DOI: https://doi.org/10.26089/NumMet.v17r328

Increasing regularity of generalized solutions to the wave equation for computing optimal boundary controls

Authors

  • D.A. Ivanov

Keywords:

wave equation
generalized solution
boundary control
subcritical interval
approximate data
approximate solution
convergence

Abstract

Problems with two-sided boundary controls of three main types are considered for the wave equation in the classes of weak generalized solutions on intervals of subcritical length. An algorithm is proposed for the stable approximation of boundary controls. This algorithm is based on the preliminary smoothing of phase trajectories, the application of a variational method in the classes of strong generalized solutions, and the final differentiation of the resulting smoothed controls. Numerical results are discussed.

New computational technologies

Downloads

Published

2016-07-25

Issue

Vol. 17 (2016): Issue 3.

Section

Section 1. Numerical methods and applications

Author Biography

D.A. Ivanov

Lomonosov Moscow State University,
Faculty of Computational Mathematics and Cybernetics
• PhD Student

References

  1. M. M. Potapov and D. A. Ivanov, “Problems of Two-Sided Boundary Control for the Wave Equation on Subcritical Intervals in Classes of Strong Generalized Solutions,” Tr. Inst. Mat. Mekh. UrO RAN 19 (4), 192-202 (2013) [Proc. Steklov Inst. Math. 287 (Suppl. 1), 145-155 (2014)].
  2. D. A. Ivanov and M. M. Potapov, “Continuous Invertibility of a Boundary Control Operator for the Wave Equation on Subcritical Intervals in Classes of Weak Generalized Solutions,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 4, 5-12 (2014) [Moscow Univ. Comput. Math. Cybern. 38 (4), 139-146 (2014)].
  3. M. M. Potapov, “A Stable Method for Solving Linear Equations with Nonuniformly Perturbed Operators,” Dokl. Akad. Nauk 365 (5), 596-598 (1999) [Dokl. Math. 59 (2), 286-288 (1999)].
  4. F. P. Vasil’ev, M. A. Kurzhanskii, M. M. Potapov, and A. V. Razgulin, Approximate Solution of Dual Control and Observation Problems (Maks Press, Moscow, 2010) [in Russian].
  5. D. A. Ivanov and M. M. Potapov, “Approximate Solution to a Time Optimal Boundary Control Problem for the Wave Equation,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 291, 112-127 (2015) [Proc. Steklov Inst. Math. 291 (1), 102-117 (2015)].
  6. M. M. Potapov, “Approximate Solutions to Dirichlet Control Problems for the Wave Equation in Sobolev Classes and Dual Observation Problems,” Zh. Vychisl. Mat. Mat. Fiz. 46 (12), 2191-2208 (2006) [Comput. Math. Math. Phys. 46 (12), 2092-2109 (2006)].
  7. M. M. Potapov, “Finite-Difference Approximation of Dirichlet Observation Problems for Weak Solutions to the Wave Equation Subject to Robin Boundary Conditions,” Zh. Vychisl. Mat. Mat. Fiz. 47 (8), 1323-1339 (2007) [Comput. Math. Math. Phys. 47 (8), 1268-1284 (2007)].