.. Schéma-illustrant-formellement-les-résultats-démontrés-dans-lapremì-ere-partie, 11 3 (a) Donnée initiale ¯ u 0 . (b) Zone d'observation ?. (c) Coefficient exact. (d) Coefficient récupéré, p.13

?. Déformation-du-domaine, 13 5 (a) Maillage. (b) Champ de déformations du cerveau soumisàsoumis`soumisà la gravité, p.15

. Schéma-rhéologique-du-modèle-viscoélastique.........., viscoplastique linéaire unidimensionnel avec ? la contrainte et ?, ? e , ? v et ? vp les déformations totale, ´ elastique, visqueuse et viscoplastique, respectivement, p.16

. Maillage-de-calcul....., Données initiale ¯ u 0 (` a droite), p.72

L. Choix-de, A gauche, certaines ondes sont perdues. ` A droite, p.81

.. Maillage-de-calcul-de-u, Déplacement initial ¯ u 0 (au centre) Zone d'observation ? en rouge (` a droite), p.84

*. Coefficient-récupéré-p, a l'itération 0 avec K 0 = 100, p.86

.. De-vue, Déformation de lapì ece mécanique observée de différents points, p.109

. Echantillon-au-repos...., ´ echantillon déformé (droite), p.113

. Echantillon-au-repos, Echantillon déformé, p.116

H. Ammari, P. Garapon, H. Kang, and H. Lee, A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements, Quarterly of Applied Mathematics, vol.66, issue.1, pp.139-175, 2008.
DOI : 10.1090/S0033-569X-07-01089-8

I. Babu?-ska, The finite element method with lagrangian multipliers, Numerische Mathematik, vol.203, pp.179-192, 1973.

M. Bellassoued, Unicité et contrôle pour le système de Lamé. ESAIM : Control, Optimisation and Calculus of Variations, pp.561-592, 2001.

M. Bellassoued, O. Imanuvilov, and M. Yamamoto, Inverse Problem of Determining the Density and Two Lam?? Coefficients by Boundary Data, SIAM Journal on Mathematical Analysis, vol.40, issue.1, pp.238-265, 2008.
DOI : 10.1137/070679971

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, Journal de Math??matiques Pures et Appliqu??es, vol.85, issue.2, pp.193-224, 2006.
DOI : 10.1016/j.matpur.2005.02.004

H. Bellout and J. Ne?as, Existence of global weak solutions for a class of quasilinear hyperbolic integro-differential equations describing visco-elastic materials, Mathematische Annalen, vol.24, issue.1, pp.275-291, 1994.
DOI : 10.1007/BF01459784

D. W. Brands, G. W. Peters, and P. H. Bovendeerd, Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact, Journal of Biomechanics, vol.37, issue.1, pp.127-134, 2004.
DOI : 10.1016/S0021-9290(03)00243-4

J. Bros and D. Iagolnitzer, Causality and local analyticity : general results and some applications to quantum field theory, Trudy Mat. Inst. Steklov, vol.135, pp.54-67, 1975.

M. Bucki, C. Lobos, and Y. Pahan, Bio-mechanical model of the brain for a peroperative image-guided neuronavigator compensating for brain-shift deformations, Computer Methods in Biomechanics and Biomedical Engineering, vol.1, pp.25-26, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00170010

A. L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, 2000.

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Soviet Math. Dokl, vol.24, pp.244-247, 1981.

T. Carleman, Sur unprobì eme d'unicité pour les systèmes d'´ equations aux dérivées partiellesàpartiellesà deux variables indépendantes, Ark. Mat. Astr. Fys, vol.2, pp.1-9, 1939.

C. Cavaterra, A. Lorenzi, and M. Yamamoto, A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation, Computational and Applied Mathematics, vol.25, pp.229-250, 2006.

P. Ciarlet, Mathematical Elasticity. North-Holland, 1988.
URL : https://hal.archives-ouvertes.fr/hal-01077424

J. Salençon, Mécanique des milieux continus, Ellipses Editeur, p.154, 1988.

K. Darvish and J. Crandall, Nonlinear viscoelastic effects in oscillatory shear deformation of brain tissue, Medical Engineering & Physics, vol.23, issue.9, pp.633-645, 2001.
DOI : 10.1016/S1350-4533(01)00101-1

R. Dautray and J. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, 1987.

M. De-buhan and P. Frey, A generalized model of nonlinear viscoelasticity : numerical issues and application to the simulation of the brain shift, IJNME, 2010.

M. De-buhan and A. Osses, Un r??sultat de stabilit?? pour la r??cup??ration d'un param??tre du syst??me de la visco??lasticit?? 3D, Comptes Rendus Mathematique, vol.347, issue.23-24, pp.1373-1378, 2009.
DOI : 10.1016/j.crma.2009.10.022

M. De-buhan and A. Osses, Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example, Inverse Problems, vol.26, issue.9, p.95006, 2010.
DOI : 10.1088/0266-5611/26/9/095006

URL : https://hal.archives-ouvertes.fr/hal-01116979

J. F. Delort, I transformation : Second microlocalization and semilinear caustics, In Lecture Notes in Mathematics, vol.1522, 1992.

T. Sayed, A. Mota, F. Fraternali, and M. Ortiz, A variational constitutive model for soft biological tissues, Journal of Biomechanics, vol.41, issue.7, pp.1458-1466, 2008.
DOI : 10.1016/j.jbiomech.2008.02.023

M. Eller, V. Isakov, G. Nakamura, and D. Tataru, Uniqueness and Stability in the Cauchy Problem for Maxwell and Elasticity Systems, Nonlinear Partial Diffrential Equations and Their Applications, pp.329-349, 2002.
DOI : 10.1016/S0168-2024(02)80016-9

M. Fortin and R. Glowinski, Méthode de Lagrangien Augmenté, 1982.

P. Frey, Generation and adaptation of computational surface meshes from discrete anatomical data, International Journal for Numerical Methods in Engineering, vol.60, issue.6, pp.1049-1074, 2004.
DOI : 10.1002/nme.992

P. Frey and P. George, Mesh generation. Application to finite elements, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00073738

M. Grasselli, Determining the relaxation tensor in linear viscoelasticity of integral type, Japan Journal of Industrial and Applied Mathematics, vol.672, issue.1, pp.131-153, 1994.
DOI : 10.1007/BF03167218

M. Grasselli, On an inverse problem for a linear hyperbolic integrodifferential equation, Forum Mathematicum, vol.6, issue.6, pp.83-100, 1994.
DOI : 10.1515/form.1994.6.83

J. Hadamard, Sur lesprobì emes aux dérivées partielles et leur signification physique, Princeton University Bulletin, pp.49-52, 1902.

R. Hill, The Mathematical Theory of Plasticity, The Oxford Engineering Science Series, 1950.

G. A. Holzapfel and T. C. Gasser, A viscoelastic model for fiber-reinforced composites at finite strains : continuum basis, computational aspects and applications The mechanical behaviour of brain tissue : Large strain response and constitutive modelling, Computer Methods in Applied Mechanics and Engineering Hörmander. Linear Partial Differential Operators. Berlin Biorheology, vol.190, issue.43, pp.4379-4403633, 1963.

A. Ibrahimbegovic, Mécanique non linéaire des solides déformables, 2006.

A. Ibrahimbegovic and L. Chorfi, Viscoplasticity model at finite deformations with combined isotropic and kinematic hardening, Computers & Structures, vol.77, issue.5, pp.509-525, 2000.
DOI : 10.1016/S0045-7949(99)00232-1

O. Imanuvilov and J. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Comptes Rendus Mathematique, vol.335, issue.1, pp.883-913, 2003.
DOI : 10.1016/S1631-073X(02)02389-0

O. Imanuvilov and M. Yamamoto, GLOBAL UNIQUENESS AND STABILITY IN DETERMINING COEFFICIENTS OF WAVE EQUATIONS, Communications in Partial Differential Equations, vol.75, issue.7-8, pp.1409-1425, 2001.
DOI : 10.1016/S0021-7824(99)80010-5

O. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional nonstationary Lamé system and applications to an inverse problem, Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math, vol.242, pp.337-374, 2005.

V. Isakov, Inverse Problems for Partial Differential Equations, 1998.

C. E. Jamison, R. D. Marangoni, and A. A. Glaser, Viscoelastic properties of soft tissue by discrete model characterization, Journal of Biomechanics, vol.1, issue.1, pp.33-46, 1968.
DOI : 10.1016/0021-9290(68)90036-5

J. Janno, Identification of Weakly Singular Relaxation Kernels in Three-Dimensional Viscoelasticity, Journal of Mathematical Analysis and Applications, vol.262, issue.1, pp.133-159, 2001.
DOI : 10.1006/jmaa.2001.7547

J. Janno and L. Von-wolfersdorf, Inverse Problems for Identification of Memory Kernels in Viscoelasticity, Mathematical Methods in the Applied Sciences, vol.12, issue.4, pp.291-314, 1997.
DOI : 10.1002/(SICI)1099-1476(19970310)20:4<291::AID-MMA860>3.0.CO;2-W

M. Kern, Probì emes inverses. Aspects numériques. Cours de l'´ ecole supérieure d'ingénieurs Léonard de Vinci, 2003.

A. Kirsch, An introduction to the mathematical theory of inverse problems, Applied Mathematical Sciences, 1991.

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, 2004.
DOI : 10.1515/9783110915549

S. G. Krantz and H. R. Parks, A primer of real analytic functions, 2002.

A. Lada and L. Sidz, Approximate boundary controllability for the system of linear elasticity, Mathematical Methods in the Applied Sciences, vol.24, issue.17, pp.2017-2026, 2004.
DOI : 10.1002/mma.535

R. Lakes, Viscoelastic solids, 1998.

M. M. Lavrent-'ev, Some Ill-Posed Problems of Mathematics Physics, 1967.

M. M. Lavrent-'ev, V. G. Romanov, and S. P. , Shishat'ski?iski?ski?i. Ill-posed Problems of Mathematical Physics and Analysis, 1986.

L. Tallec, Numerical methods for nonlinear three-dimensional elasticity In Handbook of Numerical Analysis, Lions eds, pp.465-624, 1994.

P. , L. Tallec, C. Rahier, and A. Kaiss, Three-dimensional incompressible viscoelasticity in large strains : formulation and numerical approximation, Computer methods in applied mechanics and engineering, vol.109, pp.233-258, 1993.

G. Lebeau and L. Robbiano, Contr??le Exact De L??quation De La Chaleur, Communications in Partial Differential Equations, vol.52, issue.1-2, pp.335-356, 1995.
DOI : 10.1016/0022-0396(87)90043-X

E. Lee, Elastic-Plastic Deformation at Finite Strains, Journal of Applied Mechanics, vol.36, issue.1, pp.1-6, 1969.
DOI : 10.1115/1.3564580

V. Libertiaux, Contribution to the study of the mechanical properties of brain tissue : fractional calculus-based model and experimental characterization, Thèse de Doctorat de l'Université deLì ege, 2010.

A. Lorenzi, A Multidimensional Identification Problem Related to a Hyperbolic Integro-Differential Equation, Zeitschrift f??r Analysis und ihre Anwendungen, vol.18, issue.2, pp.407-435, 1999.
DOI : 10.4171/ZAA/890

A. Lorenzi, F. Messina, and V. G. Romanov, Recovering a Lam?? kernel in a viscoelastic system, Applicable Analysis, vol.47, issue.11, pp.1375-1395, 2007.
DOI : 10.1016/S0021-7824(99)80010-5

A. Lorenzi, Z. S. Ulekova, and V. G. Yakhno, An inverse problem in viscoelasticity, Journal of Inverse and Ill-Posed Problems, vol.2, issue.2, pp.131-164, 1994.
DOI : 10.1515/jiip.1994.2.2.131

J. Lubliner, A model of rubber viscoelasticity, Mechanics Research Communications, vol.12, issue.2, pp.93-99, 1985.
DOI : 10.1016/0093-6413(85)90075-8

J. Mandel, Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, International Journal of Solids and Structures, vol.9, issue.6, pp.725-740, 1972.
DOI : 10.1016/0020-7683(73)90120-0

A. Martinez, An introduction to semiclassical and microlocal analysis, 2002.
DOI : 10.1007/978-1-4757-4495-8

M. I. Miga, K. D. Paulsen, F. E. Kennedy, P. J. Hoopes, A. Hartov et al., Modeling surgical loads to account for subsurface tissue deformation during stereotactic surgery, IEEE SPIE Proceedings of Laser-Tissue Interaction IX, Part B : Soft-tissue Modeling, pp.501-511, 1998.

