An identification problem for a linear evolution equation in a Banach space and applications

Alfredo Lorenzi; Ioan I. Vrabie

doi:10.3934/dcdss.2011.4.671

Article Contents

2011, Volume 4, Issue 3: 671-691. Doi: 10.3934/dcdss.2011.4.671

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An identification problem for a linear evolution equation in a Banach space and applications

Alfredo Lorenzi^1, and
Ioan I. Vrabie^2,

1.
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano
2.
"O. Mayer" Mathematics Institute of the Romanian Academy, Iaşi 700505

Received: February 28, 2009

Revised: November 30, 2009

Published: May 2011

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Abstract

In this paper we prove both the existence and uniqueness of a solution to an identification problem for a first order linear differential equation in a general Banach space. Namely, we extend the explicit representation for the solution of this problem previously obtained by Anikonov and Lorenzi [1] to the case of an infinitesimal generator of an analytic $C_0$-semigroup of contractions to the general nonanalytic case and also to the case of a restriction expressed in terms of an operator-valued measure. So, our abstract result handles both parabolic and hyperbolic equations and systems.

Keywords:

Identification problem; first-order linear differential equation; unknown right-hand side; representation formula; $C_0$-semigroup of contractions; heat equation; wave equation.

Mathematics Subject Classification: Primary: 35R30, 35K20, 35K90, 34G10.

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References

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