Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

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.

[1]  Yong Zhou,et al.  Existence of mild solutions for fractional neutral evolution equations , 2010, Comput. Math. Appl..

[2]  Mário N. Berberan-Santos,et al.  Properties of the Mittag-Leffler Relaxation Function , 2005 .

[3]  Yong Zhou,et al.  A class of fractional evolution equations and optimal controls , 2011 .

[4]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[5]  JinRong Wang,et al.  Nonlocal Controllability of Semilinear Dynamic Systems with Fractional Derivative in Banach Spaces , 2012, J. Optim. Theory Appl..

[6]  Nagarajan Sukavanam,et al.  Approximate controllability of fractional order semilinear systems with bounded delay , 2012 .

[7]  Dumitru Baleanu,et al.  Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems , 2011, Comput. Math. Appl..

[8]  Jin Liang,et al.  Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions , 2012 .

[9]  Yong Zhou,et al.  Existence and controllability results for fractional semilinear differential inclusions , 2011 .

[10]  D. O’Regan,et al.  On recent developments in the theory of abstract differential equations with fractional derivatives , 2010 .

[11]  Rong-Nian Wang,et al.  Abstract fractional Cauchy problems with almost sectorial operators , 2012 .

[12]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[13]  M. El-Borai.  Some probability densities and fundamental solutions of fractional evolution equations , 2002 .

[14]  Rathinasamy Sakthivel,et al.  On the approximate controllability of semilinear fractional differential systems , 2011, Comput. Math. Appl..

[15]  Jigen Peng,et al.  Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩ , 2012 .

[16]  Yong Zhou,et al.  Complete controllability of fractional evolution systems , 2012 .

[17]  Yong Zhou,et al.  Analysis of nonlinear fractional control systems in Banach spaces , 2011 .

[18]  I. Podlubny.  Fractional differential equations , 1998 .

[19]  Mahmoud M. El-Borai,et al.  The fundamental solutions for fractional evolution equations of parabolic type , 2004 .

[20]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[21]  Yong Zhou,et al.  Nonlocal Cauchy problem for fractional evolution equations , 2010 .

[22]  Mário N. Berberan-Santos,et al.  Relation between the inverse Laplace transforms of I(tβ) and I(t): Application to the Mittag-Leffler and asymptotic inverse power law relaxation functions , 2005 .

[23]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[24]  JinRong Wang,et al.  Optimal feedback control for semilinear fractional evolution equations in Banach spaces , 2012, Syst. Control. Lett..

[25]  Yong Zhou,et al.  On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay , 2011, Journal of Optimization Theory and Applications.

[26]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[27]  K. Balachandran,et al.  Controllability of fractional integrodifferential systems in Banach spaces , 2009 .

[28]  Samuel M Rankin,et al.  A partial functional differential equation of Sobolev type , 1983 .

[29]  Yong Zhou,et al.  Fractional Schrödinger equations with potential and optimal controls , 2012 .

[30]  Krishnan Balachandran,et al.  Controllability of functional differential systems of Sobolev type in Banach spaces , 1998, Kybernetika.

[31]  JinRong Wang,et al.  Mittag-Leffler-Ulam stabilities of fractional evolution equations , 2012, Appl. Math. Lett..