On the Boundary Control of a Boussinesq System
A boundary control problem is considered for determining a canal depth function optimally for a canal system modeled by a nonlinear Boussinesq equation. By determining the optimal canal depth function, it is aimed to damp out the undesired waves in canal system filled up water. For achieving this aim, the existence and uniqueness of the solutions to system and controllability properties of the system is investigated. Optimal canal depth control function is obtained by means of a maximum principle, which is an elegant tool for transferring the optimal boundary control problem to solving a system of equations including initial-terminal-boundary conditions. The reason making this paper is important that optimal control function is gained without linearization of nonlinear term in the system. In order to show the correctness of the obtained theoretical results, several numerical examples are presented by MATLAB in graphical and table forms. Observing these tables and graphics, it is concluded that introduced boundary control algorithm is effective and has the potential for extending to other nonlinear control systems.
D. Adhikari, C. Cao, J. Wu and X. Xu, Small global solutions to the damped twodimensional Boussinesq equations, Journal of Differential Equations 256 (2014), 3594 – 3613, DOI: 10.1016/j.jde.2014.02.012.
J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, Journal De Mathématiques Pures Et Appliquées 17(2) (1872), 55 – 108, URL: http://sites.mathdoc.fr/JMPA/PDF/JMPA_1872_2_17_A2_0.pdf.
W. Chen and A. Sarria, Damped infinite energy solutions of the 3D Euler and Boussinesq equations, Journal of Differential Equations 265 (2018), 3841 – 3857, DOI: 10.1016/j.jde.2018.04.009.
A. Esfahani and L. G. Farah, Local well-posedness for the sixth-order Boussinesq equation, Journal of Mathematical Analysis and Applications 385 (2012), 230 – 242, DOI: 10.1016/j.jmaa.2011.06.038.
G. W. Griffiths and W. E. Schiesser, Traveling Wave Analysis of Partial Differential Equations, Academic Press (2011), URL: https://www.elsevier.com/books/traveling-wave-analysis-of-partial-differential-equations/griffiths/978-0-12-384652-5.
H.-C. Lee and Y. Choi, Analysis and approximation of linear feedback control problems for the Boussinesq equations, Computers and Mathematics with Applications 51 (2006), 829 – 848, DOI: 10.1016/j.camwa.2006.03.012.
S. Li, Optimal controls of Boussinesq equations with state constraints, Nonlinear Analysis: Theory, Methods & Applications 60 (2005), 1485 – 1508, DOI: 10.1016/j.na.2004.11.010.
H. Liu and H. Gao, Global well-posedness and long time decay of the 3D Boussinesq equations, Journal of Differential Equations 263 (2017), 8649 – 8665, DOI: 10.1016/j.jde.2017.08.049.
V. G. Makhankov and O. K. Pashaev, Nonlinear Evolution Equations and Dynamical Systems: Needs’90, Springer-Verlag (1991), URL: https://www.springer.com/gp/book/9783540532941.
V. B. Shakhmurov, Nonlocal problems for Boussinesq equations, Nonlinear Analysis 142 (2016), 134 – 151, DOI: 10.1016/j.na.2016.04.014.
M. D. Todorov, Nonlinear Waves, Morgan-Claypool (2018), URL: https://catalog.princeton.edu/catalog/10969883.
V. N. Vu, C. Lee and T.-H. Jung, Extended Boussinesq equations for waves in porous media, Coastal Engineering 139 (2018), 85 – 97, DOI: 10.1016/j.coastaleng.2018.04.023.
A. M. Wazwaz, Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation, Ocean Engineering 94 (2015), 111 – 115, DOI: 10.1016/j.oceaneng.2014.11.024.
Z. Yang, Longtime dynamics of the damped Boussinesq equation, Journal of Mathematical Analysis and Applications 399 (2013), 180 – 190, DOI: 10.1016/j.jmaa.2012.09.042.
H. Zhang, X. Jiang, M. Zhao and R. Zheng, Spectral method for solving the time fractional Boussinesq equation, Applied Mathematics Letters 85 (2018), 164 – 170, DOI: 10.1016/j.aml.2018.06.008.
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