examples are provided to illustrate the results. (English)
Keyword: fractional delay integrodifferential equation
Keyword: Laplace transform
Keyword: controllability
Keyword: Mittag–Leffler function
Keyword: Caputo fractional derivative
MSC: 34A08
MSC: 93B05
idZBL: Zbl 06738600
idMR: MR3638562
DOI: 10.14736/kyb-2017-1-0161
.
Date available: 2017-04-03T10:54:25Z
Last updated: 2018-01-10
Stable URL: http://hdl.handle.net/10338.dmlcz/146714
.
Reference: [1] Bagley, R. L., Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity..J. Rheol. 27 (1983), 201-210. Zbl 0515.76012, 10.1122/1.549724
Reference: [2] Bagley, R. L., Torvik, P. J.: Fractional calculus in the transient analysis of viscoelastically damped structures..AIAA J. 23 (1985), 918-925. Zbl 0562.73071, 10.2514/3.9007
Reference: [3] Balachandran, K.: Global relative controllability of non-linear systems with time-varying multiple delays in control..Int. J. Control. 46 (1987), 193-200. MR 0895702, 10.1080/00207178708933892
Reference: [4] Balachandran, K., Dauer, J. P.: Controllability of perturbed nonlinear delay systems..IEEE Trans. Autom. Control. 32(1987), 172-174. Zbl 0614.93011, MR 0872593, 10.1109/tac.1987.1104536
Reference: [5] Balachandran, K., Kokila, J., Trujillo, J.J.: Relative controllability of fractional dynamical systems with multiple delays in control..Comput. Math. Appl. 64 (2012), 3037-3045. Zbl 1268.93021, MR 2989332, 10.1016/j.camwa.2012.01.071
Reference: [6] Balachandran, K., Zhou, Y., Kokila, J.: Relative controllability of fractional dynamical systems with delays in control..Commun. Nonlinear. Sci. Numer. Simul. 17 (2012), 3508-3520. Zbl 1248.93022, MR 2913988, 10.1016/j.cnsns.2011.12.018
Reference: [7] Balachandran, K., Zhou, Y., Kokila, J.: Relative controllability of fractional dynamical systems with distributive delays in control..Comput. Math. Appl. 64(2012), 3201-3209. MR 2989348, 10.1016/j.camwa.2011.11.061
Reference: [8] Bellman, R., Cooke, K. L.: Differential Difference Equations..Academic Press, New York 1963. Zbl 0163.10501, MR 0147745, 10.1002/zamm.19650450612
Reference: [9] Chow, T. S.: Fractional dynamics of interfaces between soft-nanoparticles and rough substrates..Physics Letter A 342 (2005), 148-155. 10.1016/j.physleta.2005.05.045
Reference: [10] Dauer, J. P., Gahl, R. D.: Controllability of nonlinear delay systems..J. Optimiz. Theory. App. 21 (1977), 59-68. Zbl 0325.93007, MR 0433306, 10.1007/bf00932544
Reference: [11] Dauer, J. P.: Nonlinear perturbations of quasi-linear control systems..J. Math. Anal. Appl. 54 (1976), 717-725. Zbl 0339.93004, MR 0415473, 10.1016/0022-247x(76)90191-8
Reference: [12] Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags..Academic Press, New York 1966. Zbl 0144.08701, MR 0216103, 10.1016/s0076-5392(08)x6057-6
Reference: [13] Hale, J.: Theory of Functional Differential Equations..Springer, New York 1977. Zbl 0662.34064, MR 0508721, 10.1007/978-1-4612-9892-2
Reference: [14] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media..Comput. Methods. Appl. Mech. Eng. 167 (1998), 57-68. Zbl 0942.76077, MR 1665221, 10.1016/s0045-7825(98)00108-x
Reference: [15] Nirmala, R. Joice, Balachandran, K.: Controllability of nonlinear fractional delay integrodifferential systems..J. Applied Nonlinear Dynamics 5 (2016), 59-73. MR 3577640, 10.5890/dnc.2016.03.007
Reference: [16] Nirmala, R. Joice, Balachandran, K., Germa, L. R., Trujillo, J. J.: Controllability of nonlinear fractional delay dynamical systems..Rep. Math. Phys. 77 (2016), 87-104. MR 3461800, 10.1016/s0034-4877(16)30007-6
