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 and 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), 693-698.

[2]

J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201. doi: 10.1016/0022-247X(64)90061-7.

[3]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[4]

P. Grisvard, Caractérisation de quelques espaces d'interpolation (French), Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[5]

G. Hellwig, "Partial Differential Equations: An Introduction," Blaisdell Publishing Co. Ginn and Co., New York-Toronto-London, 1964.

[6]

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

[7]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.

[8]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128. doi: 10.1080/00036810802189654.

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.

[10]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.

[11]

M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, in "Ill-Posed Problems in Natural Sciences" (Moscow, 1991), VSP, Utrecht, (1992), 439-445.

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), 693-698.

[2]

J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201. doi: 10.1016/0022-247X(64)90061-7.

[3]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[4]

P. Grisvard, Caractérisation de quelques espaces d'interpolation (French), Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[5]

G. Hellwig, "Partial Differential Equations: An Introduction," Blaisdell Publishing Co. Ginn and Co., New York-Toronto-London, 1964.

[6]

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

[7]

L. Lorenzi, An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.

[8]

A. Sh. Lyubanova, Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128. doi: 10.1080/00036810802189654.

[9]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.

[10]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.

[11]

M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, in "Ill-Posed Problems in Natural Sciences" (Moscow, 1991), VSP, Utrecht, (1992), 439-445.

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