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doi: 10.3934/mcrf.2022014
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Boundary control for transport equations

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

Received  April 2021 Revised  December 2021 Early access March 2022

Fund Project: 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

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.

Citation: Guillaume Bal, Alexandre Jollivet. Boundary control for transport equations. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022014
References:
[1]

S. Acosta, Time reversal for radiative transport with applications to inverse and control problems, Inverse Problems, 29 (2013), 085014, 19pp. doi: 10.1088/0266-5611/29/8/085014.

[2]

J. H. Albert, Genericity of simple eigenvalues for elliptic PDE's, Proc. Amer. Math. Soc., 48 (1975), 413-418.  doi: 10.2307/2040275.

[3]

G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, Société Mathématique de France, 2018.

[4]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.

[5]

G. Bal, Hybrid inverse problems and internal functionals, in Inside Out II, MSRI Publications, 60 (2013), 325–368.

[6]

G. BalF. J. Chung and J. C. Schotland, Ultrasound modulated bioluminescence tomography and controllability of the radiative transport equation, SIAM Journal on Mathematical Analysis, 48 (2016), 1332-1347.  doi: 10.1137/15M1026262.

[7]

G. Bal and A. Jollivet, Generalized stability estimates in inverse transport theory, Inverse Problems and Imaging, 12 (2018), 59-90.  doi: 10.3934/ipi.2018003.

[8]

G. BalF. Monard and G. Uhlmann, Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields, SIAM Journal on Applied Mathematics, 75 (2015), 2214-2231.  doi: 10.1137/151005269.

[9]

G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Commun. Pure Appl. Math., 66 (2013), 1629-1652.  doi: 10.1002/cpa.21453.

[10]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Springer, 1991. doi: 10.1007/978-94-011-3154-4.

[11]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 89-92. 

[12]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992.

[13]

H. Egger and M. Schlottbom, An Lp theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296.  doi: 10.1080/00036811.2013.826798.

[14]

G. Eskin, Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics, 123. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/gsm/123.

[15]

F. Golse and L. Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Math. Acad. Sci. Paris, 334 (2002), 557-562.  doi: 10.1016/S1631-073X(02)02302-6.

[16]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195.  doi: 10.1137/060652804.

[17]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.  doi: 10.1063/1.1666510.

[18]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math., 9 (1956), 747-766.  doi: 10.1002/cpa.3160090407.

[19]

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York, 1966.

[20]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, With a chapter by M. Choulli and P. Stefanov, Series on Advances in Mathematics for Applied Sciences, 46. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[21]

F. Natterer, The Mathematics of Computerized Tomography, reprint of the 1986 original. Classics in Applied Mathematics, 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284.

[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I. Functional Analysis, Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. 
[23]

F. Rellich, Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969.

[24]

P. Stefanov and A. Tamasan, Uniqueness and non-uniqueness in inverse radiative transfer, Proc. Amer. Math. Soc., 137 (2009), 2335-2344.  doi: 10.1090/S0002-9939-09-09839-6.

show all references

References:
[1]

S. Acosta, Time reversal for radiative transport with applications to inverse and control problems, Inverse Problems, 29 (2013), 085014, 19pp. doi: 10.1088/0266-5611/29/8/085014.

[2]

J. H. Albert, Genericity of simple eigenvalues for elliptic PDE's, Proc. Amer. Math. Soc., 48 (1975), 413-418.  doi: 10.2307/2040275.

[3]

G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, Société Mathématique de France, 2018.

[4]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.

[5]

G. Bal, Hybrid inverse problems and internal functionals, in Inside Out II, MSRI Publications, 60 (2013), 325–368.

[6]

G. BalF. J. Chung and J. C. Schotland, Ultrasound modulated bioluminescence tomography and controllability of the radiative transport equation, SIAM Journal on Mathematical Analysis, 48 (2016), 1332-1347.  doi: 10.1137/15M1026262.

[7]

G. Bal and A. Jollivet, Generalized stability estimates in inverse transport theory, Inverse Problems and Imaging, 12 (2018), 59-90.  doi: 10.3934/ipi.2018003.

[8]

G. BalF. Monard and G. Uhlmann, Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields, SIAM Journal on Applied Mathematics, 75 (2015), 2214-2231.  doi: 10.1137/151005269.

[9]

G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Commun. Pure Appl. Math., 66 (2013), 1629-1652.  doi: 10.1002/cpa.21453.

[10]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Springer, 1991. doi: 10.1007/978-94-011-3154-4.

[11]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 89-92. 

[12]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992.

[13]

H. Egger and M. Schlottbom, An Lp theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296.  doi: 10.1080/00036811.2013.826798.

[14]

G. Eskin, Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics, 123. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/gsm/123.

[15]

F. Golse and L. Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Math. Acad. Sci. Paris, 334 (2002), 557-562.  doi: 10.1016/S1631-073X(02)02302-6.

[16]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195.  doi: 10.1137/060652804.

[17]

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.  doi: 10.1063/1.1666510.

[18]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math., 9 (1956), 747-766.  doi: 10.1002/cpa.3160090407.

[19]

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York, 1966.

[20]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, With a chapter by M. Choulli and P. Stefanov, Series on Advances in Mathematics for Applied Sciences, 46. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[21]

F. Natterer, The Mathematics of Computerized Tomography, reprint of the 1986 original. Classics in Applied Mathematics, 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. doi: 10.1137/1.9780898719284.

[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I. Functional Analysis, Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. 
[23]

F. Rellich, Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969.

[24]

P. Stefanov and A. Tamasan, Uniqueness and non-uniqueness in inverse radiative transfer, Proc. Amer. Math. Soc., 137 (2009), 2335-2344.  doi: 10.1090/S0002-9939-09-09839-6.

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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