Physics

Nonlinear system

Mathematics

Domain (mathematical analysis)

Boundary (topology)

Partial differential equation

Distributed parameter system

Hyperbolic partial differential equation

Mathematical analysis

Voltage

Boundary value problem

Artificial intelligence

Control (management)

Geometry

Flow (mathematics)

Computer science

Quantum mechanics

Applied mathematics

Bounded function

Control theory (sociology)

Negative feedback

IEEE Transactions on Automatic Control

This paper investigates and provides results, including feedback control, for a nonlinear, hyperbolic, one-dimensional partial differential equation (PDE) system on a bounded domain. The considered model consists of two first-order PDEs with a dynamic boundary condition on the one end and actuation on the other. It is shown that, for all positive initial conditions, the system admits a globally defined, unique, classical solution that remains positive and bounded for all times; these properties are important, for example for traffic flow models. Moreover, it is shown that global stabilization can be achieved for arbitrary equilibria by means of an explicit boundary feedback law. The stabilizing feedback law depends only on collocated boundary measurements. The efficiency of the proposed boundary feedback law is demonstrated by means of a numerical example of traffic density regulation.

Feedback Control of Nonlinear Hyperbolic PDE Systems Inspired by Traffic Flow Models