Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model

Controller design for heterogeneous traffic with bottleneck and disturbances

“…In Paper [12], the linearized ARZ traffic flow model with boundary disturbances is mapped into an iISS target system by using a backstepping transformation in order to obtain a full state feedback controller, and we use backstepping method to derive an observer-based output feedback controller to dissolve traffic congestion resulting from traffic breakdown. The exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow is studied in [5]. In [13], the authors propose a new continuum model with an additional anisotropic term which ensures the characteristic velocities can be less than or equal to the macroscopic flow speed.…”

Lyapunov stabilization of a nonlocal LWR traffic flow model

“…The works [19,20] consider Lyapunov functions on a ring road. In [17] the authors prove the exact controllability towards a target state together with explicit rates of convergence. Instead of a ring road they consider a bounded domain and apply the control at the entrance and exit point of the road.…”

“…Here, we consider as in [17,19] the nonlocal LWR model of [3] for general velocity functions and want to obtain a Lyapunov function and an exponential decay rate. In particular, in contrast to [17,19], the Lyapunov function considers the L 2 distance in the nonlocal velocity and we apply, as in [18], the control on the leading vehicle.…”

A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

“…Nonlocal conservation laws have been studied and analyzed quite intensively over the last decade from an application point of view with a particular focus on traffic flow [5,20,30,45,53,37], supply chains [39,55,32], pedestrian flow/crowd dynamics [21], opinion formation [2,51], chemical engineering [50,57], sedimentation [6], conveyor belts [54] and more. For the underlying dynamics existence and uniqueness [35,40,45,44,46,12,25], (optimal) control problems [33,4,14,19,38,24], and suitable numerical schemes [1,11,13,28,52] have been analyzed.…”

“…Such a convergence result would also give additional insights into questions related to control theory (see [4]), in the spirit of [22,34,29,47]. Showing control results for nonlocal conservation laws might be easier due to the fact that these equations are invertible in time, so that one can actually go back from a current state to the initial datum.…”