Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Mourad Choulli[a]

Various stability estimates for the problem of determining an initial heat distribution from a single measurement

Pages: 279-307
Received: 22 December 2015
Accepted in revised form: 24 January 2017
Mathematics Subject Classification (2010): 35R30.
Keywords: Heat equation, fractional heat equation, initial heat distribution, Müntz's theorem, point measurement, boundary measurement, stability estimates of Hölder and logarithmic type.
Author address:
[a]: Université de Lorraine, Institut Élie Cartan de Lorraine,UMR CNRS 7502, Boulevard des Aiguillettes, BP 70239 54506 Vandoeuvre les Nancy cedex - Ile du Saulcy 57045 Metz cedex 01, France

Abstract: We consider the problem of determining the initial heat distribution in the heat equation from a point measurement. We show that this inverse problem is naturally related to the one of recovering the coefficients of Dirichlet series from its sum. Taking the advantage of existing literature on Dirichlet series, in connection with Müntz's theorem, we establish various stability estimates of Holder and logarithmic type. These stability estimates are then used to derive the corresponding ones for the original inverse problem, mainly in the case of one space dimension. In higher space dimensions, we are interested to an internal or a boundary measurement. This issue is closely related to the problem of observability arising in Control Theory. We complete and improve the existing results.

References


[1] K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, Springer, Cham, 2015. MR3308343
[2] M. Asaduzzaman and S. Saitoh, Inversion formulas for the Reznitskaya transform and stability of Lipschitz type in determination of initial heat distribution, Appl. Anal. 77 (2001), no. 3-4, 343-350. MR1975740
[3] J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 11, 2433-2475. MR3283402
[4] P. Borwein and T. Erdélyi, Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces, J. Amer. Math. Soc. 10 (1997), 327-349. MR1415318
[5] C. Castro and E. Zuazua, Unique continuation and control for the heat equation from an oscillating lower dimensional manifold, SIAM J. Control Optim. 43 (2004/05), no. 4, 1400-1434. MR2124279
[6] M. Choulli, Une introduction aux problèmes inverses elliptiques et paraboliques (French), Mathématiques & Applications, 65, Springer-Verlag, Berlin 2009. MR2554831
Recovery of an initial temperature from discrete sampling, Math. Models Methods Appl. Sci. 24 (2014), no. 12, 2487-2501. MR3248633
[8] L. D. Drager, R. L. Foote and C. F. Martin, Observing the heat equation on a torus along a dense geodesic, Systems Sci. Math. Sci. 4 (1991), no. 2, 186-192. MR1119293
[9] A. El Badia, T. Ha-Duong and A. Hamdi, Identification of a point source in a linear advection-dispersion-reaction equation: application to a pollution source problem, Inverse Problems 21 (2005), 1121-1136. MR2146825
[10] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272-292. MR0335014
[11] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations 5 (2000), no. 4-6, 465-514. MR1750109
[12] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43, (1967), 82-86.
[13] W. Gautschi, On inverse of Vandermonde and confluent Vandermonde matrices, Numer. Math. 4 (1962), 117-123. MR0139627
[14] D. S. Gilliam, J. R. Lund and C. F. Martin, A Discrete sampling inversion scheme for the heat equation, Numer. Math. 54 (1989), 493-506. MR0978604
[15] D. S. Gilliam, J. R. Lund and C. F. Martin, Inverse parabolic problems and discrete orthogonality, Numer. Math. 59 (1991), 361-383. MR1113196
[16] D. S. Gilliam, B. A. Mair and C. F. Martin, Determination of initial states of parabolic systems from discrete data, Inverse Problems 6 (1990), 737-747. MR1073864
[17] D. S. Gilliam and C. F. Martin, Discrete observability and Dirichlet series, Systems Control Lett. 9 (1987), 345-348. MR0912687
[18] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques (French), Mathématiques & Applications, 13, Springer-Verlag, Paris 1993. MR1276944
[19] Y. Li, S. Osher and R. Tsai, Heat source identification based on \(\ell_1\) constrained minimization, Inverse Probl. Imaging 8 (2014), no. 1, 199-221. MR3180418
[20] G. S. Li,Y. J. Tan, J. Cheng and X. Q. Wang, Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse Probl. Sci. Eng. 14 (2006), no. 3, 287-300. MR2203363
[21] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var. 18 (2012), no. 3, 712-747. MR3041662
[22] S. Micu, I. Roventa and M. Tucsnak, Time optimal boundary controls for the heat equation, J. Funct. Anal. 263 (2012), 25-49. MR2920839
[23] G. Nakamura, S. Saitoh and A. Syarif, Representations of initial heat distributions by means of their heat distributions as functions of time, Inverse Problems 15 (1999), 1255-1261. MR1715363
[24] Y. Privat and M. Sigalotti, The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent, ESAIM Control Optim. Calc. Var. 16 (2010), no. 3, 794-805. MR2674637
[25] S. Saitoh and M. Yamamoto, Stability of Lipschitz type in determination of initial heat distribution, J. Inequal. Appl. 1 (1997), no. 1, 73-83. MR1731742
[26] V. K. Tuan, S. Saitoh and M. Saigo, Size of support of initial heat distribution in the 1D heat equation, Appl. Anal. 74 (2000), no. 3-4, 439-446. MR1757549
[27] L. Schwartz, Étude des sommes d'exponentielles (French), Hermann, Paris 1959. MR0106383
[28] D. V. Widder, An introduction to transform theory, Academic Press, New York 1971. MR0106383


Home Riv.Mat.Univ.Parma