Impulsive control functional differential systems of fractional order: stability with respect to manifolds
Tóm tắt
In this paper, we consider nonlinear impulsive functional differential systems involving Caputo fractional derivatives. Using vector multivariable piecewise continuous Lyapunov functions and a new vector fractional comparison principle, stability criteria with respect to a manifold are established. Some examples are presented in order to illustrate our theoretical findings.
Tài liệu tham khảo
D. Baleanu, O.G. Mustafa, Asymptotic integration and stability. For ordinary, functional and discrete differential equations of fractional order, in Series on complexity, nonlinearity and chaos (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015), Vol. 4
S.R. Bernfeld, C. Corduneanu, A.O. Ignatyev, Nonlinear Anal. 55, 641 (2003)
N.P. Bhatia, V. Lakshmikantham, Mich. Math. J. 12, 183 (1965)
B. Chen, J. Chen, Appl. Math. Comput. 254, 63 (2015)
K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, in Lecture notes in mathematics (Springer-Verlag, Berlin, 2010), Vol. 2004
J.K. Hale, S.M. Verduyn Lunel, Introduction to functional-differential equations, in Applied mathematical sciences (Springer-Verlag, New York, 1993), Vol. 99
X. Hei, R. Wu, Appl. Math. Model. 40, 4285 (2016)
R. Hilfer, Applications of fractional calculus in physics (World Scientific Publishing Co., Singapore, 2000)
G.K. Kulev, D.D. Baĭnov, J. Comput. Appl. Math. 23, 305 (1988)
V. Lakshmikantham, V.M. Matrosov, S. Sivasundaram, Vector Lyapunov functions and stability analysis of nonlinear systems, in Mathematics and its applications (Kluwer Academic Publishers Group, Dordrecht, 1991), Vol. 63
X. Li, M. Bohner, C.-K. Wang, Autom. J. IFAC 52, 173 (2015)
X. Li, S.Song, IEEE Trans. Automat. Control 62, 406 (2017)
Y. Li, Y.Q. Chen, I. Podlubny, Comput. Math. Appl. 59, 1810 (2010)
I. Podlubny, 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, Inc., San Diego, CA, 1999), Vol. 198
K. Shah, H. Khalil, R.A. Khan, Chaos Solitons Fractals 77, 240 (2015)
G. Stamov, Methods Appl. Anal. 16, 291 (2009)
I.M. Stamova, Q. Appl. Math. 73, 525 (2015)
I. Stamova, J. Henderson, ISA Trans. 64, 77 (2016)
I. Stamova, G. Stamov, Commun. Nonlinear Sci. Numer. Simul. 19, 702 (2014)
I. Stamova, G. Stamov, CMS books in mathematics/Ouvrages de Mathématiques de la SMC (Springer, Cham, 2016)
J. Suo, J. Sun, Autom. J. IFAC 51, 302 (2015)
G. Teschl, Ordinary differential equations and dynamical systems, in Graduate studies in mathematics (American Mathematical Society, Providence, RI, 2012), Vol. 140
H. Wang, J. Appl. Math. Comput. 38, 85 (2012)
J.R. Wang, M. Fečkan, Y. Zhou, J. Optim. Theory Appl. 156, 13 (2013)
L. Wang, L. Chen, J.J. Nieto, Nonlinear Anal. Real World Appl. 11, 1374 (2010)
R. Wu, M. Fečkan, Nonlinear Dyn. 82, 2007 (2015)
X. Yang, C. Li, T. Huang, Q. Song, Appl. Math. Comput. 293, 416 (2017)
K. Zhang, J. Xu, J. Nonlinear Sci. Appl. 9, 4628 (2016)
Y. Zhang, CSSP 35, 3882 (2016)
Y. Zhang, Math. Comput. Simul. 132, 183 (2017)
K. Zhao, P. Gong, Adv. Differ. Equ. 255, 19 (2014)