In this paper we study the problem of minimizing the weighted amplitude of the time response dire to a given fixed input signal for single-input/single-output (SISO) continuous-time systems and focus on obtaining rational suboptimal solutions, An EAS (Euler approximating system)-based method is proposed for designing a rational L(x)-suboptimal controller for SISO systems, It is shown that this rational approximation is the best one among a set of rational approximations, in the sense of providing the tightest upper bound and that it can approximate the optimal cost arbitrarily close.