# American Institute of Mathematical Sciences

June  2011, 4(3): 671-691. doi: 10.3934/dcdss.2011.4.671

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

 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, Romania

Received  March 2009 Revised  December 2009 Published  November 2010

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.
Citation: Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671
##### References:
 [1] Yu. Anikonov and A. Lorenzi, Explicit representation for the solution to a parabolic differential identification problem in a Banach space, J. Inverse Ill Posed Problems, 15 (2007), 669-683.  Google Scholar [2] J. Diestel and J. J. Uhl, Jr., "Vector Measures," Mathematical Surveys, 15, American Mathematical Society, 1977.  Google Scholar [3] I. Dobrakov, On integration in Banach spaces. I, Czechoslovak Math. J., 20 (1970), 511-536. Google Scholar [4] I. Dobrakov, On integration in Banach spaces. II, Czechoslovak Math. J., 20 (1970), 680-695. Google Scholar [5] A. I. Prilepko and A. B. Kostin, An estimate for the spectral radius of an operator and the solvability of inverse problems for evolution equations, Mat. Zametki, 53 (1993), 89-94.  Google Scholar [6] A. I. Prilepko and I. V. Tikhonov, Reconstruction of the inhomogeneous term in an abstract evolution equation, Izv. Ross. Akad. Nauk Ser. Mat., 58 (1994), 167-188.  Google Scholar [7] A. I. Prilepko, S. Piskarev and S.-Y. Shaw, On approximation of inverse problem for abstract parabolic differential equation in Banach spaces, J. Inv. Ill-Posed Problems, 15 (2007), 831-851.  Google Scholar [8] I. V. Tikhonov and Yu. S. Eidel'man, Problem of correctness of ordinary and inverse problems for evolutionary equations in special form, Mat. Zametki, 56 (1994), 99-113. (Russian) (English Translation: Mathematical Notes, pp. 830-839).  Google Scholar [9] I. V. Tikhonov and Yu. S. Eidel'man, The unique solvability of a two-point inverse problem for an abstract differential equation with unknown parameter, Differential'nye Uravneniya, 36 (2000), 1132-1133.  Google Scholar [10] I. V. Tikhonov and Yu. S. Eidel'man, Theorems of the mapping point spectrum for $C_0$-semigroups and their application to uniqueness problems for abstract differential equations, Dokl. Akad. Nauk, 394 (2004), 32-35.  Google Scholar [11] I. Vrabie, "Compactness Methods for Nolinear Evolutions. Second Edition," Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman Scientific & Technical, Harlow; John Wiley & Sons Inc., New York, 1995.  Google Scholar [12] I. I. Vrabie, "$C_0$-Semigroups and Applications," North-Holland Publishing Co. Amsterdam, 2003.  Google Scholar

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##### References:
 [1] Yu. Anikonov and A. Lorenzi, Explicit representation for the solution to a parabolic differential identification problem in a Banach space, J. Inverse Ill Posed Problems, 15 (2007), 669-683.  Google Scholar [2] J. Diestel and J. J. Uhl, Jr., "Vector Measures," Mathematical Surveys, 15, American Mathematical Society, 1977.  Google Scholar [3] I. Dobrakov, On integration in Banach spaces. I, Czechoslovak Math. J., 20 (1970), 511-536. Google Scholar [4] I. Dobrakov, On integration in Banach spaces. II, Czechoslovak Math. J., 20 (1970), 680-695. Google Scholar [5] A. I. Prilepko and A. B. Kostin, An estimate for the spectral radius of an operator and the solvability of inverse problems for evolution equations, Mat. Zametki, 53 (1993), 89-94.  Google Scholar [6] A. I. Prilepko and I. V. Tikhonov, Reconstruction of the inhomogeneous term in an abstract evolution equation, Izv. Ross. Akad. Nauk Ser. Mat., 58 (1994), 167-188.  Google Scholar [7] A. I. Prilepko, S. Piskarev and S.-Y. Shaw, On approximation of inverse problem for abstract parabolic differential equation in Banach spaces, J. Inv. Ill-Posed Problems, 15 (2007), 831-851.  Google Scholar [8] I. V. Tikhonov and Yu. S. Eidel'man, Problem of correctness of ordinary and inverse problems for evolutionary equations in special form, Mat. Zametki, 56 (1994), 99-113. (Russian) (English Translation: Mathematical Notes, pp. 830-839).  Google Scholar [9] I. V. Tikhonov and Yu. S. Eidel'man, The unique solvability of a two-point inverse problem for an abstract differential equation with unknown parameter, Differential'nye Uravneniya, 36 (2000), 1132-1133.  Google Scholar [10] I. V. Tikhonov and Yu. S. Eidel'man, Theorems of the mapping point spectrum for $C_0$-semigroups and their application to uniqueness problems for abstract differential equations, Dokl. Akad. Nauk, 394 (2004), 32-35.  Google Scholar [11] I. Vrabie, "Compactness Methods for Nolinear Evolutions. Second Edition," Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman Scientific & Technical, Harlow; John Wiley & Sons Inc., New York, 1995.  Google Scholar [12] I. I. Vrabie, "$C_0$-Semigroups and Applications," North-Holland Publishing Co. Amsterdam, 2003.  Google Scholar
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