Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces

Abstract This paper introduces a new concept called impulsive control inclusion condition, i.e., the impulsive condition is presented, in the first time, as inclusion related to multivalued maps and controls. The notion of approximate controllability of a class of semilinear Hilfer fractional differential control inclusions in Banach spaces is established. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. An appropriate set of sufficient conditions for the considered system to be approximately controllable is studied. Finally, we give an illustrated example to provide the obtained theory.

[1]  S. S. Ezz-Eldien,et al.  A new Legendre operational technique for delay fractional optimal control problems , 2016 .

[2]  Roberto Triggiani,et al.  A Note on the Lack of Exact Controllability for Mild Solutions in Banach Spaces , 1977 .

[3]  Juan J. Nieto,et al.  Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls , 2014, Appl. Math. Comput..

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

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

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

[7]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[8]  JinRong Wang,et al.  Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces , 2015, Appl. Math. Comput..

[9]  Zuomao Yan,et al.  Approximate controllability of impulsive fractional stochastic partial neutral integrodifferential inclusions with infinite delay , 2016, IMA J. Math. Control. Inf..

[10]  Yong Zhou,et al.  Controllability for fractional evolution inclusions without compactness , 2015 .

[11]  N. Mahmudov,et al.  Approximate Controllability of Second-Order Evolution Differential Inclusions in Hilbert Spaces , 2015, 1502.00080.

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

[13]  R. Sakthivel,et al.  Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions , 2016 .

[14]  Juan J. Trujillo,et al.  Existence of mild solution for evolution equation with Hilfer fractional derivative , 2015, Appl. Math. Comput..

[15]  JinRong Wang,et al.  Nonlocal initial value problems for differential equations with Hilfer fractional derivative , 2015, Appl. Math. Comput..

[16]  Delfim F. M. Torres,et al.  Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces , 2013, Int. J. Control.

[17]  Yong Zhou,et al.  Fractional Evolution Equations and Inclusions: Analysis and Control , 2016 .

[18]  Soovoojeet Jana,et al.  Optimal control and stability analysis of an epidemic model with population dispersal , 2016 .

[19]  Xianlong Fu,et al.  Approximate controllability for neutral impulsive differential inclusions with nonlocal conditions , 2011 .

[20]  Delfim F. M. Torres,et al.  Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions , 2014, Appl. Math. Comput..

[21]  Zuomao Yan,et al.  Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay , 2017, Appl. Math. Comput..

[22]  Mouffak Benchohra,et al.  Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions , 2009 .

[23]  Michal Fečkan,et al.  A survey on impulsive fractional differential equations , 2016 .

[24]  Min Yang,et al.  Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions , 2017 .

[25]  Nasser-eddine Tatar,et al.  Existence and uniqueness for a problem involving Hilfer fractional derivative , 2012, Comput. Math. Appl..

[26]  R. Agarwal,et al.  A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , 2010 .

[27]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[28]  Sohrab Effati,et al.  A Neural Network Approach for Solving a Class of Fractional Optimal Control Problems , 2017, Neural Processing Letters.

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

[30]  Agnieszka B. Malinowska,et al.  Introduction to the Fractional Calculus of Variations , 2012 .

[31]  Ravi P. Agarwal,et al.  Nonlocal nonlinear integrodifferential equations of fractional orders , 2012 .

[32]  Jinrong Wang,et al.  Analysis of nonlinear Hadamard fractional differential equations via properties of Mittag–Leffler functions , 2016 .

[33]  Ilknur Koca,et al.  Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order , 2016 .

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

[35]  Nikolaos S. Papageorgiou.  On the theory of Banach space valued multifunctions. 1. Integration and conditional expectation , 1985 .

[36]  K. Deimling.  Multivalued Differential Equations , 1992 .

[37]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[38]  Rabah Rabah,et al.  Exact observability and controllability for linear neutral type systems , 2016, Syst. Control. Lett..