August  2011, 5(3): 695-714. doi: 10.3934/ipi.2011.5.695

Identification of a real constant in linear evolution equations in Hilbert spaces

1. 

Dipartimento di Matematica “F. Enriques”, Universitá di Milano, via C. Saldini 50, 20133 Milano, Italy

Received  November 2010 Revised  February 2011 Published  August 2011

Let $H$ be a real separable Hilbert space and $A:\mathcal{D}(A) \to H$ be a positive and self-adjoint (unbounded) operator, and denote by $A^\sigma$ its power of exponent $\sigma \in [-1,1)$. We consider the identification problem consisting in searching for a function $u:[0,T] \to H$ and a real constant $\mu$ that fulfill the initial-value problem $$ u' + Au = \mu \, A^\sigma u, \quad t \in (0,T), \quad u(0) = u_0, $$ and the additional condition $$ \alpha \|u(T)\|^{2} + \beta \int_{0}^{T}\|A^{1/2}u(\tau)\|^{2}d\tau = \rho, $$ where $u_{0} \in H$, $u_{0} \neq 0$ and $\alpha, \beta \geq 0$, $\alpha+\beta > 0$ and $\rho >0$ are given. By means of a finite-dimensional approximation scheme, we construct a unique solution $(u,\mu)$ of suitable regularity on the whole interval $[0,T]$, and exhibit an explicit continuous dependence estimate of Lipschitz-type with respect to the data $u_{0}$ and $\rho $. Also, we provide specific applications to second and fourth-order parabolic initial-boundary value problems.
Citation: Alfredo Lorenzi, Gianluca Mola. Identification of a real constant in linear evolution equations in Hilbert spaces. Inverse Problems & Imaging, 2011, 5 (3) : 695-714. doi: 10.3934/ipi.2011.5.695
References:
[1]

E. A. Artyukhin and A. S. Okhapkin, Determination of the parameters in the generalized heat-conduction equation from transient experimental data,, J. Eng. Phys. Thermophys., 42 (1982), 693. Google Scholar

[2]

J. R. Cannon, Determination of certain parameters in heat conduction problems,, J. Math. Anal. Appl., 8 (1964), 188. doi: 10.1016/0022-247X(64)90061-7. Google Scholar

[3]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

[4]

P. Grisvard, Caractérisation de quelques espaces d'interpolation (French),, Arch. Rational Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar

[5]

G. Hellwig, "Partial Differential Equations: An Introduction,", Blaisdell Publishing Co. Ginn and Co., (1964). Google Scholar

[6]

A. Lorenzi, Recovering two constants in a parabolic linear equation,, Journal of Physics: Conference Series, 73 (2007). Google Scholar

[7]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator,, Journal of Inverse and Ill-posed Problems, 19 (2011), 293. Google Scholar

[8]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation,, Appl. Anal., 87 (2008), 1121. doi: 10.1080/00036810802189654. Google Scholar

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. Google Scholar

[10]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988). Google Scholar

[11]

M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations,, in, (1992), 439. Google Scholar

show all references

References:
[1]

E. A. Artyukhin and A. S. Okhapkin, Determination of the parameters in the generalized heat-conduction equation from transient experimental data,, J. Eng. Phys. Thermophys., 42 (1982), 693. Google Scholar

[2]

J. R. Cannon, Determination of certain parameters in heat conduction problems,, J. Math. Anal. Appl., 8 (1964), 188. doi: 10.1016/0022-247X(64)90061-7. Google Scholar

[3]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

[4]

P. Grisvard, Caractérisation de quelques espaces d'interpolation (French),, Arch. Rational Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar

[5]

G. Hellwig, "Partial Differential Equations: An Introduction,", Blaisdell Publishing Co. Ginn and Co., (1964). Google Scholar

[6]

A. Lorenzi, Recovering two constants in a parabolic linear equation,, Journal of Physics: Conference Series, 73 (2007). Google Scholar

[7]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator,, Journal of Inverse and Ill-posed Problems, 19 (2011), 293. Google Scholar

[8]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation,, Appl. Anal., 87 (2008), 1121. doi: 10.1080/00036810802189654. Google Scholar

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. Google Scholar

[10]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Applied Mathematical Sciences, 68 (1988). Google Scholar

[11]

M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations,, in, (1992), 439. Google Scholar

[1]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039

[2]

Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002

[3]

Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030

[4]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143

[5]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395

[6]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455

[7]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[8]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171

[9]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[10]

Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457

[11]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005

[12]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[13]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15

[14]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

[15]

Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030

[16]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[17]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865

[18]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[19]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[20]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]