Hiperbolik Denklem İçeren Bir Optimal Kontrol Probleminin Nümerik Çözümü Üzerine

Yıl 2020, Cilt: 9 Sayı: 3, 1091 - 1101, 26.09.2020

Seda İğret Araz

https://doi.org/10.17798/bitlisfen.665651

Öz

Bu makalede, hiperbolik denklem içeren optimal kontrol problemlerinin bir sınıfını çözmek için bir nümerik algoritma sunuyoruz. Bir regüler uzayda optimal çözümün var ve tek olduğunu gösteriyoruz. Eşlenik problemi elde ettikten ve amaç fonksiyonelinin türevini hesapladıktan sonra, Gradyen metoduyla nümerik yaklaşımlar elde edilir. Hesaplanan sonuçlar, önerilen metodun optimal kontrol problemleri için iyi nümerik yaklaşımlar üretebildiğini gösterir.

Anahtar Kelimeler

Optimal Kontrol, Hiperbolik denklemler

Kaynakça

• Deiveegan A., Prakash P., Nieto P.P., 2017. Optimization Method for Identifying The Source Term in an Inverse Wave Equation. Electronic Journal of Differential Equations, Vol. 2017, 1-15.
• Tagiyev R.K., 2012. On Optimal Control of the Hyperbolic Equation Coefficients. Automation and Remote Control, 1145-1155.
• Kröner A., 2011. Adaptive Finite Element Methods for Optimal Control of Second Order Hyperbolic Equations. Computational Methods in Applied Mathematics, 214-240.
• Bahaa G.M., 2012. Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags. Intelligent Control and Automation, 3, 211-221.
• Ju EY., Jeong J.M., 2013. Optimal control problems for hyperbolic equations with damping terms involving p-Laplacian. Journal of Inequalities and Applications, 92.
• Hwang J., Nakagiri S., 2006. Optimal control problems for the equation of motion of membrane with strong viscosity. Journal of Mathematical Analysis and Applications, 321 (1) 327-342.
• Lions J.L., 1971. Optimal Control of Systems Governed by Partial Differential Equations. Springer, 273-291 p, Berlin.
• Subaşı M., İğret Araz S., Numerical Regularization of Optimal Control for the Coefficient Function in a Wave Equation, Iranian Journal of Science and Technology, Transactions A: Science, 2019, 1-9.
• Bahaa G.M.,Tharwat M.M., 2011. Optimal control problem for infinite variables hyperbolic systems with time lags. Archives of Control Sciences, 21 (4), 373-393.
• Mordukhovich B.S., Raymond J.P., 2004. Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim. 2004; 49, 145.157.
• Lagnese J.E., Leugering G., 2003. Time-domain decomposition of optimal control problems for the wave equation. Systems Control Lett., 48, 229.242.
• Barbu V., Pavel N.H., 1997. Determining the acoustic impedance in the 1-D wave equation via an optimal control problem. SIAM. J. Control. Optim., 35, 1544.1556.
• Liang M., 1999. Bilinear optimal control for a wave equation. Math. Models Methods Appl. Sci., 9, 45.68.
• Ton B.A., 2003. An inverse source problem for the wave equation. Nonlinear Anal., 55, 269.284.
• Ladyzhenskaya O. A., 1985. Boundary Value Problems in Mathematical Physics. Springer-Verlag, 322 p, New York.
• Goebel M., 1979. On Existence of Optimal Control. Math. Nachr., Vol 93, 67-73.
• Yosida K., 1980. Functional Analysis. Springer-Verlag, 624 p, New York.
• Vasilyev F.P., 1981. Ekstremal problemlerin çözüm metotları, Nauka, 400.
• Subaşı M., 2004. A Variational method of optimal control problems for nonlinear Schrödinger equation. Numerical Methods for Partial Differential Equations, 20(1), 82-89.
• Li Q.H., Wang J., 2013. Weak Galerkin Finite Element methods for parabolic equations, Numerical Methods for Partial Differential Equations, 29,2004-2024.