In this paper we present a Lyapunov based feedback design strategy, by employing the sum-of-squares polynomials framework, to maximize the bootstrap current in tokamaks. The bootstrap current may play an important role in reducing the external energy input required for tokamak operation. The sum-of-squares polynomials framework allows us to algorithmically construct controllers. Additionally, we provide a heuristic to take into account the control input shape constraints which arise due to limitations on the actuators.