This paper deals with the problem of designing a robust controller with an adaptation mechanism in order to reduce conservatism inherent in a robust controller with a fixed gain. It is assumed that the linear state-space model of the plant contains unknown parameters which vary within certain prescribed ranges. To begin with, a parameter-dependent state feedback gain is constructed so as to minimize an upper bound of an appropriately selected quadratic cost function. This design procedure can be reduced to a convex optimization problem. Next, an adaptive law to adjust parameters in the parameter-dependent state feedback gain is presented, by which the adjustable parameters are selected on-line on the boundary surface of the possible parameter space. As a result, asymptotic stability of the proposed control system is ensured. The effectiveness of the proposed method is shown through some simulation results.