In this paper, we are concerned with the existence of mild solution and controllability for a class of nonlinear fractional control systems with damping in Hilbert spaces. Our first step is to give the representation of mild solution for this control system by utilizing the general method of Laplace transform and the theory of (α,γ)-regularized families of operators. Next, we study the solvability and controllability of nonlinear fractional control systems with damping under some suitable sufficient conditions. Finally, two examples are given to illustrate the theory.
Xiuwen LI
,
Zhenhai LIU
,
Jing LI
,
Chris TISDELL
. EXISTENCE AND CONTROLLABILITY FOR NONLINEAR FRACTIONAL CONTROL SYSTEMS WITH DAMPING IN HILBERT SPACES[J]. Acta mathematica scientia, Series B, 2019
, 39(1)
: 229
-242
.
DOI: 10.1007/s10473-019-0118-5
[1] Arendt W, Batty C, Hieber M, Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, Vol 96. Basel:Birkhäuser, 2001
[2] Balachandran K, Govindaraj V, Reiver M, Trujillo J J. Controllability of fractional damped dynamical systems. Appl Math Comput, 2015, 257:66-73
[3] Baleanu D, Golmankhaneh A K. On electromagnetic field in fractional space. Nonlinear Anal RWA, 2010, 11(1):288-292
[4] Curtain R F, Zwart H J. An Introduction to Infinite Dimensional Linear Systems Theorem. New York:Springer-Verlag, 1995
[5] Debbouche A, Torres D F M. Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions. Appl Math Comput, 2014, 243:161-175.
[6] Galucio A C, Deu J F, Ohayon R. A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain-application to sandwich beams. J Intell Mater Syst Struct, 2005, 16(1):33-45
[7] Heymans N, Podlubny I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol Acta, 2006, 45:765-771
[8] Hilfer R. Applications of Fractional Calculus in Physics. Singapore:World Scientific Publ Co, 2000
[9] Jia J H, Shen X Y, Hua H X. Viscoelastic behavior analysis and application of the fractional derivative Maxwell model. J Vib Control, 2007, 13(4):385-401
[10] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol 204. Amsterdam:Elservier Science BV, 2006
[11] Koeller R C. Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech, 1984, 51(2):299-307
[12] Kumar S, Sukavanam N. Approximate controllability of fractional order semilinear systems with bounded delay. J Diff Equat, 2012, 252:6163-6174
[13] Li J, Liu F, Feng L, Turner I. A novel finite volume method for the Riesz space distributed-order diffusion equation. Comput Math Appl, 2017, 74:772-783
[14] Li X W, Liu Z H, Tisdell C C. Existence and exact controllability of fractional evolution inclusions with damping. Math Meth Appl Sci, 2017, 40(12):4548-4559
[15] Liu X Y, Liu Z H, Fu X. Relaxation in nonconvex optimal control problems described by fractional differential equations. J Math Anal Appl, 2014, 409(1):446-458
[16] Liu Z H, Li X W. On the controllability of impulsive fractional evolution inclusions in Banach spaces. J Optim Theory Appl, 2013, 156:167-182
[17] Liu Z H, Li X W. Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J Control Optim, 2015, 53(4):1920-1933
[18] Liu Z H, Zeng S D, Bai Y R. Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications. Fract Calc Appl Anal, 2016, 19(1):188-211
[19] Lizama C. An operator theoretical approach to a class of fractional order differential equations. Appl Math Lett, 2011, 24:184-190
[20] Mei Z D, Peng J G. Riemann-Liouville abstract fractional Cauchy problem with damping. Indagat Math, 2014, 25:145-161
[21] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag, 1983
[22] Podlubny I. Fractional Differential Equations. San Diego:Academic Press, 1999
[23] Rüdinger F. Tuned mass damper with fractional derivative damping. Eng Struct, 2006, 28(13):1774-1779
[24] Wang J, Fěckan M, Zhou Y. Center stable manifold for planar fractional damped equations. Appl Math Comput, 2017, 296:257-269
[25] Liu Z H, Zeng S D. Differential variational inequalities in infinite Banach spaces. Acta Math Sci, 2017, 37B(1):26-32
