Numerical Solution of the Positional Boundary Control Problem for the Wave Equation with Unknown Initial Data

  • A. A. Dryazhenkov
  • M. M. Potapov


The problem of one-sided boundary Neumann control is considered for the onedimensional wave equation. Information about the initial state of the process is absent. Instead, the values of Dirichlet observations are received in real time at the controlled boundary. The aim is to bring the process to a complete rest by means of positional boundary controls. To solve this problem, we propose an efficient numerical algorithm with an optimal guaranteed damping time. Some results of numerical experiments are presented.


wave equation positional control problem numerical solution. 


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  1. 1.
    F. P. Vasil’ev, M. A. Kurzhanskii, M. M. Potapov, and A. V. Razgulin, Approximate Solution of Dual Control and Observation Problems (MAKS, Moscow, 2010) [in Russian].Google Scholar
  2. 2.
    M. M. Potapov and A. A. Dryazhenkov, “Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition,” Proc. Steklov Inst. Math. 277, 206–220 (2012).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].MATHGoogle Scholar
  4. 4.
    V. A. Il’in, “The solvability of mixed problems for hyperbolic and parabolic equations,” Russ. Math. Surv. 15 (2), 85–142 (1960).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Yu. S. Osipov, “Control packages: An approach to solution of positional control problems with incomplete information,” Russ. Math. Surv. 61 (4), 611–661 (2006).CrossRefMATHGoogle Scholar
  6. 6.
    A. V. Kryazhimskii and Yu. S. Osipov, “Idealized program packages and problems of positional control with incomplete information,” Proc. Steklov Inst. Math. 268 (Suppl. 1), S155–S174 (2010).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    L. V. Kritskov and M. F. Abdukarimov, “Boundary control of the displacement at one end with the other end free for a process described by the telegraph equation with a variable coefficient,” Dokl. Math. 87 (3), 351–353 (2013).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976) [in Russian].MATHGoogle Scholar
  9. 9.
    D. A. Ivanov and M. M. Potapov, “Time-optimal boundary controls for the wave equation,” in Stability and Oscillations of Nonlinear Control Systems: Proceedings of the 13th International Conference Workshop, Moscow, Russia, 2016 (Inst. Probl. Upravl., Moscow, 2016), pp. 166–168.Google Scholar
  10. 10.
    A. A. Dryazhenkov and M. M. Potapov, “Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint,” Comp. Math. Math. Phys. 54 (6), 939–952 (2014).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. A. Dryazhenkov, “Observability inequality for a wave equation with elastic fastening in the case of critical time interval,” Moscow Univ. Comp. Math. Cybernet. 38 (3), 105–109 (2014).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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