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

  • * Corresponding author: Gabriela Marinoschi

    * Corresponding author: Gabriela Marinoschi
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  • 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.

    Mathematics Subject Classification: Primary: 35R30, 47Dxx, 49J27, 49K15, 49K20, 65N21, 92D25.

    Citation:

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  • [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.
    [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.
    [3] M. Al HoraniA. 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.
    [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.
    [5] M. Al HoraniA. 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.
    [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
    [7] C. CusulinM. 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.
    [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.
    [9] A. FaviniA. 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.
    [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.
    [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.
    [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.
    [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. 
    [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.
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