In this paper, a continuous-time adaptive actor-critic reinforcement learning (RL) controller is developed for drift-free nonlinear systems. Practical examples of such systems are image-based visual servoing (IBVS) and wheeled mobile robots (WMR), where the system dynamics includes a parametric uncertainty in the control effectiveness matrix with no drift term. The uncertainty in the input term poses a challenge for developing a continuous-time RL controller using existing methods. In this paper, an actor-critic or synchronous policy iteration (PI)-based RL controller is presented with a concurrent learning (CL)-based parameter update law for estimating the unknown parameters of the control effectiveness matrix. An infinite-horizon value function minimization objective is achieved by regulating the current states to the desired with near-optimal control efforts. The proposed controller guarantees closed-loop stability and simulation results validate the proposed theory using IBVS and WMR examples.