The time-varying network topology can significantly affect the stability of multi-agent systems. This paper examines the stability of leader–follower multi-agent systems with general linear dynamics and switching network topologies, which have applications in the platooning of connected vehicles. The switching interaction topology is modeled as a class of directed graphs in order to describe the information exchange between multi-agent systems, where the eigenvalues of every associated matrix are required to be positive real. The Hurwitz criterion and the Riccati inequality are used to design a distributed control law and estimate the convergence speed of the closed-loop system. A sufficient condition is provided for the stability of multi-agent systems under switching topologies. A common Lyapunov function is formulated to prove closed-loop stability for the directed network with switching topologies. The result is applied to a typical cyber–physical system—that is, a connected vehicle platoon—which illustrates the effectiveness of the proposed method.