In the paper, the problem of source function reconstruction in a differential equation of the parabolic type is investigated. Using the semigroup representation of trajectories of dynamical systems, we build a finite-step iterative procedure for solving this problem. The algorithm originates from the theory of closed-loop control (the method of extremal shift). At every step of the algorithm, the sum of a quality criterion and a linear penalty term is minimized. This procedure is robust to perturbations in problems data.
@article{bwmeta1.element.bwnjournal-article-amcv20i2p239bwm,
author = {Vyacheslav Maksimov},
title = {On one algorithm for solving the problem of source function reconstruction},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {20},
year = {2010},
pages = {239-247},
zbl = {1196.93032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p239bwm}
}
Vyacheslav Maksimov. On one algorithm for solving the problem of source function reconstruction. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 239-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p239bwm/
[000] Bensoussan, A., Prato, G.D., Delfour, M. and Mitter, S. (1992). Representation and Control of Infinite Dimensional Systems, Vol. I, Birkhäuser, Boston, MA. | Zbl 0781.93002
[001] Blizorukova, M.S. and Maksimov, V. I. (1998). On the reconstruction of an extremal input in a system with hereditary, Vestnik PGTU. Matematika i Prikladnaya Matematika (Mathematics and Applied Mathematics) 4(4): 51-61, (in Russian).
[002] Digas, B.V., Maksimov, V.I., Lander, A.V. and Bukchin, B.G. (2003). On an algorithm for solving the inverse problem of ray seismics, in D. Chowdhury (Ed.), Computational Seismology and Geodynamics, American Geophysical Union, Washington, DC, pp. 84-92.
[003] Korbicz, J. and Zgurowski, M. (1991). Estimation and Control of Stochastic Distributed-Parameter Systems, Polish Scientific Publishers, Warsaw, (in Polish). | Zbl 0748.93049
[004] Krasovskii, N. and Subbotin, A. (1988). Game-Theoretical Control Problems, Springer, Berlin.
[005] Kryazhimskii, A.V., Maksimov, V.I. and Osipov, Yu.S. (1997). Reconstruction of extremal disturbances in parabolic equations, Journal of Computational Mathematics and Mathematical Physics 37(3): 119-125, (in Russian).
[006] Kryazhimskii, A.V. and Osipov, Yu.S. (1987). To a regularization of a convex extremal problem with inaccurately given constraints. An application to an optimal control problem with state constraints, in A.I. Korotkii and V.I. Maksimov (Eds.), Some Methods of Positional and Program Control, Ural Scientific Center, Sverdlovsk, pp. 34-54, (in Russian).
[007] Omatu, S. and Seinfeld, J. (1989). Distributed Parameter Systems: Theory and Applications, Oxford Mathematical Monographs, Oxford University Press, New York, NY. | Zbl 0675.93001
[008] Uciński, D. (1999). Measurement Optimization for Parameter Estimation in Distributed Systems, Technical University Press, Zielona Góra.
[009] Vasiliev, F. (1981). Solution Methods to Extremal Problems, Nauka, Moscow, (in Russian).