On Controllability For Sobolev Type Hilfer Fractional Evolution Equations In Hilbert Spaces

Fractional differential equations and its applications have gained much importance during the past three decades,mainly because fractional differential equations has become a powerful tool in modeling several complex phenomena of science and engineering.Controllability is one of the important fundamental concepts in mathematical control theory and plays a vital role in both deterministic and stochastic control systems.In this paper,we study controllability for Sobolev type Hilfer fractional evolution equations in Hilbert spaces.In section 1,we introduce the background and content of research.In section 2,we give some definitions and lemmas,which will be used to obtain our main results.In section 3,we discuss the approximate controllability of Hilfer fractional nonautonomous evolution equations of Sobolev type with delay.The sufficient conditions for the approximate controllability of the control system are obtained with the help of the theory of fractional calculus,semigroup theory and the Schauder fixed point theorem under the assumption that the corresponding linear system is approximately controllable.In section 4,we discuss the finite-approximate controllability of Hilfer fractional evolution equations of Sobolev type with nonlocal conditions.The sufficient conditions for the finiteapproximate controllability of the control system are obtained with the help of variational approach and the Schaefer fixed point theorem under the assumption that the corresponding linear system is approximately controllable.In section 5,we discuss the partial-approximate controllability of Hilfer fractional evolution equations of Sobolev type with nonlocal conditions.By using variational approach and approximating method,we give the approximate solution operator of the control system and get the compactness of approximate solution set,the sufficient conditions for the partial-approximate controllability of the control system are obtained when Lipschitz conditions for the nonlocal function are not required.The obtained results are more general to known results on this issue.