In this study, an indirect method is proposed based on the Chebyshev pseudo-spectral method for solving optimal control problems governed by Burgers’ equation. Pseudo-spectral methods are one of the most accurate methods for solving nonlinear continuous-time problems, specially optimal control problems. By using optimality conditions, the original optimal control problem is first reduced to a system of partial differential equations with boundary conditions. Control and state functions are then approximated by interpolating polynomials. The convergence is analyzed, and some numerical examples are solved to show the efficiency and capability of the method.
Burgers’ equation;, Optimal control;, Chebyshev-Gauss-Lobatto nodes.
2. Allahverdi, N., Pozo, A. and Zuazua, E. Numerical aspects of large-time optimal control of Burgers’ equation, M2AN Math. Model. Numer. Anal. 50(5) (2016), 1371–1401.
3. Baumann, M. Nonlinear Model Order Reduction using POD/DEIM for Optimal Control of Burgers’ Equation, Thesis submitted to the Delft In stitute of Applied Mathematics, 2013.
4. Chorin, A.J. and Marsden, J.E. A Mathematical Introduction to Fluid Mechanics, Springer-Verlag New York, 1979.
5. De los Reyes, J.C. and Kunisch, K. A comparison of algorithms for control constrained optimal control of the Burgers’ equation, Calcolo, 41 (2004), 203–225.
6. Fernandez-Cara, E. and Guerrero, S. Null controllability of the Burgers’ system with distributed controls, Systems Control Lett. 56 (5) (2007), 366–372.
7. Gong, Q., Ross, I.M., Kang, W. and Fahroo, F. Connections between the covector mapping theorem and convergence of pseudo-spectral methods for optimal control, Comput Optim Appl. 41 (2008), 307–335.
8. Hashemi, S.M. and Werner, H. LPV Modelling and Control of Burg ers’ Equation, The International Federation of Automatic Control Milano (Italy) August 28–September 2, 2011.
9. Hinze, M. and Volkwein, S. Analysis of instantaneous control for the Burgers’ equation, Nonlinear Anal. 50 (2002), 1–26.
10. Jaluria, Y. and Torrance, K.K.E. Computational heat transfer, Series in Computational and Physical Processes in Mechanics and Thermal Sciences Series. Taylor and Francis Group, 2003.
11. Karasozen, B. and Yilmaz, F. Optimal boundary control of the unsteady Burgers’ equation with simultaneous space-time discretization, Optimal Control Appl. Methods. 35 (2014), 423–434.
12. Kobayashi, T. Adaptive regulator design of a viscous Burgers’ system by boundary control, IMA J. Math. Control Inform. 18 (3) (2001), 427–437.
13. Kucuk, I. and Sadek, I. A numerical approach to an optimal boundary control of the viscous Burgers’ equation, Int. J. Appl. Math. Comput. 210 (2009), 126–135.
14. Kunisch, K. and Volkwein, S. Control of the Burgers’ equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (1999), 345–371.
15. Marburger, J. and Pinnau, R. Optimal control for Burgers’ equation using particle methods, arxiv:1309.7619, 2013.
16. Noack, A. and Walther, A. Adjoint concepts for the optimal control of Burgers’ equation, Comput. Optim. Appl. 36 (2007), 109–133.
17. Peterson, A.F., Ray, S.L. and Mittra, R. Computational Methods for Electromagnetics, IEEE Press Series on Electromagnetic Wave Theory. Wiley, 1998.
18. Press, W.H., Flannery, B.P., Teutolsky, S.A. and Vetterling, W.T. Numerical Recipes, The Art of Scientific Computing. Cambridge University Press, Cambridge, 1990.
19. Ragozin, D.L. Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150, 1970.
20. Sabeh, Z., Shamsi, M. and Dehghan, M. Distributed optimal control of the viscous Burgers’ equation via a Legendre pseudo-spectral approach, Math. Methods Appl. Sci. 39 (12) (2016), 3350–3360.
21. Sadek, I. and Kucuk, I. A robust technique for solving optimal control of coupled Burgers’ equations, IMA J. Math. Control Inform. 28 (3) (2011), 239–250.
22. Singh, S., Patel, V.K., Singh, V.K. and Tohidi, E. Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions, Comput. Math. Appl. 75 (7) (2018), 2280–2294.
23. Smaoui, N., Zribi, M. and Almulla, A. Sliding mode control of the forced generalized Burgers’ equation, IMA J. Math. Control Inform. 23 (3) (2006), 301–323.
24. Taflove, A. and Hagness, S.C. Computational Electrodynamics: The Finite- Difference Time- Domain Method, Artech House, third edition, 2005.
25. Troltzsch, F. and Volkwein, S. The SQP method for control constrained optimal control of the Burgers’ equation, ESAIM Control Optim. Calc. Var. 6 (2001) 649–674.
26. Volkwein, S. Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces and control problems for the Burgers’ equation, Dissertation at the University of Technology Berlin, 1997.
27. Volkwein, S. Distributed control problems for the Burgers’ equation, Com put. Optim. Appl. 18 (2) (2001), 115–140.
28. Wesseling, P. Principles of Computational Fluid Dynamics, Springer Verlag Berlin Heidelberg, 2001.
29. Yilmaz, F. and Karasozen, B. Solving optimal control problems for the unsteady Burgers’ equation in COMSOL Multiphysics, J. Comput. Appl. Math. 235 (2011), 4839–4850.
30. Yilmaz, F. and Karasozen, B. An all-at-once approach for the optimal control of the unsteady Burgers’ equation, J. Comput. Appl. Math. 259 (2014), 771–779.
31. Zeng, M.L. and Zhang, G.F. Preconditioning optimal control of the unsteady Burgers’ equations with H1 regularized term, Appl. Math. Comput. 254 (2015), 133–147.
32. Zogheib, B. and Tohidi, E. An Accurate Space-Time Pseudo-spectral Method for Solving Nonlinear Multi-Dimensional Heat Transfer Problems, Mediterr. J. Math. 14 (1) (2017).
33. Zogheib, B. and Tohidi, E. Modal Hermite spectral collocation method for solving multi-dimensional hyperbolic telegraph equations, Comput. Math. Appl. 75 (2018), 3571–3588.
This work is licensed under a Creative Commons Attribution 4.0 International License.