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Optimal boundary control by a force at one end of a string with a given force mode at the other end. (English. Russian original) Zbl 1235.35190

Differ. Equ. 47, No. 10, 1508-1513 (2011); translation from Differ. Uravn. 47, No. 10, 1492-1497 (2011).
Summary: We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35Q93 PDEs in connection with control and optimization
35L05 Wave equation
74K05 Strings
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References:

[1] Moiseev, E.I. and Kholomeeva, A.A., Optimal Boundary Control by the Displacement at One End of the String for a Given Elastic Force at the Other End, Tr. Inst. Mat. Mekh., 2011, vol. 17, no. 2, pp. 151–158. · Zbl 1251.49005
[2] Moiseev, E.I. and Kholomeeva, A.A., Optimal Boundary Control by a Force at One End of the String for a Given Displacement Mode at the Other End, Differ. Uravn., 2011, vol. 47, no. 7, pp. 1018–1023. · Zbl 1251.49005
[3] Moiseev, E.I. and Kholomeeva, A.A., Optimal Boundary Control by Displacement of String Vibrations with a Nonlocal Oddness Condition of the First Kind, Differ. Uravn., 2010, vol. 46, no. 11, pp. 1623–1630. · Zbl 1210.49021
[4] Moiseev, E.I. and Kholomeeva, A.A., Optimal Boundary Control by Displacements of String Vibrations with a Nonlocal Evenness Condition of the Second Kind, Differ. Uravn., 2011, vol. 47, no. 1, pp. 127–134. · Zbl 1287.74017
[5] Kholomeeva, A.A., Optimal Boundary Control of String Vibrations with Model Nonlocal Conditions of One of Two Types, Dokl. Akad. Nauk, 2011, vol. 437, no. 2, pp. 164–167. · Zbl 1234.35268
[6] Il’in, V.A. and Moiseev, E.I., Optimization of Boundary Controls of String Vibrations, Uspekhi Mat. Nauk, 2005, vol. 60, no. 6, pp. 89–114. · doi:10.4213/rm1678
[7] Zygmund, A., Trigonometric Series, Cambridge, 1959. Translated under the title Trigonometricheskie ryady, Moscow, 1965, vol. 1. · Zbl 0085.05601
[8] Il’in, V.A. and Moiseev, E.I., Optimization of Control by Elastic Boundary Forces at Two Ends of a String on an Arbitrary Sufficiently Large Time Interval, Differ. Uravn., 2008, vol. 44, no. 1, pp. 89–110.
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