K. Miller, Constitutive modelling of brain tissue: Experiment and theory, Journal of Biomechanics, vol.30, issue.11-12, pp.1115-1121, 1997.
DOI : 10.1016/S0021-9290(97)00092-4

K. Miller and K. Chinzei, Mechanical properties of brain tissue in tension, Journal of Biomechanics, vol.35, issue.4, pp.483-490, 2002.
DOI : 10.1016/S0021-9290(01)00234-2

A. Modecai, Nonlinear Programming : Analysis and Methods, 2003.

X. Ning, Q. Zhu, Y. Lanir, and S. S. Margulies, A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation, Journal of Biomechanical Engineering, vol.128, issue.6, pp.925-933, 2006.
DOI : 10.1115/1.2354208

J. T. Oden, Finite Elements for Nonlinear Continua, 1972.

A. Osses and J. Puel, Boundary Controllability of a Stationary Stokes System with Linear Convection Observed on an Interior Curve, Journal of Optimization Theory and Applications, vol.82, issue.1, pp.201-234, 1998.
DOI : 10.1023/A:1021760429614

A. Osses and J. Puel, On the controllability of the Laplace equation observed on an interior curve, Revista Matem??tica Complutense, vol.11, issue.2, pp.403-441, 1998.
DOI : 10.5209/rev_REMA.1998.v11.n2.17271

E. Pagnacco, A. Moreau, and D. Lemoss, Inverse strategies for the identification of elastic and viscoelastic material parameters using full-field measurements, Materials Science and Engineering: A, vol.452, issue.453, pp.452-453737, 2007.
DOI : 10.1016/j.msea.2006.10.122

E. Pena, B. Calvo, M. A. Martinez, and M. Doblare, An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects, Recent Advances in Applied Mechanics, pp.760-778, 1966.
DOI : 10.1016/j.ijsolstr.2006.05.018

P. P. Provenzano, R. S. Lakes, D. T. Corr, and R. Vanderby, Application of nonlinear viscoelastic models to describe ligament behavior, Biomechanics and Modeling in Mechanobiology, vol.1, issue.1, pp.45-47, 2002.
DOI : 10.1007/s10237-002-0004-1

L. Robbiano, Th??or??me d'unicit?? adapt?? au contr??le des solutions des probl??mes hyperboliques, Communications in Partial Differential Equations, vol.5, issue.4-5, pp.789-800, 1991.
DOI : 10.1080/03605309108820778

L. Robbiano, Fonction de coût et contrôle des solutions deséquationsdeséquations hyperboliques, Asymptot. Anal, vol.10, pp.95-115, 1995.

L. Robbiano and C. Zuily, Microlocal analytic smoothing effect for the Schrödinger equation. Duke mathematical journal, pp.93-129, 1999.

S. Shaw, M. K. Warby, J. R. Whiteman, C. Dawson, and M. F. Wheeler, Numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids, Computer Methods in Applied Mechanics and Engineering, vol.118, issue.3-4, pp.211-237, 1994.
DOI : 10.1016/0045-7825(94)90001-9

J. C. Simo, On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects, Computer Methods in Applied Mechanics and Engineering, vol.60, issue.2, pp.153-173, 1987.
DOI : 10.1016/0045-7825(87)90107-1

R. Sinkus, M. Tanter, T. Xydeas, S. Catheline, J. Bercoff et al., Viscoelastic shear properties of in vivo breast lesions measured by MR elastography, Magnetic Resonance Imaging, vol.23, issue.2, pp.159-165, 2005.
DOI : 10.1016/j.mri.2004.11.060

J. Sjöstrand, Singularités analytiques microlocales. Astérisque, pp.1-166, 1982.

D. Tataru, Unique continuation for solutions to PDEs : between Hörmander theorem and Holmgren theorem, Comm. Partial Differential Equations, vol.20, pp.855-884, 1995.

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl, vol.75, pp.367-408, 1996.

A. Tychonoff, On the stability of inverse problems, Doklady Akademii Nauk SSSR, vol.395, pp.195-198, 1943.

K. Ueno, J. Melvin, L. Li, and J. Lighthall, Development of Tissue Level Brain Injury Criteria by Finite Element Analysis, Journal of Neurotrauma, vol.12, issue.4, pp.695-706, 1995.
DOI : 10.1089/neu.1995.12.695

L. Wolfersdorf, On Identification of Memory Kernels in Linear Viscoelasticity, Mathematische Nachrichten, vol.672, issue.1, pp.203-217, 1993.
DOI : 10.1002/mana.19931610115

A. Wittek, T. Hawkins, and K. Miller, On the unimportance of constitutive models in computing brain deformation for image-guided surgery, Biomechanics and Modeling in Mechanobiology, vol.5, issue.1, pp.77-84, 2009.
DOI : 10.1007/s10237-008-0118-1

A. Wittek, R. Kikinis, S. Warfield, and K. Miller, Brain Shift Computation Using a Fully Nonlinear Biomechanical Model, J. Duncan and G. Gerig Edition, pp.583-590, 2005.
DOI : 10.1007/11566489_72