Controllability of Sobolev type fractional evolution systems

The main purpose of this paper is to investigate a class of Sobolev type semilinear fractional evolution systems in a separable Banach space. Applying a suitable fixed point theorem as well as condensing mapping, controllability results for two class of control sets are established by means of the theory of propagation family and technique of measure of noncompactness. An application involving a partial differential equation with a Caupto fractional derivative is considered.

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

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

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

[4]  Rathinasamy Sakthivel,et al.  Controllability for a class of fractional-order neutral evolution control systems , 2012, Appl. Math. Comput..

[5]  Valeri Obukhovskii,et al.  Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces , 2011 .

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

[7]  JinRong Wang,et al.  Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations , 2012, Journal of Optimization Theory and Applications.

[8]  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 .

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

[10]  Krishnan Balachandran,et al.  Relative controllability of fractional dynamical systems with delays in control , 2012 .

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

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

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

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

[15]  Jin Liang,et al.  Abstract Degenerate Cauchy Problems in Locally Convex Spaces , 2001 .

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

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

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

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

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

[21]  JinRong Wang,et al.  Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators , 2013, J. Optim. Theory Appl..

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

[23]  Rathinasamy Sakthivel,et al.  Approximate controllability of fractional stochastic evolution equations , 2012, Comput. Math. Appl..

[24]  Dhirendra Bahuguna,et al.  Existence of solutions to Sobolev-type partial neutraldifferential equations , 2006 .

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

[26]  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.

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

[28]  Krishnan Balachandran,et al.  On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces , 2011, Comput. Math. Appl..

[29]  N. Lloyd.  MEASURES OF NONCOMPACTNESS IN BANACH SPACES , 1981 .

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

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

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

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

[34]  A. Rodkina,et al.  Measures of noncompactness and condensing operators , 1992 .

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

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

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

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

[39]  Colin Atkinson,et al.  Rational Solutions for the Time-Fractional Diffusion Equation , 2011, SIAM J. Appl. Math..

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