Data di Pubblicazione:
2011
Citazione:
Identification of a real constant in linear evolution equations in Hilbert spaces / A. Lorenzi, G. Mola. - In: INVERSE PROBLEMS AND IMAGING. - ISSN 1930-8337. - 5:3(2011 Aug), pp. 695-714. [10.3934/ipi.2011.5.695]
Abstract:
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.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Faedo-Galerkin approximation; Identification problems; Linear evolution equa- tions in Hilbert spaces; Linear parabolic equations; Unknown constants; Well-posedness results
Elenco autori:
A. Lorenzi, G. Mola
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