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. Google Scholar

[2]

J. Diestel and J. J. Uhl, Jr., "Vector Measures,", Mathematical Surveys, 15 (1977). Google Scholar

[3]

I. Dobrakov, On integration in Banach spaces. I,, Czechoslovak Math. J., 20 (1970), 511. Google Scholar

[4]

I. Dobrakov, On integration in Banach spaces. II,, Czechoslovak Math. J., 20 (1970), 680. 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. 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. 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. 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. 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. 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. Google Scholar

[11]

I. Vrabie, "Compactness Methods for Nolinear Evolutions. Second Edition,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 75 (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. Google Scholar

[2]

J. Diestel and J. J. Uhl, Jr., "Vector Measures,", Mathematical Surveys, 15 (1977). Google Scholar

[3]

I. Dobrakov, On integration in Banach spaces. I,, Czechoslovak Math. J., 20 (1970), 511. Google Scholar

[4]

I. Dobrakov, On integration in Banach spaces. II,, Czechoslovak Math. J., 20 (1970), 680. 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. 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. 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. 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. 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. 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. Google Scholar

[11]

I. Vrabie, "Compactness Methods for Nolinear Evolutions. Second Edition,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 75 (1995). Google Scholar

[12]

I. I. Vrabie, "$C_0$-Semigroups and Applications,", North-Holland Publishing Co. Amsterdam, (2003). Google Scholar

[1]

Angela Alberico, Teresa Alberico, Carlo Sbordone. Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1053-1067. doi: 10.3934/dcds.2011.31.1053

[2]

Ansgar Jüngel, Ingrid Violet. First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 861-877. doi: 10.3934/dcdsb.2007.8.861

[3]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361

[4]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[5]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020081

[6]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[7]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[8]

Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5253-5271. doi: 10.3934/dcds.2013.33.5253

[9]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044

[10]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758

[11]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[12]

Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934/cpaa.2009.8.457

[13]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065

[14]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[15]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[16]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878

[17]

Liming Ling. The algebraic representation for high order solution of Sasa-Satsuma equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1975-2010. doi: 10.3934/dcdss.2016081

[18]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43

[19]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[20]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]