In this paper, we investigate a robust $H_{infty}$ state feedback control technique for continuous time bilinear systems with an additive disturbance input. The nonlinear robust $H_{infty}$control for bilinear systems requires a solution to the state dependent algebraic Riccati equation (SDARE). We present a new robust $H_{infty}$control technique based on the successive approximation method for solving the SDARE by converting bilinear systems into time-varying linear systems. The proposed control method guarantees robust stability for closed loop bilinear systems. The proposed algorithm is verified by numerical examples.