Geodesic
Boundary control of wave system in the space of generalized solutions on a graph
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2013), pp. 112-120 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A boundary value problem for the wave equation on an arbitrary geometrical graph is considered. To analyze this problem a spectral technique is used (Fourier analysis): the difficulties are generated by the geometry of the graph which can be relatively easely overcome especially in the case where the graph contains cycles. On the other hand, the possibility of expansion in generalized eigenfunctions of the corresponding boundary value problem is effectively used in the proofs of existence theorems by the methods represented in the known works of S. L. Sobolev, O. A. Ladyzhenskaya, V. I. Smirnov. On the example of a model problem on a star-graph the existence of boundary control actions is substantiated and the method for finding them is presented. To simplify the above formulas the length of edges is multiple $\pi$, the wave equation is used in its simplest form: $u_{tt}=u_{xx}$. The main results of the work are presented as formulas that determine unknown boundary control as a function of time. Bibliogr. 5.
Keywords: wave equation on a graph, boundary problem, unique solvability, boundary control. 
@article{VSPUI_2013_3_a11,
     author = {V. V. Provotorov and Yu. A. Gnilitskaya},
     title = {Boundary control of wave system in the space of generalized solutions on~a~graph},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {112--120},
     year = {2013},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2013_3_a11/}
}
TY  - JOUR
AU  - V. V. Provotorov
AU  - Yu. A. Gnilitskaya
TI  - Boundary control of wave system in the space of generalized solutions on a graph
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2013
SP  - 112
EP  - 120
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2013_3_a11/
LA  - ru
ID  - VSPUI_2013_3_a11
ER  - 
%0 Journal Article
%A V. V. Provotorov
%A Yu. A. Gnilitskaya
%T Boundary control of wave system in the space of generalized solutions on a graph
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2013
%P 112-120
%N 3
%U http://geodesic.mathdoc.fr/item/VSPUI_2013_3_a11/
%G ru
%F VSPUI_2013_3_a11
V. V. Provotorov; Yu. A. Gnilitskaya. Boundary control of wave system in the space of generalized solutions on a graph. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2013), pp. 112-120. http://geodesic.mathdoc.fr/item/VSPUI_2013_3_a11/

[1] Provotorov V. V., “The construction of boundary controls in the problem of vibration damping system of strings”, Vestnik S.-Peterb. un-ta, ser. 10: pricladnaya matematica, informatica, processes upravleniya, 2012, no. 1, 62–71 | MR

[2] Gnilitskaya Yu. A., “Boundary control of vibrations of a stringsystem”, Processes control and stability, Works of the 43rd Intern. conference of graduate students and students, eds. A. S. Eremin, N. V. Smirnov, Izdat. Dom S.-Peterb. gos. un-ta, S.-Peterburg, 2012, 21–25

[3] Ladyzhenskaya O. A., The boundary value problems of mathematical physics, Nauka, M., 1973, 407 pp. | MR

[4] Mikhlin S. G., Linear partial differential equations, Vysshaija shkola, M., 1977, 431 pp. | MR

[5] Volkova A. S., “Fredholm solvability in the class of $W_2^2$ of the Dirichlet problem for equation of elliptic type on the star graph”, Mathematics and its applications, 1:8 (2011), 15–28