Geodesic
Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed
Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 117-123.

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

For a large time interval $T$, we study the boundary control of an elastic force at the endpoint $x=0$ of a string under the condition that the second endpoint $x=l$ of the string is fixed. We determine and present in an analytical form the optimal boundary control $u_x(0,t)=\mu (t)$ that minimizes the elastic boundary energy integral $\int _0^T\mu ^2(t)\,dt$ over the set of all functions $\mu (t)$ from the class $L_2[0,T]$ under the condition that the oscillation process transfers the string from an arbitrary given initial state into an arbitrary given final state. 
@article{TRSPY_2005_248_a11,
     author = {V. A. Il'in},
     title = {Optimization of the {Boundary} {Control} of an {Elastic} {Force} at {One} {Endpoint} of a {String} with the {Other} {Endpoint} {Fixed}},
     journal = {Informatics and Automation},
     pages = {117--123},
     publisher = {mathdoc},
     volume = {248},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a11/}
}
TY  - JOUR
AU  - V. A. Il'in
TI  - Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed
JO  - Informatics and Automation
PY  - 2005
SP  - 117
EP  - 123
VL  - 248
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a11/
LA  - ru
ID  - TRSPY_2005_248_a11
ER  - 
%0 Journal Article
%A V. A. Il'in
%T Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed
%J Informatics and Automation
%D 2005
%P 117-123
%V 248
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a11/
%G ru
%F TRSPY_2005_248_a11
V. A. Il'in. Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed. Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 117-123. http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a11/

[1] Ilin V.A., “Granichnoe upravlenie protsessom kolebanii na dvukh kontsakh v terminakh obobschennogo resheniya volnovogo uravneniya s konechnoi energiei”, Dif. uravneniya, 36:11 (2000), 1513–1528 | MR | Zbl

[2] Ilin V.A., “Granichnoe upravlenie protsessom kolebanii na odnom kontse pri zakreplennom vtorom kontse v terminakh obobschennogo resheniya volnovogo uravneniya s konechnoi energiei”, Dif. uravneniya, 36:12 (2000), 1670–1686 | MR | Zbl

[3] Ilin V.A., “O razreshimosti smeshannykh zadach dlya giperbolicheskogo i parabolicheskogo uravnenii”, UMN, 15:2 (1960), 97–154 | MR | Zbl