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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

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, 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.
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, 34G1.
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

show all references

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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