This paper focuses on the design of H-infinity memoryless state feedback controllers satisfying some a-stability constraints on the closed-loop poles for a class of linear systems with delayed state, subject to disturbance inputs. A sufficient condition for feasibility is derived in terms of linear matrix inequalities. Furthermore, we also consider a particular optimization problem : the maximal bound allowed on time-delay such that the prescribed level on disturbance attenuation and the a-stability constraints are still preserved. This problem is conversed into a linear matrix inequality (LMT) optimization problem and we give a suboptimal bound on the delay.