Boundary Control for Transport Equations

TL;DR

This paper investigates boundary control problems for linear transport equations, demonstrating exact controllability under convexity assumptions and highlighting limitations in controlling outgoing solutions, contrasting with elliptic equations.

Contribution

It establishes conditions for exact boundary control of transport solutions and shows the non-existence of control in certain cases, revealing fundamental differences from elliptic equations.

Findings

Exact control of transport solutions from boundary conditions under convexity.

Control of outgoing solutions may be impossible for specific coefficients.

Existence of non-trivial incoming conditions with zero outgoing conditions in certain scenarios.

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 X can be controlled exactly from incoming boundary conditions for X 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.