Approximation Methods in Optimal Control Problems for Nonlinear Infinite-Dimensional Systems
Matematičeskie zametki, Tome 94 (2013) no. 4, pp. 600-619.

Voir la notice de l'article provenant de la source Math-Net.Ru

Some notions related to approximate solutions and to the approximation of extremum problems for nonlinear infinite-dimensional systems are proposed. Optimization problems for nonlinear parabolic equations with a fixed terminal state and on an infinite time interval, as well as for singular stationary systems with phase constraints, are illustrated by several examples.
Keywords: optimization problem, nonlinear parabolic equation, approximate solution, phase constraint, fixed terminal state.
@article{MZM_2013_94_4_a10,
     author = {S. Ya. Serovaǐskiǐ},
     title = {Approximation {Methods} in {Optimal} {Control} {Problems} for {Nonlinear} {Infinite-Dimensional} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {600--619},
     publisher = {mathdoc},
     volume = {94},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_94_4_a10/}
}
TY  - JOUR
AU  - S. Ya. Serovaǐskiǐ
TI  - Approximation Methods in Optimal Control Problems for Nonlinear Infinite-Dimensional Systems
JO  - Matematičeskie zametki
PY  - 2013
SP  - 600
EP  - 619
VL  - 94
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2013_94_4_a10/
LA  - ru
ID  - MZM_2013_94_4_a10
ER  - 
%0 Journal Article
%A S. Ya. Serovaǐskiǐ
%T Approximation Methods in Optimal Control Problems for Nonlinear Infinite-Dimensional Systems
%J Matematičeskie zametki
%D 2013
%P 600-619
%V 94
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2013_94_4_a10/
%G ru
%F MZM_2013_94_4_a10
S. Ya. Serovaǐskiǐ. Approximation Methods in Optimal Control Problems for Nonlinear Infinite-Dimensional Systems. Matematičeskie zametki, Tome 94 (2013) no. 4, pp. 600-619. http://geodesic.mathdoc.fr/item/MZM_2013_94_4_a10/

[1] F. P. Vasilev, Metody optimizatsii, Faktorial, M., 2002

[2] T. Zolezzi, “A characterizations of well-posed optimal control systems”, SIAM J. Control Optim., 19:5 (1981), 604–616 | DOI | MR | Zbl

[3] Dzh. Varga, Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1987 | MR | Zbl

[4] M. I. Sumin, “Suboptimalnoe upravlenie polulineinymi ellipticheskimi uravneniyami s fazovymi ogranicheniyami, I: printsip maksimuma dlyaminimiziruyuschikh posledovatelnostei, normalnost”, Izv. vuzov. Matem., 2000, no. 6, 33–44 | MR | Zbl

[5] I. Ekeland, “The $\varepsilon$-variational principle”, Methods of Nonconvex Analysis, Lecture Notes in Math., 1446, Springer-Verlag, Berlin, 1990, 1–15 | DOI | MR | Zbl

[6] A. V. Fursikov, Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Universitetskaya seriya, 5, Nauchnaya kniga, Novosibirsk, 1999 | Zbl

[7] Zh.-L. Lions, Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987 | MR | Zbl

[8] S. Ya. Serovaiskii, “Priblizhennoe reshenie optimizatsionnykh zadach dlya singulyarnykh beskonechnomernykh sistem”, Sib. matem. zhurn., 44:3 (2003), 660–673 | MR | Zbl

[9] S. Ya. Serovaiskii, “Priblizhennoe reshenie zadachi optimalnogo upravleniya dlya singulyarnogo uravneniya ellipticheskogo tipa s negladkoi nelineinostyu”, Izv. vuzov. Matem., 2004, no. 1, 80–86 | MR | Zbl

[10] V. F. Krotov, V. I. Gurman, Metody i zadachi optimalnogo upravleniya, Nauka, M., 1973 | MR | Zbl

[11] Kh. Gaevskii, K. Greger, K. Zakharias, Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR | Zbl

[12] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Math. Sci. Engrg., 190, Academic Press, Boston, MA, 1993 | MR | Zbl

[13] P. Neittaanmäki, D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms, and Applications, Monogr. Textbooks Pure Appl. Math., 179, Marcel Dekker, New York, 1994 | MR | Zbl

[14] E. Casas, “Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations”, SIAM J. Control Optim., 35:4 (1997), 1297–1327 | DOI | MR | Zbl

[15] J. P. Raymond, H. Zidani, “Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls”, SIAM J. Control Optim., 36:6 (1998), 1853–1879 | DOI | MR | Zbl

[16] I. Ekland, R. Temam, Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979 | MR | Zbl

[17] M. Bergounioux, “A penalization method for optimal control of elliptic problems with state constraints”, SIAM J. Control Optim., 30:2 (1992), 305–323 | DOI | MR | Zbl

[18] S. Ya. Serovaiskii, “Optimalnoe upravlenie dlya singulyarnogo evolyutsionnogo uravneniya s negladkim operatorom i zakreplennym konechnym sostoyaniem”, Differents. uravneniya, 43:2 (2007), 251–258 | MR | Zbl

[19] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl

[20] Funktsionalnyi analiz, ed. S. G. Krein, Nauka, M., 1972 | MR | Zbl

[21] S. Ya. Serovajsky, “Calculation of functional gradients and extended differentiation of operators”, J. Inverse Ill-Posed Probl., 13:4 (2005), 383–396 | DOI | MR | Zbl

[22] Zh.-L. Lions, Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972 | MR | Zbl

[23] V. I. Averbukh, O. G. Smolyanov, “Teoriya differentsirovaniya v lineinykh topologicheskikh prostranstvakh”, UMN, 22:6(138) (1967), 201–260 | MR | Zbl

[24] V. Barbu, “Optimal feedback controls for a class of nonlinear distributed parameter systems”, SIAM J. Control Optim., 21:6 (1983), 871–894 | DOI | MR | Zbl

[25] P. Cannarsa, G. Da Prato, “Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton–Jacobi equations”, SIAM J. Control Optim., 27:4 (1989), 861–875 | DOI | MR | Zbl

[26] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, Nauka, M., 1977 | MR | Zbl

[27] J. F. Bonnans, E. Casas, “Optimal control of semilinear multistate systems with state constraints”, SIAM J. Control Optim., 27:2 (1989), 446–455 | DOI | MR | Zbl

[28] E. Casas, F. Tröltzsch, A. Unger, “Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations”, SIAM J. Control Optim., 38:5 (2000), 1369–1391 | DOI | MR | Zbl

[29] A. Rösch, F. Tröltzsch, “On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints”, SIAM J. Control Optim., 46:3 (2007), 1098–1115 | DOI | MR | Zbl