Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

TL;DR

This paper develops an explicit feedback law using backstepping to exponentially stabilize a 1-D linear transport equation with scalar control, assuming controllability in a higher-order Sobolev space.

Contribution

It introduces a method to design explicit feedback controls for rapid stabilization of a 1-D transport equation based on controllability in Sobolev spaces.

Findings

Exponential stabilization is achievable under controllability assumptions.

An explicit feedback law is constructed for any desired decay rate.

The approach applies to systems with piecewise regular source functions.

Abstract

We use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate.