American Institute of Mathematical Sciences

May  2020, 13(5): 1429-1440. doi: 10.3934/dcdss.2020081

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, 2020, 13 (5) : 1429-1440. 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

show all references

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
 [1] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [2] Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 [3] Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 [4] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [5] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [6] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [7] Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 [8] Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053 [9] Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128 [10] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [11] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [12] Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426 [13] Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288 [14] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [15] Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104 [16] Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 [17] Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 [18] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [19] Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171 [20] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

2019 Impact Factor: 1.233