In this paper, we study mixed traffic flow in which some cars are under car-following control and others are under bilateral control. We also provide the necessary (modular string) stability condition for this type of mixed traffic, which can be viewed as an extension of the condition for pure car-following control based traffic. This necessary stability condition provides some indication of how the introduction of self-driving cars (under bilateral control) will affect today's traffic.

Traffic congestion wastes fuel and commuters' time, and adds to CO₂ emissions. Stop-and-go traffic instabilities can be suppresses using bilateral control--which differs from ``car following'' and adaptive cruise control in that, counter-intuitively, it uses information about the following vehicle (as well as about the leading vehicle). Stability can be proven mathematically, and can be demonstrated in simulation. A physical analog of a sequence of vehicles using bilateral control is a chain of masses connected by springs and dampers--a system which is inherently stable, since it lacks an external energy source. Here, in order to further understand bilateral control and its capacity to suppress instabilities, we move from a microscopic view (interaction of individual vehicles) to a macroscopic view (densities and flow rates). This leads us to the damped wave equation governing traffic under bilateral control. That equation allows us to determine the speed of propagation of disturbances, as well as their rate of decay. The equation is also useful in fine tuning parameters of bilateral control systems.

Many authors have recognized that traffic under the traditional car‐following model (CFM) is subject to flow instabilities. A recent model achieves stability using bilateral control (BCM)—by looking both forward and backward [1]. (Looking back may be difficult or distracting for human drivers, but is not a problem for sensors.) We analyze the underlying systems of differential equations by studying their eigenvalues and eigenvectors under various boundary conditions. Simulations further confirm that bilateral control can avoid instabilities and reduce the chance of collisions.

In this paper, we study the chain stability of vehicles under bilateral control (BCM), and prove that vehicles under bilateral control are chain stable: i.e. $all$ input perturbations to the chain decay exponentially (with the length of the chain). Chain stability analysis tells us how vehicles under bilateral control will act in traffic when mixed with cars driven by human drivers. It shows that self-driving cars using bilateral control can reduce traffic flow instabilities in mixed traffic. Indeed, chains of BCM vehicles become perturbation-consuming dampers when inserted in traffic, since they split chains of human-driven vehicles and prevent perturbations from being transmitted from one chain of car-following cars to the next. Thus today's traffic can be improved greatly by the insertion of BCM vehicles. The simulation results validate the theoretical analysis.