Boundary Control for Transport Equations

Guillaume Bal (1, 2, 3) , Alexandre Jollivet (4)

1 University of Chicago
2 Department of Statistics
3 Department of Mathematics
4 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Mots clés: Transport theory; boundary control; albedo operator; diffusion approximation; unique continuation

Domaines: Equations aux dérivées partielles [math.AP]

Guillaume Bal, Alexandre Jollivet. Boundary Control for Transport Equations. Mathematical Control and Related Fields, 2023, pp.721-770. ⟨10.3934/mcrf.2022014⟩. ⟨hal-03199743⟩