Boundary control for transport equations

Guillaume Bal; Alexandre Jollivet

doi:10.3934/mcrf.2022014

Article Contents

2023, Volume 13, Issue 2: 721-770. Doi: 10.3934/mcrf.2022014

This issue Previous Article Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation Next Article Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

Boundary control for transport equations

Guillaume Bal^1, and
Alexandre Jollivet^2, ,

1.
Departments of Statistics and Mathematics, University of Chicago, Chicago, IL 60637, USA
2.
Laboratoire de Mathématiques Paul Painlevé, CNRS UMR 8524/Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France

^*Corresponding author: Alexandre Jollivet
^*Corresponding author: Alexandre Jollivet

Received: April 2021

Revised: December 2021

Early access: March 2022

Published: June 2023

GB's research was partially supported by the National Science Foundation, Grants DMS-1908736 and EFMA-1641100 and by the Office of Naval Research, Grant N00014-17-1-2096

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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.

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Mathematics Subject Classification: Primary:35R30;Secondary:35Q20.

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Figure 1. Geometry of the $ k $-th layer $ Y_{k,N} $. Here, $ (x_0,v_0)\in \mathcal G_-\big({k\over N},{k-1\over N}\big) $, $ (x_1,v_1)\in \Gamma_+(Z_{k-1\over N}) $, while $ (x_2,v_2)\in \mathcal C_{+,k} $

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Figure 2. Layering of the domain $ B(0,2){\backslash}\overline{B(0,1)} $ with layers $ Y_{k,N} $ of equal thickness

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