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Boundary control for transport equations

  • *Corresponding author: Alexandre Jollivet

    *Corresponding author: Alexandre Jollivet

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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  • 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.

    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} $

    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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