Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation
Анотація
We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution ud 2 L2( ) and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero.
Ключові слова
Повний текст:
PDF (English)Посилання
L. Afraites, A. Atlas, F. Karami, D. Meskine, Some class of parabolic systems applied to image processing, Discrete and Continuous Dynamical Systems, Series B, 21 (6) (2016), 1671–1687.
L. Afraites, A. Hadri, A. Laghrib, M. Nachaoui, A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification, Inverse Problems and Imaging, (2022), doi: 10.3934/ipi.2022001, 1– 44.
L. Alvarez, P.-L. Lions, J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal,, 29 (1992), 845– 66.
L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.
H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, SIAM, Philadelphia, 2006.
L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Analysis, Theory, Methods and Applications, 19 (6) (1992), 581–597.
F. Catt´e, P.L. Lions, J-M. Morel, T. Coll, Image Selective Smoothing and Edge Detection by Nonlinear Diffusion, SIAM Journal on Numerical Analysis, 29 (1) (1992), 182–193.
C. D’Apice, P.I. Kogut, R. Manzo, On coupled two-level variational problem in Sobolev-Orlicz space, submitted for publication.
C. D’Apice, U. De Maio, P.I. Kogut, Suboptimal boundary control for elliptic equations in critically perforated domains, Ann. Inst. H. Poincar´e Anal. Non Lineaire, 25(2008), 1073–1101.
C. D’Apice, U. De Maio, P. Kogut, An indirect approach to the existence of quasi-optimal controls in coefficients for multi-dimensional thermistor problem, in “Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics”, Editors: Sadovnichiy, Victor A., Zgurovsky, Michael (Eds.). Springer. Chapter 24, (2020), 489–522.
U. De Maio, P.I. Kogut, R. Manzo, Asymptotic Analysis of an Optimal Boundary Control Problem for Ill-Posed Elliptic Equation in Domains with Rugous Boundary, Asymptotic Analysis, 118 (3) (2020), 209–234.
T. Horsin, P.I. Kogut, Optimal L2-Control Problem in Coefficients for a Linear Elliptic Equation. I. Existence Result, Mathematical Control and Related Fields, 5 (1) (2015), 73–96.
L. Evans, M. Portilheiro, Irreversibility and hysteresis for a forward- ackward dffusion equation, Mathematical Models and Methods in Applied Sciences, 14 (11) (2004), 1599–1620.
P. Guidotti, Anisotropic diffusions of image processing from Peronaâ“Malik on, Advanced Studies in Pure Mathematics, 67 (2015), 131–156.
F. Karami, L. Ziad, K. Sadik, A splitting algorithm for a novel regularization of Perona-Malik and application to image restoration, EURASIP Journal on Advances in Signal Processing, 2017 (46) (2017), 1– 9.
P. Kogut, On optimal and quasi-optimal controls in coefficients for multi-dimensional thermistor problem with mixed Dirichlet-Neumann boundary conditions, Control and Cybernetics, 48(1) (2019), 31–68.
P. Kogut, G. Leugering, On S-homogenization of an optimal control problem with control and state constraints, Zeitschrift fuer Analysis und ihre Anwendungen, 20 (2001), 395–429.
P. Kogut, G. Leugering Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Series: Systems and Control, Birkh¨auser Verlag, Boston, 2011.
P.I. Kogut, R. Manzo, On Vector-Valued Approximation of State Constrained Optimal Control Problems for Nonlinear Hyperbolic Conservation Laws, Journal of Dynamical and Control Systems, 19 (2) (2013), 381–404.
O.A. Ladyzhenskaja, V.A. Solonnikov, N.N. Uralâ™ceva, Linear and quasilinear equations of parabolic type, Translations of the American Mathematical Society, American Mathematical Society, Providence, 1968.
S. Lecheheb, M. Maouni, H. Lakhal, Image Restoration Using a Novel Model Combining the Perona-Malik Equation and the Heat Equation, International Journal of Analysis and Applications, 19 (2) (2021), 228–238.
J.-L. Lions, Contrˆolabilit´e exacte. Perturbations et stabilisation de syst`emes distribu´es, Masson, Paris, 1988.
P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence, 12 (1990), 161– 192.
J Simon, Compact sets in the space Lp(0; T;B), Ann. Mat. pura Appl., 146 (1987), 65–96.
R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.
V.B Surya Prasath, J.M. Urbano, D. Vorotnikov, Analysis of adaptive forward-backward diffusion flows with applications in image processing, Inverse Problems, 31 (2015), Id 105008, 1–30.
DOI: http://dx.doi.org/10.15421/142202
Посилання
- Поки немає зовнішніх посилань.
Індексування журналу
Журнал розміщено у наукометричних базах, репозитаріях та пошукових системах:
Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І.
email: p.kogut@i.ua
Це видання має доступ за ліцензією Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.