Microscopic models describe the behavior of individual vehicles within a traffic stream. Analyzing the stability of such models, particularly under the influence of cruise control systems, is a challenging task. These systems are inherently nonlinear and evolve on sets that are not linear spaces while the set of equilibrium points may be non-compact and may lack uniform attractivity properties.

In [J8] and [J9] we dealt with the two-dimensional movement of automated vehicles on lane-free roads whose movement is described by the bicycle kinematic model:

The bicycle kinematic model was used to model the dynamics of the vehicles and a Control Lyapunov methodology was employed to design the Cruise Controllers. The Lyapunov functions are based on measures of the energy of the system, with the kinetic energy expressed in ways similar to Newtonian or relativistic mechanics. 

It was proved  that the closed-loop system is well-posed, i.e., its solution is well-defined for all times. Namely, it was shown that the proposed controllers guarantee that  vehicles do not collide with each other or with the boundary of the road, their speeds are always positive and remain within the road speed limits and converge to a desired value.  Applications: \href{ https://www.youtube.com/watch?v=tmTJ0z3HV68}{[video 3]}.   \blfootnote{[video 3] \url{ https://www.youtube.com/watch?v=tmTJ0z3HV68}}

Then, in  paper [C10] we studied the application of the above controllers through sampling (sampled-data control). In particular, we derived sufficient conditions for the sampling time $T$ to ensure collision avoidance between vehicles and with road boundaries, as well as bounded speed and bounded orientation of vehicles. Furthermore, we have shown that it is possible for each vehicle to have its own independent sampling time $T_i$.

In [J10] we designed cruise controllers for the movement of automated vehicles on ring-roads that are free of traffic lanes.

The design of the feedback laws is based on Lyapunov control functions which are based on the energy of the system, with the kinetic energy expressed in ways similar to Newtonian or relativistic mechanics. It is shown that there are no collisions between vehicles or with the inner and outer boundaries of the roundabout, the velocities of all vehicles never exceed a given speed limit and always remain positive, and the angular velocities of all vehicles converge to a given value. Having designed course controllers for both the case of a circular road and the case of a straight road, it is now possible to consider any road represented by a non-intersecting closed or open curve by choosing an appropriate coordinate change so that: (i) convert any curved road of constant width and infinite length into a straight road of constant width, and (ii) convert any road of constant width represented by a closed curve into a circular road of constant width. Application: \href{https://www.youtube.com/watch?v=7tY28TTz4PQ}{[video 4]}, \href{https://www.youtube.com/watch?v=ToKZfOTSilo}{[video 5]} \blfootnote{[video 4] \url{https://www.youtube.com/watch?v=7tY28TTz4PQ}}\blfootnote{[video 5] \url{https://www.youtube.com/ watch?v=ToKZfOTSilo}}

Finally, in [J14] we studied abstract dynamical systems defined on open sets (systems with state constraints) and novel properties of forward completeness (existence of solutions for all times). These properties were then exploited to derive feedback laws (Cruise Controllers) for vehicles in lane free roads with non-constant width that may include on-ramps and off-ramps.  Applications: \href{ https://youtu.be/h4nIUz2nl9s?si=_UTJmXa8MYnLfQgn}{[video 6]}.   \blfootnote{[video 6] \url{ https://youtu.be/h4nIUz2nl9s?si=_UTJmXa8MYnLfQgn}}

{Future and Ongoing Research:} Currently, my focus on this subject lies on the study of mixed-traffic, where in addition to fully automated vehicles, there exist several human-driven vehicles. 

Finally, in paper [J7] we presented a new bidirectional microscopic Adaptive Cruise Control (ACC)  model without viscosity terms that uses only distance information from the preceding and following vehicles in order to choose the correct collision avoidance action and maintain a desired speed. In addition, $KL$ estimates are provided that guarantee uniform convergence of the ACC model to the set of equilibria of the system.