Abstract
We consider the problem of optimal boundary control of string vibrations by a displacement at one endpoint with the other endpoint being fixed; the problem is considered in the space
and then, more generally, in the space
, p ≥ 1. The control brings the vibration process from the quiescent state to an arbitrarily prescribed state in a time that is a multiple of the double string length.
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Il’in, V.A. and Moiseev, E.I., Optimization of Boundary Controls at One Endpoint of the String by a Displacement or an Elastic Force for an Arbitrary Sufficiently Large Time Interval T, Uspekhi Mat. Nauk, 2005, vol. 60, no. 6, pp. 89–114.
Il’in, V.A. and Moiseev, E.I., Minimization of the L p-Norm for p ≥ 1 of the Derivative of the Boundary Displacement Control on an Arbitrary Sufficiently Large Time Interval [0, T], Differ. Uravn., 2006, vol. 42, no. 11, pp. 1558–1570.
Il’in, V.A., On the Solvability of Mixed Problems for Hyperbolic and Parabolic Equations, Uspekhi Mat. Nauk, 1960, vol. 15, no. 2, pp. 97–154.
Moiseev, E.I., Optimal Boundary Control by a Displacement in W p′ of a String with a Free End, Differ. Uravn., 2008, vol. 44, no. 5, pp. 709–711.
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Original Russian Text © E.I. Moiseev, A.A. Kholomeeva, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 5, pp. 741–745.
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Moiseev, E.I., Kholomeeva, A.A. Optimization of the boundary control of string vibrations in the excitation problem with a fixed endpoint in the class W 22 . Diff Equat 45, 757–761 (2009). https://doi.org/10.1134/S0012266109050140
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DOI: https://doi.org/10.1134/S0012266109050140