Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

Angelo Favini; Gianluca Mola; Silvia Romanelli

doi:10.3934/dcdss.2022017

Previous Article
Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping
DCDS-S Home
This Issue
Next Article
Polynomial stability in viscoelastic network of strings

June 2022, 15(6): 1439-1454. doi: 10.3934/dcdss.2022017

Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

Angelo Favini ^1, , Gianluca Mola ^2, and Silvia Romanelli ^3,

1.	Dipartimento di Matematica, Universitá di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna BO, Italy
2.	Department of Science and Engeneering, Sorbonne Unversity Abu Dhabi, Al Reem Island, 51133 Abu Dhabi, United Arab Emirates
3.	Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via Edoardo Orabona 4, 70125 Bari BA, Italy

Received September 2021 Revised December 2021 Published June 2022 Early access February 2022

Let

$ \left(H, \langle \cdot, \cdot \rangle \right) $

be a separable Hilbert space and

$ A_{i}:D(A_i) \to H $

(

$ i = 1, \cdots, n $

) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function

$ u:[0, T] \to H $

and

$ n $

time-dependent diffusion coefficients

$ \alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+ $

that fulfill the initial-value problem

$ u'(t) + \alpha_{1}(t) A_{1}u(t) + \cdots + \alpha_{n}(t) A_{n}u(t) = 0, \quad s \leq t \leq T, \quad u(s) = x, $

and the additional conditions

$ \langle A_{1} u(t), u(t)\rangle = \varphi_{1}(t), \quad \cdots \quad, \langle A_{n} u(t), u(t)\rangle = \varphi_{n}(t), \quad s \leq t \leq T. $

Under suitable assumptions on the operators

$ A_i $

$ i = 1, \cdots, n $

, on the initial data

$ x\in H $

and on the given functions

$ \varphi_i $

$ i = 1, \cdots, n $

, we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lamé parameters in an elasticity problem on a plate.

Keywords: Identification problems, unknown coefficients, linear evolution equations in Hilbert spaces, nonautonomous parabolic equations, wellposedness results.

Mathematics Subject Classification: Primary: 35R30, 35K90, 47B25; Secondary: 35K20, 35K25.

Citation: Angelo Favini, Gianluca Mola, Silvia Romanelli. Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1439-1454. doi: 10.3934/dcdss.2022017

References:

[1]	M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354.
[2]	K.-C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin and New York, 2005.
[3]	D. Huang, Y. Li and D. Pei, Identification of a time-dependent coefficient in heat conduction problem by new iteration method, Adv. Math. Phys., 2018 (2018), Art. ID 4918256, 7 pp. doi: 10.1155/2018/4918256.
[4]	M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publications, L'viv, Ukraine, 2003.
[5]	N. I. Ivanchov, On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity, Sib. Math. J., 35 (1994), 547-555. doi: 10.1007/BF02104818.
[6]	N. I. Ivanchov and N. V. Pabyrivska, On determination of two time-dependent coefficients in a parabolic equation, Sib. Math. J., 43 (2002), 323-329. doi: 10.1023/A:1014749222472.
[7]	T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234. doi: 10.2969/jmsj/00520208.
[8]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[9]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in nonlinear differential equations and their application Vol. 16, Birkhauser, Basel-Boston-Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.
[10]	G. Mola, Recovering a large number of diffusion constants in a parabolic equations from energy measurements, Inverse Problems and Imaging, 12 (2018), 527-543. doi: 10.3934/ipi.2018023.
[11]	G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. doi: 10.1088/0266-5611/32/11/115016.
[12]	G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, Amer. J. Math., 115 (1993), 1161-1187. doi: 10.2307/2375069.
[13]	A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, M. Dekker, New York, 2000.
[14]	D. Trucu, D. B. Ingham and D. Lesnic, Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser., 124 (2008), 012050.
[15]	S. J. L. van Eijndhoven and J. de Graaf, A fundamental approach to the generalized eigenvalue problem for self-adjoint operators, J. Functional Analysis, 63 (1985), 74-85. doi: 10.1016/0022-1236(85)90098-9.

show all references

References:

[1]	M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354.
[2]	K.-C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin and New York, 2005.
[3]	D. Huang, Y. Li and D. Pei, Identification of a time-dependent coefficient in heat conduction problem by new iteration method, Adv. Math. Phys., 2018 (2018), Art. ID 4918256, 7 pp. doi: 10.1155/2018/4918256.
[4]	M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publications, L'viv, Ukraine, 2003.
[5]	N. I. Ivanchov, On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity, Sib. Math. J., 35 (1994), 547-555. doi: 10.1007/BF02104818.
[6]	N. I. Ivanchov and N. V. Pabyrivska, On determination of two time-dependent coefficients in a parabolic equation, Sib. Math. J., 43 (2002), 323-329. doi: 10.1023/A:1014749222472.
[7]	T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234. doi: 10.2969/jmsj/00520208.
[8]	T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976.
[9]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in nonlinear differential equations and their application Vol. 16, Birkhauser, Basel-Boston-Berlin, 1995. doi: 10.1007/978-3-0348-9234-6.
[10]	G. Mola, Recovering a large number of diffusion constants in a parabolic equations from energy measurements, Inverse Problems and Imaging, 12 (2018), 527-543. doi: 10.3934/ipi.2018023.
[11]	G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. doi: 10.1088/0266-5611/32/11/115016.
[12]	G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, Amer. J. Math., 115 (1993), 1161-1187. doi: 10.2307/2375069.
[13]	A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, M. Dekker, New York, 2000.
[14]	D. Trucu, D. B. Ingham and D. Lesnic, Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser., 124 (2008), 012050.
[15]	S. J. L. van Eijndhoven and J. de Graaf, A fundamental approach to the generalized eigenvalue problem for self-adjoint operators, J. Functional Analysis, 63 (1985), 74-85. doi: 10.1016/0022-1236(85)90098-9.

[1]	Alfredo Lorenzi, Gianluca Mola. Identification of a real constant in linear evolution equations in Hilbert spaces. Inverse Problems and Imaging, 2011, 5 (3) : 695-714. doi: 10.3934/ipi.2011.5.695
[2]	Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure and Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039
[3]	Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations and Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[4]	Jin Liang, James H. Liu, Ti-Jun Xiao. Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2006, 5 (3) : 529-535. doi: 10.3934/cpaa.2006.5.529
[5]	Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661
[6]	Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817
[7]	Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations and Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002
[8]	A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003
[9]	Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[10]	Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
[11]	Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations and Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009
[12]	Hildebrando M. Rodrigues, J. Solà-Morales, G. K. Nakassima. Stability problems in nonautonomous linear differential equations in infinite dimensions. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3189-3207. doi: 10.3934/cpaa.2020138
[13]	Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
[14]	Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[15]	Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
[16]	Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050
[17]	Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
[18]	Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11
[19]	Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[20]	Behzad Azmi, Karl Kunisch, Sérgio S. Rodrigues. Stabilization of nonautonomous parabolic equations by a single moving actuator. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5789-5824. doi: 10.3934/dcds.2021096

American Institute of Mathematical Sciences

Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors