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## An identification problem for a linear evolution equation in a banach space

 1 Octav Mayer Institute of Mathematics of the Romanian Academy, Bdul Carol I, 8, Iasi, Romania 2 Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics, and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711 Bucharest, Romania

* Corresponding author: Gabriela Marinoschi

Received  January 2018 Revised  April 2018 Published  June 2019

We study a problem of a parameter identification related to a linear evolution equation in a Banach space, using an additional information about the solution. For sufficiently regular data we provide an exact solution given by a Volterra integral equation, while for less regular data we obtain an approximating solution by an optimal control approach. Under certain hypotheses, the characterization of the limit of the sequence of the approximating solutions reveals that it is a solution to the original identification problem. An application to an inverse problem arising in population dynamics is presented.

Citation: Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020081
##### References:
 [1] M. Al Horani and A. Favini, Inverse problems for singular differential-operator equations with higher order polar singularities, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2159-2168. doi: 10.3934/dcdsb.2014.19.2159. Google Scholar [2] M. Al Horani and A. Favini, First-order inverse evolution equations, Evolution Equations and Control Theory, 3 (2014), 355-361. doi: 10.3934/eect.2014.3.355. Google Scholar [3] M. Al Horani, A. Favini and H. Tanabe, Parabolic first and second order differential equations, Milan Journal of Mathematics, 84 (2016), 299-315. doi: 10.1007/s00032-016-0260-7. Google Scholar [4] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Identification problems for degenerate integro-differential equations, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (eds. A. Favini, P. Colli, E. Rocca, G. Schimperna and J. Sprekels), Springer, 22 (2017), 55–75. Google Scholar [5] M. Al Horani, A. Favini and H. Tanabe, Inverse problems for evolution equations with time dependent operator-coefficients, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 737-744. doi: 10.3934/dcdss.2016025. Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. Google Scholar [7] C. Cusulin, M. Iannelli and G. Marinoschi, Convergence in a multi-layer population model with age-structure, Nonlinear Anal. Real World Appl., 8 (2007), 887-902. doi: 10.1016/j.nonrwa.2006.03.012. Google Scholar [8] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electronic Journal of Differential Equations, 2015 (2015), 22pp. Google Scholar [9] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: $L^{p}$-theory, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665. doi: 10.1016/j.jmaa.2016.10.029. Google Scholar [10] A. Favini and G. Marinoschi, Identification of the time derivative coefficient in a fast diffusion degenerate equation, Journal of Optimization Theory and Applications, 145 (2010), 249-269. doi: 10.1007/s10957-009-9635-z. Google Scholar [11] A. Favini and G. Marinoschi, Identification for degenerate problems of hyperbolic type, Applicable Analysis, 91 (2012), 1511-1527. doi: 10.1080/00036811.2011.630665. Google Scholar [12] A. Favini and G. Marinoschi, Degenerate Nonlinear Diffusion Equations, Lecture Notes in Mathematics, 2049, Springer-Verlag, 2012. doi: 10.1007/978-3-642-28285-0. Google Scholar [13] M. Iannelli and G. Marinoschi, Well-posedness for a hyperbolic-parabolic Cauchy problem arising in population dynamics, Differential Integral Equations, 21 (2008), 917-934. Google Scholar [14] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

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##### References:
 [1] M. Al Horani and A. Favini, Inverse problems for singular differential-operator equations with higher order polar singularities, Discrete and Continuous Dynamical Systems, Series B, 19 (2014), 2159-2168. doi: 10.3934/dcdsb.2014.19.2159. Google Scholar [2] M. Al Horani and A. Favini, First-order inverse evolution equations, Evolution Equations and Control Theory, 3 (2014), 355-361. doi: 10.3934/eect.2014.3.355. Google Scholar [3] M. Al Horani, A. Favini and H. Tanabe, Parabolic first and second order differential equations, Milan Journal of Mathematics, 84 (2016), 299-315. doi: 10.1007/s00032-016-0260-7. Google Scholar [4] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Identification problems for degenerate integro-differential equations, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (eds. A. Favini, P. Colli, E. Rocca, G. Schimperna and J. Sprekels), Springer, 22 (2017), 55–75. Google Scholar [5] M. Al Horani, A. Favini and H. Tanabe, Inverse problems for evolution equations with time dependent operator-coefficients, Discrete and Continuous Dynamical Systems, Series S, 9 (2016), 737-744. doi: 10.3934/dcdss.2016025. Google Scholar [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. Google Scholar [7] C. Cusulin, M. Iannelli and G. Marinoschi, Convergence in a multi-layer population model with age-structure, Nonlinear Anal. Real World Appl., 8 (2007), 887-902. doi: 10.1016/j.nonrwa.2006.03.012. Google Scholar [8] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electronic Journal of Differential Equations, 2015 (2015), 22pp. Google Scholar [9] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: $L^{p}$-theory, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665. doi: 10.1016/j.jmaa.2016.10.029. Google Scholar [10] A. Favini and G. Marinoschi, Identification of the time derivative coefficient in a fast diffusion degenerate equation, Journal of Optimization Theory and Applications, 145 (2010), 249-269. doi: 10.1007/s10957-009-9635-z. Google Scholar [11] A. Favini and G. Marinoschi, Identification for degenerate problems of hyperbolic type, Applicable Analysis, 91 (2012), 1511-1527. doi: 10.1080/00036811.2011.630665. Google Scholar [12] A. Favini and G. Marinoschi, Degenerate Nonlinear Diffusion Equations, Lecture Notes in Mathematics, 2049, Springer-Verlag, 2012. doi: 10.1007/978-3-642-28285-0. Google Scholar [13] M. Iannelli and G. Marinoschi, Well-posedness for a hyperbolic-parabolic Cauchy problem arising in population dynamics, Differential Integral Equations, 21 (2008), 917-934. Google Scholar [14] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar
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