Approximate controllability of nonlinear fractional dynamical systems

In this paper, we consider the controllability problems for a class of nonlinear fractional differential equations of order 1 < q < 2 with nonlocal conditions. In particular, a set of sufficient conditions are derived for the approximate controllability of nonlinear fractional dynamical systems by assuming the associated linear system is approximately controllable. Further, the result is extended to study the approximate controllability result for the nonlocal fractional control system with infinite delay. Also, as a remark, the conditions for the exact controllability results are obtained. The results are established by using solution operator theory, fractional calculations and fixed point techniques. Finally, an example is provided to illustrate the obtained theory.