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

Abstract

<p style='text-indent:20px;'>We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval <inline-formula><tex-math id="M1">\begin{document}$ (0,L) $\end{document}</tex-math></inline-formula>, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than <inline-formula><tex-math id="M2">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>, 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. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.</p>