×

Some results of the degenerate fractional differential system with delay. (English) Zbl 1228.34023

Summary: An analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented. We discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems. The exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained. Finally, an example is provided to illustrate the effectiveness of the presented analytical approaches.

MSC:

34A08 Fractional ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Miller, K. S.; Boss, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002
[2] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[3] Podlubny, I., Fractional Differential Equations (1999), San Diego Academic Press · Zbl 0918.34010
[4] Kilbas, A. A.; Hari, M.; Srivastava, J.; Trujillo, Juan, Theory and Applications of Fractional Differential Equations (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam · Zbl 1092.45003
[5] Das, Shantanu, Functional Fractional Calculus for System Identification and Controls (2008), Springer Verlag: Springer Verlag Berlin Heidelberg · Zbl 1154.26007
[6] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of Fractional Dynamic Systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002
[7] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Analysis, 69, 2677-2682 (2008) · Zbl 1161.34001
[8] Lakhmikantham, V., Theory of fractional functional differential equations, Nonlinear Analysis, 69, 3337-3343 (2008) · Zbl 1162.34344
[9] Bonilla, B.; Rivero, M.; Trujillo, J. J., On systems of linear fractional differential equations with constant coefficients, Applied Mathematics and Computation, 187, 68-78 (2007) · Zbl 1121.34006
[10] Odibat, Zaid M., Analytic study on linear systems of fractional differential equations, Computers and Mathematics with Applications, 59, 1171-1183 (2010) · Zbl 1189.34017
[11] P Lazarevic, Mihailo; Spasic, Aleksandar M., Finite time stability analysis of fractional order time delay systems: Gronwall’s approach, Mathematical and Computer Modeling, 49, 475-481 (2009) · Zbl 1165.34408
[12] Zhang, Xiuyun, Some results of linear fractional order time-delay system, Applied Mathematics and Computation, 197, 407-411 (2008) · Zbl 1138.34328
[13] zhou, Yong; Jiao, Feng; Li, Jing, Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Analysis, 71, 7-8, 2724-2733 (2009) · Zbl 1175.34082
[14] Zhou, Yong; Jiao, Feng, Existence of mild solutions for fractional neutral evolution equations, Computers and Mathematics with Applications, 59, 3, 1063-1077 (2010) · Zbl 1189.34154
[15] Wei, Jiang, The constant variation formulae for singular fractional differential systems with delay, Computers and Mathematics with Applications, 59, 3, 1184-1190 (2010) · Zbl 1189.34153
[16] Kunkel, Peter; Mehrmann, Volker, Differential Algebraic Equations (2006), European Mathematical Society · Zbl 1095.34004
[17] Dai, L., Singular Control Systems (1989), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0669.93034
[18] Campbell, S. L., Singular Systems of Differential Equations (1980), Pitman advanced publishing Program: Pitman advanced publishing Program Sanfrancisco London Melbourne · Zbl 0419.34007
[19] Campbell, S. L.; Linh, Vu Hong, Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions, Applied Mathematics and Computation, 208, 397-415 (2009) · Zbl 1169.65079
[20] Hale, Jack K.; Verduyn Lunel, Sjoerd M., Introduction to Functional Differential Equations (1992), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0787.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.