Partial-approximate controllability of nonlocal fractional evolution equations via approximating method

In this paper we study partial-approximate controllability of semilinear nonlocal fractional evolution equations in Hilbert spaces. By using fractional calculus, variational approach and approximating technique, we give the approximate problem of the control system and get the compactness of approximate solution set. Then new sufficient conditions for the partial-approximate controllability of the control system are obtained when the compactness conditions or Lipschitz conditions for the nonlocal function are not required. Finally, we apply our abstract results to the partial-approximate controllability of the semilinear heat equation and delay equation.