This paper proposes an approach to approximately solve the optimal tracking control problem of a class of continuous-time nonlinear systems with input constraints. For a class of single-input single-output nonlinear systems with the tracking cost function in finite time, the proposed optimal tracking control method will make the corresponding cost function of the target system approximately optimal. Firstly, the cost function of the optimal control problem is expanded along the time by Taylor series, and an approximate cost function is obtained. Then, based on the observability condition, the problem is further transformed into a nonlinear programming problem. Then, the neurodynamic approach is applied to solve the approximate optimal control problem, where the control input is the decision variable with inequality constraints and the corresponding recurrent neural network module is given. In addition, the stability of the closed-loop system is proved under certain conditions. Finally, two examples are provided to verify the effectiveness of the proposed method.