Geodesic
Boundary Control of Spherically Symmetric Oscillations of a~Three-Dimensional Ball
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 144-155.

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The problem of boundary control of radially symmetric oscillations of a 3-ball that are described by a wave equation whose solutions $u(r, t)$ admit the existence of finite energy at every moment of time is studied. The state of an oscillating ball at every fixed moment of time $t$ is characterized by a pair of functions $\{ u (r, t), u_t (r, t) \}$. A minimal time interval $T$ is determined that is sufficient for changing an arbitrary initial state $\{ u (r, 0), u_t (r, 0) \}$ of the oscillation process to an arbitrary preset state $\{ u (r, T), u_t (r, T) \}$ with the use of a boundary control on the ball surface. 
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     author = {V. A. Il'in},
     title = {Boundary {Control} of {Spherically} {Symmetric} {Oscillations} of {a~Three-Dimensional} {Ball}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {144--155},
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     url = {http://geodesic.mathdoc.fr/item/TM_2001_232_a12/}
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V. A. Il'in. Boundary Control of Spherically Symmetric Oscillations of a~Three-Dimensional Ball. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 144-155. http://geodesic.mathdoc.fr/item/TM_2001_232_a12/

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[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