Based on the stabilizability of a nominal system (i.e. a system in the absence of uncertainty), by making use of the Lyapunov stability criterion and combining with the algebraic Riccati equation, a new approach for designing a robust linear state feedback controller for uncertain linear dynamical systems is presented. Using this approach, the BIBO stability of uncertain linear dynamical systems is also discussed, Some analytical methods and the Bellman-Gronwall inequality are employed to investigate the robust stabilization conditions on the feedback controller. The main features of this approach are that no matching condition about uncertainty is needed and the uncertain systems can be asymptotically stabilized. An example is given to demonstrate the validity of our results.