Geodesic
Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 6, pp. 1015-1032 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Mutually dual difference approximations are constructed for the problems of boundary control and observation for the differential equation describing the transverse vibrations of an inhomogeneous bar. (The equation is second-order in time and fourth-order in space.) The controllability and observability condition requiring that the length of the underlying time interval be larger than a certain critical value is assumed to be fulfilled. It is shown how the a priori information available in this case can be used to construct approximate solutions to both difference problems that converge strongly for arbitrary objective and observed states in the classes under consideration. 
@article{ZVMMF_2005_45_6_a5,
     author = {M. M. Potapov},
     title = {Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1015--1032},
     year = {2005},
     volume = {45},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_6_a5/}
}
TY  - JOUR
AU  - M. M. Potapov
TI  - Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2005
SP  - 1015
EP  - 1032
VL  - 45
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_6_a5/
LA  - ru
ID  - ZVMMF_2005_45_6_a5
ER  - 
%0 Journal Article
%A M. M. Potapov
%T Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2005
%P 1015-1032
%V 45
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_6_a5/
%G ru
%F ZVMMF_2005_45_6_a5
M. M. Potapov. Approximate solution to boundary control and observation problems for the equation describing transverse vibrations of a bar. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 6, pp. 1015-1032. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_6_a5/

[1] Lions J.-L., “Exact controllability, stabilizability and perturbations for distributed systems”, SIAM Rev., 30:1 (1988), 1–68 | DOI | MR | Zbl

[2] Znamenskaya L. H., Upravlenie uprugimi kolebaniyami, Fizmatlit, M., 2004 | Zbl

[3] Vasilev F. P., “O dvoistvennosti v lineinykh zadachakh upravleniya i nablyudeniya”, Differents. ur-niya, 31:11 (1995), 1894–1901

[4] Potapov M. M., “Ustoichivyi metod resheniya lineinykh uravnenii s neravnomerno vozmuschennym operatorom”, Dokl. RAN, 365:5 (1999), 596–598 | MR | Zbl

[5] Potapov M. M., “O silnoi skhodimosti raznostnykh approksimatsii zadach granichnogo upravleniya i nablyudeniya dlya volnovogo uravneniya”, Zh. vychisl. matem. i matem. fiz., 38:3 (1998), 387–397 | MR | Zbl

[6] Potapov M. M., Kostikova O. P., “Raznostnaya approksimatsiya zadach upravleniya i nablyudeniya dlya uravneniya kolebanii chetvertogo poryadka”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibernetika, 2003, no. 1, 33–37 | MR | Zbl

[7] Krein S. G., Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971 | MR

[8] Korneichuk N. P., Splainy v teorii priblizheniya, Nauka, M., 1984 | MR

[9] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl