In this paper, to synthesize stabilizing controllers for nonlinear systems we develop nonlinear controller design methods based on polynomial representation and nonpolynomial representation using particle swarm optimization( PSO). To do this we derive the stability conditions of polynomial systems and non-polynomial systems for estimating domain of attraction(DOA) with input magnitude constraints. From these conditions stabilizing controllers of polynomial systems and non-polynomial systems are synthesized by using sum of squares (SOS) and PSO techniques respectively. Finally, obtained controllers are applied to Furuta pendulum and validity of these controllers is demonstrated by numerical simulations.