In this paper, we consider fractional systems with quadratic cost functional over the infinite time horizon. By using the Mittag-Leffler matrix function and its properties, we show that if $\alpha\in(0,1/2]$, the finiteness of the cost functional implies the initial value must be equal to zero which results in that there is no optimal control whence the initial value is not zero, while the cost functional must be finite if $\alpha\in(1/2,1)$. Moreover, the existence and uniqueness of the optimal control for $\alpha\in(1/2,1)$ is characterized by maximum principle type necessary conditions, and based on this characterization, an additional interesting finding that the optimal control for linear quadratic optimal control of the fractional system cannot be obtained by linear feedback with a constant gain satisfying an algebraic Riccatic equation is concluded.