# American Institute of Mathematical Sciences

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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