This paper presents the design and simulation test of an automated vehicle string longitudinal control. The vehicle string is constituted of three highly cooperative vehicles following a leader. The string follows a leading vehicle which is supposed manually driven or at least not cooperative. The first vehicle of the string uses only its on-board sensor to obtain relative speed and inter-distance to the leader. Based on Lyapunov theory and invariant sets, the control law of the string is able to guarantee that the state trajectories remain bounded under bounded disturbance inputs, such as leading vehicle acceleration. The control law is designed in such a way that variable time headway strategy is naturally handled. The control law is computed using Linear and Bilinear Matrix Inequalities (LMI-BMI) methods. Some design parameters can be adjusted to handle the trade-off between safety constraints and comfort specifications.