<p>In this paper, we study infinite horizon overtaking optimal control problems for fractional system involving linear quadratic performance index which is allowed to be unbounded. When the controllability or stabilizability condition is not assumed and the nonhomogeneous term and the weight function are global integrability, it is shown that the solution of the overtaking optimal control can be derived by solving the linear quadratic optimal control problem with the state equation and running cost rate function in the controllable subspace based on the orthogonal projection. When the nonhomogeneous term and the weight function are not global integrability, both the existence and nonexistence of solutions for overtaking optimal control in various cases are established by characterizing the weight function. Several examples are also included to verify the above-mentioned theory. There results reveal that the overtaking optimality approach can be adopted to solve some optimal control problems with unbounded performance index and at the same time, the limitation of this approach is also exposed.</p>