Geodesic
A terminal--boundary value problem that describes the process of damping the vibrations of a~rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times
Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 133-142.

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In this paper, in terms of a finite-energy weak solution, we study a terminal–boundary value problem describing the complete damping, at a terminal time $T$, of the longitudinal vibrations of a rod consisting of two segments with different densities and elasticity coefficients under the condition that the lengths of the segments are such that the wave travel times along these segments are equal. We find an explicit analytic expression for a solution to this problem and prove its uniqueness. This problem is important for the design of acoustic systems in which one can completely damp an acoustic signal by a terminal time instant by applying boundary controls at the ends of a vibrating rod. 
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V. A. Il'in. A terminal--boundary value problem that describes the process of damping the vibrations of a~rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times. Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 133-142. http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a9/

[1] Ilin V.A., Luferenko P.V., “Obobschennye resheniya smeshannykh zadach dlya razryvnogo volnovogo uravneniya pri uslovii ravenstva impedansov”, DAN, 429:3 (2009), 317–321 | MR | Zbl

[2] Ilin V.A., “O prodolnykh kolebaniyakh sterzhnya, sostoyaschego iz dvukh uchastkov raznoi plotnosti i uprugosti, v sluchae sovpadeniya vremeni prokhozhdeniya volny po kazhdomu iz etikh uchastkov”, DAN, 429:6 (2009), 742–745 | MR | Zbl

[3] 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

[4] 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

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

[6] Ilin V.A., Izbrannye trudy, v. 1, Maks-Press, M., 2008