. A. Ae-]-k, J. F. Ames, and . Epperson, A kernel-based method for the approximate solution of backward parabolic problems, SIAM J. Numer. Anal, vol.34, issue.4, pp.1357-1390, 1997.

. J. Aewz, L. Apraiz, G. Escauriaza, C. Wang, and . Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc, vol.16, issue.11, pp.2433-2475, 2014.

. A. Ap-]-k, L. E. Ames, and . Payne, Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation, Math. Models Methods Appl. Sci, vol.8, issue.1, pp.187-202, 1998.

. Bp-]-c, K. D. Bardos, and . Phung, Observation estimate for kinetic transport equation by diffusion approximation

. Db-]-m, K. Denche, and . Bessila, A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl, vol.301, issue.2, pp.419-426, 2005.

[. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions, Mathematical Control and Related Fields, vol.1, issue.2, pp.177-187, 2011.
DOI : 10.3934/mcrf.2011.1.177

URL : https://hal.archives-ouvertes.fr/hal-00992036

. [. Franklin, On Tikhonov's Method for ill-posed problems, Math. Comput, vol.28, issue.128, pp.889-907, 1974.

E. [. Fernández-cara and . Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.17, issue.5, pp.583-616, 2000.
DOI : 10.1016/S0294-1449(00)00117-7

. V. Fi-]-a, O. Y. Fursikov, and . Imanuvilov, Controllability of evolution equations, lecture notes Ser, Seoul National Univ, vol.34, 1996.

. C. Got-]-g, A. García, M. Osses, and . Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, J. Inverse and Ill-posed Probl, pp.755-779, 2013.

. C. Gt-]-g, T. García, and . Takahashi, Inverse problem and null-controllability for parabolic systems, J. Inverse and Ill-Posed Probl, vol.19, issue.3, pp.379-405, 2011.

J. Hadamard, Lecturers on Cauchy Problem in linear partial differential equation, 1923.

. J. Hn, B. Hunter, and . Nachtergaele, Applied Analysis, 2001.

. S. Hx, Y. Hapuarachchi, and . Xu, Backward heat equation with time dependent variable coefficient, Math. Meth. Appl. Sci, vol.40, issue.4, pp.928-938, 2017.

. [. Isakov, Inverse problems for Partial Differential Equations, 2006.

V. [. Jidesh, S. Shubha, and . George, A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations, Applied Mathematics and Computation, vol.254, pp.148-156, 2015.
DOI : 10.1016/j.amc.2014.12.090

. [. Kabanikhin, Inverse and Ill-posed Problems: theory and applications, 2011.
DOI : 10.1515/9783110224016

. D. Kt-]-c, N. H. Khanh, and . Tuan, On a multi-dimensional initial inverse heat problem with a time-dependent coefficient, Applied Mathematics in Engineering and Reliability, pp.255-268, 2016.

J. [. Li and . Liu, Solution of backward heat problem by Morozov discrepancy principle and conditional stability, Numer. Math. J. Chinese Univ, vol.14, issue.2, pp.180-192, 2005.

D. [. Liu and . Luo, ON STABILITY AND REGULARIZATION FOR BACKWARD HEAT EQUATION, Chinese Annals of Mathematics, vol.33, issue.01, pp.35-44, 2003.
DOI : 10.1007/978-1-4612-5338-9

R. [. Lebeau and . Robbiano, Contrôle exact de l'équation de la chaleur, Comm. P.D.E, vol.20, issue.12, pp.336-356, 1995.

. J. Lyz, M. Li, J. Yamamoto, and . Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal, vol.8, issue.1, pp.361-382, 2009.

. [. Mair, Tikhonov Regularization for Finitely and Infinitely Smoothing Operators, SIAM Journal on Mathematical Analysis, vol.25, issue.1, pp.135-147, 1994.
DOI : 10.1137/S0036141092238060

. T. Mfh, W. Min, Q. Fu, and . Huang, Inverse Estimates for Nonhomogeneous Backward Heat Problems, J. Appl. Math. 2014, Special Issue, 2013.

