Abstract and Applied Analysis

On the Regularized Solutions of Optimal Control Problem in a Hyperbolic System

Yeşim Saraç and Murat Subaşı

Full-text: Open access

Abstract

We use the initial condition on the state variable of a hyperbolic problem as control function and formulate a control problem whose solution implies the minimization at the final time of the distance measured in a suitable norm between the solution of the problem and given targets. We prove the existence and the uniqueness of the optimal solution and establish the optimality condition. An iterative algorithm is constructed to compute the required optimal control as limit of a suitable subsequence of controls. An iterative procedure is implemented and used to numerically solve some test problems.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 156541, 12 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174057

Digital Object Identifier
doi:10.1155/2012/156541

Mathematical Reviews number (MathSciNet)
MR2955024

Zentralblatt MATH identifier
1248.49009

Citation

Saraç, Yeşim; Subaşı, Murat. On the Regularized Solutions of Optimal Control Problem in a Hyperbolic System. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 156541, 12 pages. doi:10.1155/2012/156541. https://projecteuclid.org/euclid.aaa/1365174057


Export citation

References

  • J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer, New York, NY, USA, 1971.
  • M. Negreanu and E. Zuazua, “Uniform boundary controllability of a discrete 1-D wave equation,” Systems & Control Letters, vol. 48, no. 3-4, pp. 261–279, 2003.
  • L. I. Bloshanskaya and I. N. Smirnov, “Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem,” Differential Equations, vol. 45, no. 6, pp. 878–888, 2009.
  • A. Smyshlyaev and M. Krstic, “Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary,” Systems & Control Letters, vol. 58, no. 8, pp. 617–623, 2009.
  • X. Feng, S. Lenhart, V. Protopopescu, L. Rachele, and B. Sutton, “Identification problem for the wave equation with Neumann data input and Dirichlet data observations,” Nonlinear Analysis, vol. 52, no. 7, pp. 1777–1795, 2003.
  • A. Münch, P. Pedregal, and F. Periago, “Optimal design of the damping set for the stabilization of the wave equation,” Journal of Differential Equations, vol. 231, no. 1, pp. 331–358, 2006.
  • J. D. Benamou, “Domain decomposition, optimal control of systems governed by partial differential equations, and synthesis of feedback laws,” Journal of Optimization Theory and Applications, vol. 102, no. 1, pp. 15–36, 1999.
  • F. Periago, “Optimal shape and position of the support for the internal exact control of a string,” Systems & Control Letters, vol. 58, no. 2, pp. 136–140, 2009.
  • O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, vol. 49 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1985.
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization, Springer, New York, NY, USA, 1985.