This paper treats the control problem of a class of monodimensional (1D) hyperbolic differential models with nonlinear components by using the boundary controller, the state measuring, and the control action on the boundary of the system. This controller is easy to implement from point of view of measuring techniques and actuation. The proposed algorithm provides the exponential convergence to the desired reference trajectory and rejects the effect of the nonlinear components by using the constraints in state space. A maximum principle of this class of system is inferred in order to evaluate the effect of boundary control. A constructive Lyapunov-based proof of convergence of the control algorithm is carried out. Numerical simulations of a technical model are presented.