Abstract
In [2]we have considered the problem of controlling a vibrating string at the right-hand side where the left end is kept fixed. The results of this paper can be easily extended to vibrating systems of several dimensions which are described by the wave equation. This is the purpose of the present note. In [2] we also discussed the problem of controllability that has been previously studied by Butkovskiy [1], Russell [4], [5], and Roxin [3]. In [6],[7] controllability for hyperbolic equations in several space dimensions has been investigated by Russell too.
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References
Butkovskiy,A.G.: Theory of optimal control of distributed parameter systems. Elsevier Publ.Comp.,New York-London-Amsterdam 1969
Krabs,W.: Ein Kontroll-Approximationsproblem für die schwingende Saite. To appear in ISNM
Roxin,E.: Optimierungsprobleme mit der Wellengleichung. Preprint Nr. 146 des Fachbereichs Mathematik der TH Darmstadt, Juli 1974
Russell,D.L.: On boundary-value controllability of linear symmetric hyperbolic systems. “Proceedings of the Conference on the Mathematical Theory of Control”, edited by A.V. Balakrishnan and L.W. Neustadt, Univ. of Southern Calif., Academic Press, New York 1967, pp. 312–321
Russell,D.L.: Nonharmonic Fourier series in the control of distributed parameter systems. J.Mathem.Anal.App1. 18 (1967), 542–560
Russell,D.L.: Boundary value control of the higher-dimensional wave equation. SIAM J.Control 9 (1971), 29–42
Russell,D.L.: Boundary value control theory of the higher-dimensional wave equation, Part II. SIAM J.Control 9 (1971),401–419
Triebel,H.: Höhere Analysis. VEB Deutscher Verlag der Wissenschaften, Berlin 1972
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Krabs, W. (1976). Boundary Control of the Higher-Dimensional Wave Equation. In: Oettli, W., Ritter, K. (eds) Optimization and Operations Research. Lecture Notes in Economics and Mathematical Systems, vol 117. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46329-7_16
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DOI: https://doi.org/10.1007/978-3-642-46329-7_16
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