Boundary control for transport equations

Abstract

This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain \begin{document}$ X $\end{document} can be controlled exactly from incoming boundary conditions for \begin{document}$ X $\end{document} under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.