. T. Nt-]-m, U. Nair, and . Tautenhahn, Lavrentiev regularization for linear ill-posed problems under general source conditions, Z. Anal. Anwendungen, vol.23, issue.1, pp.167-185, 2004.

. T. Ntt-]-p, D. D. Nam, N. H. Trong, and . Tuan, The truncation method for a twodimensional nonhomogeneous backward heat problem, Appl. Math. Comput, vol.216, pp.3423-3432, 2010.

. [. Payne, Improperl y posed problems in partial differential equations, Regional Conference Series in Applied Mathematics Society for Industrial and Applied Mathematics, issue.22, 1975.
DOI : 10.1137/1.9781611970463

G. [. Phung and . Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, Journal of Functional Analysis, vol.259, issue.5, pp.1230-1247, 2010.
DOI : 10.1016/j.jfa.2010.04.015

G. [. Phung and . Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, Journal of the European Mathematical Society, vol.15, issue.2, pp.681-703, 2013.
DOI : 10.4171/JEMS/371

URL : https://hal.archives-ouvertes.fr/hal-00625082

. D. Pwz-]-k, L. Phung, C. Wang, and . Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, issue.3, pp.31-477, 2014.

. H. Qttt-]-p, D. D. Quan, L. M. Trong, N. H. Triet, and . Tuan, A modified quasiboundary value method for regularizing of a backward problem with timedependent coefficient, Inverse Probl. Sci. Eng, vol.19, issue.3, pp.409-423, 2011.

. H. Qw-]-h, T. Qin, and . Wei, Some filter regularization methods for a backward heat conduction problem, Appl. Math. Comput, vol.217, issue.24, pp.10317-10327, 2011.

. [. Seidman, Optimal Filtering for the Backward Heat Equation, SIAM Journal on Numerical Analysis, vol.33, issue.1, pp.162-170, 1996.
DOI : 10.1137/0733010

. H. Tklt-]-n, M. Tuan, L. D. Kirane, N. V. Long, and . Thinh, A new general filter method for a multi-dimensional initial inverse heat problem with a time-dependent coefficient, Electronic Journal of Differential Equations, vol.2016, issue.3, pp.1-15, 2016.

. D. Tqkt-]-d, P. H. Trong, T. V. Quan, N. H. Khanh, and . Tuan, A nonlinear case of the 1D backward heat problem: Regularization and error estimate, Z. Anal. Anwendungen, vol.26, issue.2, pp.231-245, 2007.

. H. Tqtt-]-n, P. H. Tuan, D. D. Quan, L. M. Trong, and . Triet, On a backward heat problem with time-dependent coefficient: Regularization and error estimates, Appl. Math. Comput, vol.219, issue.1, pp.432-441, 2013.

T. [. Tautenhahn and . Schröter, On Optimal Regularization Methods for the Backward Heat Equation, Zeitschrift f??r Analysis und ihre Anwendungen, vol.15, issue.2, pp.475-493, 1996.
DOI : 10.4171/ZAA/711

[. Zhang, C. Fu, and Y. Ma, An a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems, Journal of Computational and Applied Mathematics, vol.255, pp.255-150, 2014.
DOI : 10.1016/j.cam.2013.04.046

. Z. Zm, Z. Zhao, and . Meng, A modified Tikhonov regularization method for a backward heat equation, Inverse Probl. Sci. Eng, vol.19, issue.8, pp.1175-1182, 2011.

]. S. Ve and . Vessella, Handbook of differential equations: Evolutionary equations, pp.423-500, 2009.

]. T. Vo and . Vo, Construction of a Control for the Cubic Semilinear Heat Equation, Vietnam Journal of Mathematics, vol.31, issue.3, pp.587-601, 2016.
DOI : 10.1016/S1874-5717(07)80010-7

URL : https://hal.archives-ouvertes.fr/hal-01225203

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, vol.11, issue.2, pp.481-496, 1995.
DOI : 10.1088/0266-5611/11/2/013