×

zbMATH — the first resource for mathematics

Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. (English) Zbl 1263.93031
Summary: The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed-point theorem. Finally, an example is given to illustrate our theoretical results.

MSC:
93B05 Controllability
35R11 Fractional partial differential equations
47H10 Fixed-point theorems
93C25 Control/observation systems in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York (2010) · Zbl 1215.34001
[2] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[3] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[4] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[5] Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media Springer, New York (2010) · Zbl 1214.81004
[6] El-Borai, M.M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals, 14, 433-440, (2002) · Zbl 1005.34051
[7] El-Borai, M.M., The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 3, 197-211, (2004) · Zbl 1081.34053
[8] Balachandran, K.; Park, J.Y., Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal. Hybrid Syst., 3, 363-367, (2009) · Zbl 1175.93028
[9] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59, 1063-1077, (2010) · Zbl 1189.34154
[10] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., Real World Appl., 11, 4465-4475, (2010) · Zbl 1260.34017
[11] Hernández, E.; O’Regan, D.; Balachandran, K., On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., Theory Methods Appl., 73, 3462-3471, (2010) · Zbl 1229.34004
[12] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear Anal., Real World Appl., 12, 262-272, (2011) · Zbl 1214.34010
[13] Wang, J.; Zhou, Y., Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal., Real World Appl., 12, 3642-3653, (2011) · Zbl 1231.34108
[14] Wang, J.; Zhou, Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., Theory Methods Appl., 74, 5929-5942, (2011) · Zbl 1223.93059
[15] Sakthivel, R.; Ren, Y.; Mahmudov, N.I., On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl., 62, 1451-1459, (2011) · Zbl 1228.34093
[16] Debbouchea, A.; Baleanu, D., Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62, 1442-1450, (2011) · Zbl 1228.45013
[17] Wang, J.; Zhou, Y.; Medved’, M., On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. Optim. Theory Appl., 152, 31-50, (2012) · Zbl 1357.49018
[18] Wang, J.; Zhou, Y., Mittag-leffer-Ulam stabilities of fractional evolution equations, Appl. Math. Lett., 25, 723-728, (2012) · Zbl 1246.34012
[19] Wang, J.; Zhou, Y.; Wei, W., Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Control Lett., 61, 472-476, (2012) · Zbl 1250.49035
[20] Wang, J.; Fan, Z.; Zhou, Y., Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, J. Optim. Theory Appl., 154, 292-302, (2012) · Zbl 1252.93028
[21] Wang, J.; Zhou, Y., Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simul., 17, 4346-4355, (2012) · Zbl 1248.93029
[22] Wang, J.; Zhou, Y.; Wei, W., Fractional Schrödinger equations with potential and optimal controls, Nonlinear Anal., Real World Appl., 13, 2755-2766, (2012) · Zbl 1253.35205
[23] Wang, R.N.; Chen, D.H.; Xiao, T.J., Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equs., 252, 202-235, (2012) · Zbl 1238.34015
[24] Kumar, S.; Sukavanam, N., Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equs., 252, 6163-6174, (2012) · Zbl 1243.93018
[25] Li, K.; Peng, J.; Jia, J., Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263, 476-510, (2012) · Zbl 1266.47066
[26] Balachandran, K.; Dauer, J.P., Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika, 34, 349-357, (1998) · Zbl 1274.93031
[27] Li, F.; Liang, J.; Xu, H.K., Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 391, 510-525, (2012) · Zbl 1242.45009
[28] Lightbourne, J.H.; Rankin, S.M., A partial functional differential equation of Sobolev type, J. Math. Anal. Appl., 93, 328-337, (1983) · Zbl 0519.35074
[29] Berberan-Santos, M.N., Relation between the inverse Laplace transforms of \(I\)(\(t\)\^{}{\(β\)}) and \(I\)(\(t\)): application to the Mittag-Leffler and asymptotic inverse power law relaxation functions, J. Math. Chem., 38, 265-270, (2005) · Zbl 1217.44003
[30] Berberan-Santos, M.N., Properties of the Mittag-Leffler relaxation function, J. Math. Chem., 38, 629-635, (2005) · Zbl 1101.33015
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.