This paper focuses on the linear quadratic control (LQC) design of systems corrupted by both stochastic noise and bounded noise simultaneously. When only of these noises are considered, the LQC strategy leads to stochastic or robust controllers, respectively. However, there is no LQC strategy that can simultaneously handle stochastic and bounded noises efficiently. This limits the scope where existing LQC strategies can be applied. In this work, we look into the LQC problem for discrete-time systems that have both stochastic and bounded noises in its dynamics. We develop a state estimation for such systems by efficiently combining a Kalman filter and an ellipsoid set-membership filter. The developed state estimation can recover the estimation optimality when the system is subject to both kinds of noise, the stochastic and the bounded. Upon the estimated state, we derive a robust state-feedback optimal control law for the LQC problem. The control law derivation takes into account both stochastic and bounded-state estimation errors, so as to avoid over-conservativeness while sustaining stability in the control. In this way, the developed LQC strategy extends the range of scenarios where LQC can be applied, especially those of real-world control systems with diverse sensing which are subject to different kinds of noise. We present numerical simulations, and the results demonstrate the enhanced control performance with the proposed strategy.