Applied and Computational Mathematics
Volume 7, Issue 3, June 2018, Pages: 75-82
Received: Apr. 24, 2018;
Accepted: May 10, 2018;
Published: May 25, 2018
Views 1094 Downloads 119
Fatma Toyoğlu, Department of Mathematics, Faculty of Art and Science, Erzincan University, Erzincan, Turkey
In this study, the problem of determining the control function that is at the right hand side of a hyperbolic system from the final observation is investigated. Using the Fourier-Galerkin method, the weak solution of this hyperbolic system is obtained. The necessary conditions for the existence and uniqueness of the optimal solution are proved. We also find the approximate solutions of the test problems in numerical examples by a MAPLE® program. Finally, the numerical results are presented in the form of tables.
On the Solution of a Optimal Control Problem for a Hyperbolic System, Applied and Computational Mathematics.
Vol. 7, No. 3,
2018, pp. 75-82.
Ladyzhenskaya, O. A., Boundary Value Problems in Mathematical Physics, Springer, New York, 1985.
Vasilyev, F. P., Numerical Methods for Solving Extremal Problems, Nauka, Moskow, 1988.
Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.
Periago, F., Optimal shape and position of the support for the internal exact control of a string, Systems & Control Letters, 58, 136-140, 2009.
Yamamoto, M., Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Probl., 11, 481-496, 1995.
Benamou, J. D., 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. 15-36, 1999.
Kim J., Pavol., N. H., Optimal control problem for the periodic one-dimensional wave equation, Nonlinear Analysis Forum 3, 89-110, 1998.
Lopez, A., Zhang, X., Zuazua, E., Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations, J. Math. Pures. Appl., 79, 8, 741-808, 2000.
Privat Y., Trelat., E., Zuazua, E., Optimal location of controllers for the one-dimensional wave equation, Annales de L’Institut Henri Poincare (C) Analyse Non Lineaire, 30, 1097-1126, 2013.
Subaşı, M., and Saraç, Y., A Minimizer for Optimizing the Initial Velocity in a Wave Equation, Optimization, 61, 3, 327-333, 2012.
Saraç, Y., and Şener, S. Ş., “Identification of the transverse distributed loan in Euler-Bernollu beam equation from boundary measurement”, International Journal of Modelling and Optimization, 8 (1), 2018.
Saraç, Y., “Symbolic and numeric computation of optimal initial velocity in a wave equation”, Journal of Computational and Nonlinear Dynamics, 8 (1), 2013.
İskenderov, A. D., Tagiyev, R. Q. and Yagubov, Q. Y, Optimization Methods, Çaşıoğlu, Bakü, 2002.
Şener, S. Ş, Saraç, Y and Subaşı, M, “Weak solutions to hyperbolic problems with inhomogeneous Dirichlet and Neumann boundary conditions”, Applied Mathematical Modelling, 37, pp. 2623-2629, 2013.
Hasanov, A., “Simultaneous Determination of the Source Terms in a Linear Hyperbolic Problem from the Final Overdetermination: Weak Solution Approach,” IMA J. Appl. Math., 74, pp. 1-19, 2009.
Alemdar Hasanov and Alexandre Kawano, Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data, Inverse Problems, Vol:32, 1-31, 2016.