Reference: [17] Kaczorek, T.: Selected Problems of Fractional Systems Theory: Lecture Notes in Control and Information Science..Springer-Verlag, Berlin 2011. MR 2798773, 10.1007/978-3-642-20502-6
Reference: [18] Klamka, J.: Controllability of linear systems with time variable delay in control..Int. J. Control 24(1976), 869-878. MR 0424300, 10.1080/00207177608932867
Reference: [19] Klamka, J.: Relative controllability of nonlinear systems with delay in control..Automatica 12(1976), 633-634. MR 0452869, 10.1016/0005-1098(76)90046-7
Reference: [20] Kilbas, A., Srivastava, H. M., Trujillo, J. J.: Theory and Application of Fractional Differential Equations..Elsevier, Amsterdam 2006. MR 2218073
Reference: [21] Machado, J. T.: Analysis and design of fractional order digital control systems..Systems Analysis, Modelling and Simulation 27 (1997), 107-122. Zbl 0875.93154
Reference: [22] Machado, J. T., Costa, A. C., Quelhas, M. D.: Fractional dynamics in DNA..Commun. Nonlinear. Sci. Numer. Simul. 16 (2011), 2963-2969. Zbl 1218.92038, 10.1016/j.cnsns.2010.11.007
Reference: [23] Magin, R. L: Fractional calculus in bioengineering..Critical Rev. Biomed. Eng. 32 (2004), 1-377. 10.1615/critrevbiomedeng.v32.i1.10
Reference: [24] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics..In: Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, eds.), Springer-Verlag 1997, pp. 291-348. MR 1611587, 10.1007/978-3-7091-2664-6_7
Reference: [25] Manzanilla, R., Marmol, L. G., Vanegas, C. J.: On the controllability of differential equation with delayed and advanced arguments..Abstr. Appl. Anal. 2010 (2010), 1-16. MR 2660394, 10.1155/2010/307409
Reference: [26] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations..Wiley and Sons, New York 1993. Zbl 0789.26002, MR 1219954
Reference: [27] Mur, T., Henriquez, H. R.: Relative controllability of linear systems of fractional order with delay..Math. Control. Relat. F 5(2015), 845-858. Zbl 1332.93061, MR 3485753, 10.3934/mcrf.2015.5.845
Reference: [28] Oguztoreli, M. N.: Time-Lag Control Systems..Academic Press, New York 1966. Zbl 0143.12101, MR 0217394, 10.1016/s0076-5392(08)x6192-2
Reference: [29] Oldham, K. B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order..Academic Press, New York 1974. MR 0361633, 10.1016/s0076-5392(09)x6012-1
Reference: [30] Ortigueira, M. D.: On the initial conditions in continuous time fractional linear systems..Signal Process 83 (2003), 2301-2309. Zbl 1145.94367, 10.1016/s0165-1684(03)00183-x
Reference: [31] Podlubny, I.: Fractional Differential Equations..Academic Press, New York 1999. Zbl 1160.65308, MR 1658022
Reference: [32] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations to Methods of their Solution and Some of their Applications..Academic Press, 1999. Zbl 0924.34008, MR 1658022, 10.1016/s0076-5392(99)x8001-5
Reference: [33] Sabatier, J., Agrawal, O. P., (eds.), J. A. Tenreiro-Machado: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering..Springer-Verlag, New York 2007. Zbl 1116.00014, MR 3184154, 10.1007/978-1-4020-6042-7
Reference: [34] Schiff, J. L.: The Laplace Transform: Theory and Applications..Springer, New York 1999. Zbl 0934.44001, MR 1716143, 10.1007/978-0-387-22757-3
Reference: [35] 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: [36] Smith, H.: An Introduction to Delay Differential Equations with Application to the Life Sciences..Springer, New York 2011. MR 2724792, 10.1007/978-1-4419-7646-8
Reference: [37] 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
.

Files

Files Size Format View
Kybernetika_53-2017-1_9.pdf 367.1Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo