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Estimates for normal solutions in problems with irregular zonal controls for the wave equation

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Abstract

For the wave equation with variable coefficients and boundary conditions of the first kind, we consider mutually dual problems with irregular zonal controls and regular zonal observations. Constructive estimates of well-posed solvability are obtained for the observation problem with strong generalized solutions on sufficiently large time intervals. These estimates contain information necessary for the construction of stable approximations to solutions of both problems with the use of the earlier suggested variational method.

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References

  1. Ho, L.F., Exact Controllability of the One-DimensionalWave Equation with Locally Distributed Control, SIAM J. Control Optim., 1990, vol. 28, no. 3, pp. 733–748.

    Article  MATH  MathSciNet  Google Scholar 

  2. Il’in, V.A., Boundary Control of Vibrations at Two Ends in Terms of the Generalized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2000, vol. 36, no. 11, pp. 1523–1528.

    Google Scholar 

  3. Il’in, V.A., Boundary Control of Vibrations at One End with the Other End Fixed in Terms of the Generalized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2000, vol. 36, no. 12, pp. 1670–1686.

    MathSciNet  Google Scholar 

  4. Lasiecka, I., Lions, J.-L., and Triggiani, R., Non Homogeneous Boundary Value Problems for Second Order Hyperbolic Operators, J. Math. Pures Appl., 1986, vol. 65, no. 2, pp. 149–192.

    MATH  MathSciNet  Google Scholar 

  5. Vasil’ev, F.P., On Duality in Linear Problems of Control and Observation, Differ. Uravn., 1995, vol. 31, no. 11, pp. 1893–1900.

    Google Scholar 

  6. Potapov, M.M., A Stable Method for Solving Linear Equations with a Nonuniformly Perturbed Operator, Dokl. Akad. Nauk, 1999, vol. 365, no. 5, pp. 596–598.

    MATH  MathSciNet  Google Scholar 

  7. Komornik, V., Exact Controllability and Stabilization. The Multiplier Method, Paris, 1994.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    M. M. Potapov

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  1. M. M. Potapov
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Original Russian Text © M.M. Potapov, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 10, pp. 1473–1479.

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Potapov, M.M. Estimates for normal solutions in problems with irregular zonal controls for the wave equation. Diff Equat 45, 1507–1513 (2009). https://doi.org/10.1134/S0012266109100140

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  • DOI: https://doi.org/10.1134/S0012266109100140

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