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Boundary control of forced string vibrations by first directional derivatives on a short time interval

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Abstract

We solve a problem on the boundary control of forced vibrations of a homogeneous string by two first directional derivatives with time-dependent coefficients and noncharacteristic directions in the boundary conditions on a short time interval in the set of classical solutions. We obtain smoothness, coordination, and controllability conditions on the right-hand side of the equation and the initial and terminal data necessary and sufficient for the unique boundary control of the wave process.

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References

  1. Il’in, V.A. and Tikhomirov, V.V., The Wave Equation with Boundary Control at Two Ends and the Problem of Complete Damping of a Vibration Process, Differ. Uravn., 1999, vol. 35, no. 5, pp. 692–704.

    MathSciNet  Google Scholar 

  2. Moiseev, E.I. and Kholomeeva, A.A., Optimal Boundary Displacement Control at One End of a String with a Medium Exerting Resistance at the Other End, Differ. Uravn., 2013, vol. 49, no. 10, pp. 1350–1355.

    MathSciNet  Google Scholar 

  3. Sergeev, S.A., Necessary and Sufficient Conditions for the Existence of a Boundary Control of the Vibrations of a String and a Spherical Layer for Small Time Values, Differ. Uravn., 2011, vol. 47, no. 5, pp. 744–755.

    MathSciNet  Google Scholar 

  4. Lomovtsev, F.E. and Novikov, E.N., Necessary and Sufficient Conditions for the Vibrations of a Bounded String with Dirichlet Derivatives in the Boundary Conditions, Differ. Uravn., 2014, vol. 50, no. 1, pp. 126–129.

    MathSciNet  Google Scholar 

  5. Lomovtsev, F.E. and Novikov, E.N., On Necessary Conditions for the Right-Hand Side of the Equation of Vibrations of a Bounded String under Nonstationary Boundary Conditions with the First Directional Derivative, in Mater. Voronezhs. vesennei mat. shk. “Sovremennye metody teorii kraevykh zadach” (Voronezh, 3–9 maya 2014 g.) (Proc. Voronezh Spring Math. School “Modern Method of Theory of Boundary Value Problems” (Voronezh, May 3–9, 2014)), Voronezh, 2014, pp. 109–112.

    Google Scholar 

  6. Petrovskii, I.G., Lektsii po teorii integral’nykh uravnenii (Lectures on the Theory of Integral Equations), Moscow, 2003.

    Google Scholar 

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    F. E. Lomovtsev

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  1. F. E. Lomovtsev
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Correspondence to F. E. Lomovtsev.

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Original Russian Text © F.E. Lomovtsev, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 12, pp. 1669–1675.

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Lomovtsev, F.E. Boundary control of forced string vibrations by first directional derivatives on a short time interval. Diff Equat 51, 1649–1655 (2015). https://doi.org/10.1134/S0012266115120125

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  • DOI: https://doi.org/10.1134/S0012266115120125

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