Saturated boundary feedback control of LWR traffic flow models with lane-changing

Abstract

This paper investigates the boundary feedback control for the two-lane traffic flow with lane-changing interactions in the presence of actuator saturations. The macroscopic traffic dynamics are described by the Lighthill-Whitham-Richards (LWR) model for both two lanes. Two variable speed limits (VSLs) are applied at boundaries for controlling the vehicle velocity so as to stabilize the traffic density of each lane. A saturated boundary feedback controller is proposed to drive the traffic densities of both lanes to the steady states. In contrast to the fruitful results in saturated control of ordinary differential equation (ODE) systems, there are few related studies for hyperbolic partial differential equation (PDE) systems. We define the exponential stability for the hyperbolic PDEs under saturated control. Then sufficient conditions for ensuring exponential stability of the two-lane traffic flow system are developed in terms of matrix inequalities under both the linear and nonlinear cases, by employing the Lyapunov function method along with a sector condition in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -norm and H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -norm, respectively. Finally, the effectiveness of the proposed conditions is validated by numerical simulations.