Geodesic
Forward dynamical problem for Timoshenko beam
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 38, Tome 369 (2009), pp. 16-47 
M. I. Belishev; A. L. Pestov. Forward dynamical problem for Timoshenko beam. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 38, Tome 369 (2009), pp. 16-47. http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a1/
@article{ZNSL_2009_369_a1,
     author = {M. I. Belishev and A. L. Pestov},
     title = {Forward dynamical problem for {Timoshenko} beam},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {16--47},
     year = {2009},
     volume = {369},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a1/}
}
TY  - JOUR
AU  - M. I. Belishev
AU  - A. L. Pestov
TI  - Forward dynamical problem for Timoshenko beam
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2009
SP  - 16
EP  - 47
VL  - 369
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a1/
LA  - ru
ID  - ZNSL_2009_369_a1
ER  - 
%0 Journal Article
%A M. I. Belishev
%A A. L. Pestov
%T Forward dynamical problem for Timoshenko beam
%J Zapiski Nauchnykh Seminarov POMI
%D 2009
%P 16-47
%V 369
%U http://geodesic.mathdoc.fr/item/ZNSL_2009_369_a1/
%G ru
%F ZNSL_2009_369_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We deal with an initial boundary value problem of the form \begin{align*} &\rho u_{tt}-(\Gamma u_x) _x+Au_x+Bu=0,\qquad x>0,\quad 0<t<T,\\ &u|_{t=0}=u_t|_{t=0}=0,\qquad x\geq0,\\ &u|_{x=0}=f,\qquad0\leq t\leq T, \end{align*} where $\rho=\mathrm{diag}\{\rho_1,\rho_2\}$, $\Gamma=\mathrm{diag}\{\gamma_1,\gamma _2\}$, $A$, and $B$ are smooth $2\times2$-matrix functions of $x$, whereas $\rho_i,\gamma_i$ are smooth positive functions provided $0<\frac{\rho_1(x)}{\gamma_1(x)}<\frac{\rho_2(x)}{\gamma_2(x)}$, $x\geq0$; $f=\mathrm{col}\{f_1(t),f_2(t)\}$ is a boundary control; $u=u^f(x,t)=\mathrm{col}\{u_1^f(x,t),u_2^f(x,t)\}$ is a solution (wave). Such a problem describes the wave processes in a system, where two different wave modes occur and propagate with different velocities. The modes interact that implies interesting physical effects but, on the other hand, complicates the picture of waves. For controls $f\in L_2((0,T);\mathbb R^2)$, we reduce the problem to the relevant integral equation, define the the generalized solutions $u^f$, and establish the well-possedness of the problem. Also, the fundamental matrix-valued solution is introduced and its leading singularities are studied. The existence of the “slow waves” that are the certain mixture of modes, which propagate with the slow mode velocity, is established. Bibl. – 11 titles.

[1] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[2] M. I. Belishev, S. A. Ivanov, “Granichnoe upravlenie i kanonicheskie realizatsii dvukhskorostnoi dinamicheskoi sistemy”, Zap. nauchn. semin. POMI, 222, 1995, 18–44 | MR | Zbl

[3] M. I. Belishev, A. S. Blagovestchenskii, S. A. Ivanov, “Erratum to ‘The two-velocity dynamical system: boundary control of waves and inverse problems [Wave Motion 25 (1997), 83–107]’ ”, Wave Motion, 26 (1997), 99 | DOI | MR | Zbl

[4] M. I. Belishev, S. A. Ivanov, “Kharakterizatsiya dannykh dinamicheskoi obratnoi zadachi dlya dvukhskorostnoi sistemy”, Zap. nauchn. cemin. POMI, 259, 1999, 19–45 | MR | Zbl

[5] M. I. Belishev, S. A. Ivanov, “O edinstvennosti “v malom” v dinamicheskoi obratnoi zadache dlya dvukhskorostnoi dinamicheskoi sistemy”, Zap. nauchn. semin. POMI, 275, 2001, 41–54 | MR | Zbl

[6] M. I. Belishev, S. A. Ivanov, “Vosstanovlenie parametrov sistemy svyazannykh balok po dinamicheskim granichnym dannym”, Zap. nauchn. cemin. POMI, 324, 2005, 20–42 | MR | Zbl

[7] E. I. Grigolyuk, I. G. Selezov, Neklassicheskie modeli teorii kolebanii sterzhnei, plastin i obolochek, Itogi nauki i tekhniki. Ser. Mekhanika tverdykh deformiruemykh tel, 5, M., 1973 | Zbl

[8] A. V. Zurov, “Effekty, svyazannye s sovpadeniem skorostei v dvukhskorostnoi dinamicheskoi sisteme”, Zap. nauchn. semin. POMI, 297, 2003, 49–65 | MR | Zbl

[9] L. P. Nizhnik, Obratnye zadachi rasseyaniya dlya giperbolicheskikh uravnenii, Izd. Naukova Dumka, Kiev, 1991 | MR

[10] A. Morassi, G. Nakamura, M. Sini, “An inverse dynamical problem for connected beams”, European J. Appl. Math., 16:1 (2005), 83–109 | DOI | MR | Zbl

[11] Rakesh, P. Sacs, “Stability for an inverse problem for a two speed hyperbolic pde in one space dimension”, Inverse Problems, 25 (2009) (to appear)