Solving a Class of Nonlinear Optimal Control Problems Using Haar Wavelets and Hybrid GA

[1] Abo‐Hammour, Z.S., Ghaleb Asasfeh, A., Al‐Smadi, A.M., Alsmadi O.M.K. (2011). “A novel continuous genetic algorithm for the solution of optimal control problems”, Optimal Control Applications and Methods, 32, 414-432.

[2] Ast, J.M.V., Babuska, R., De Schutter, B. (2009). “Novel ant colony optimization approach to optimal control”, International Journal of Intelligent Computing and Cybernetics, 2, 414-434.

[3] Aziz, I., Ul-Islam, S., Khan, F. (2014). “A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations”, Journal of Computational and Applied Mathematics, 272, 70-80.

[4] Babaie-Kafaki, S., Ghanbari, R., Mahdavi-Amiri, N. (2012). “An efficient and practically robust hybrid metaheuristic algorithm for solving fuzzy bus terminal location problems”, Asia-Pacific Journal of Operational Research, 29, 1-25.

[5] Betts, J.T. (2010). “Practical methods for optimal control and estimation using nonlinear programming”, Society for Industrial and Applied Mathematics.

[6] Bonnans, J.J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A. (2006). “Numerical optimization: Theoretical and practical aspects”, Springer London, Limited.

[7] Chen, W.L., Shih, Y.P. (1978). “Analysis and optimal control of time-varying linear systems via Walsh functions”, International Journal of Control, 27, 917-932.

[8] Chui, C.K. (1992). “An Introduction to wavelets”, Academic Press.

[9] Cruz, I.L., Van Willigenburg, L.G., Van Straten, G. (2003). “Efficient differential evolution algorithms for multimodal optimal control problems”, Applied Soft Computing, 3, 97-122.

[10] Dai, R., Cochran Jr, J.E. (2009). “Wavelet collocation method for optimal control problems”, Journal of Optimization Theory and Applications, 143, 265-278.

[11] El-Kady, M. (2012). “Efficient reconstructed Legendre algorithm for solving linear-quadratic optimal control problems”, Applied Mathematics Letters, 25, 1034-1040.

[12] Engelbrecht, A.P. (2007). “Computational intelligence: An introduction”, Wiley.

[13] Erfanian, M., Mansoori, A. (2019). “Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet”, Mathematics and Computers in Simulation, 165, 223-237.

[14] Erfanian, M., Gachpazan, M., Beiglo, H. (2015). “Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis”, Applied Mathematics and Computation, 265, 304-312.

[15] Erfanian, M., Gachpazan, M., Beiglo, H. (2017). “A new sequential approach for solving the integro-differential equation via Haar wavelet bases”, Computational Mathematics and Mathematical Physics, 57, 297-305.

[16] Erfanian, M., Gachpazan, M., Kosari, S. (2017). “A new method for solving of Darboux problem with Haar wavelet”, SeMA Journal, 74, 475-487.

[17] Ghanbari, R., Ghorbani-Moghadam, K., Nezhadhosein, S. (2021). “A numerical indirect method for solving a class of optimal control problems”, Ulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science 1.

[18] Hsiao, C.H. (2004). “Haar wavelet direct method for solving variational problems”, Mathematics and Computers in Simulation, 64, 569-585.

[19] Hsiao, C.H., Wang, W.J. (1998). “State analysis and optimal control of linear time-varying systems via Haar wavelets”, Optimal Control Applications and Methods, 19, 423-433.

[20] Kafash, B., Delavarkhalafi, A., Karbassi, S.M. (2012). “Application of Chebyshev polynomials to derive eficient algorithms for the solution of optimal control problems”, Scientia Iranica, 19, 795-805.

[21] Kaur, H., Mittal, R.C., Mishra, V. (2014). “Haar wavelet solutions of nonlinear oscillator equations”, Applied Mathematical Modelling.

[22] Kirk, D.E. (2004). “Optimal control theory: An introduction”, Dover Publications.

[23] Kosmol, P., Pavon, M. (2001). “Solving optimal control problems by means of general Lagrange functionals”, Automatica, 37, 907-913.

[24] Lee, M.H., Han, C., Chang, K.S. (1999). “Dynamic optimization of a continuous polymer reactor using a modified differential evolution algorithm”, Industrial and Engineering Chemistry Research, 38, 4825-4831.

[25] Modares, H., BagherNaghibi Sistani, M. (2011). “Solving nonlinear optimal control problems using a hybrid IPSO - SQP algorithm”, Engineering Applications of Artificial Intelligence, 24, 476-484.

[26] Nezhadhosein, S. (2017). “Haar matrix equations for solving time-variant linear-quadratic optimal control problems”, Computer and Optimization in Applied Mathematics, 2, 1-17.

[27] Nezhadhosein, S., Ghanbari, R., Ghorbani-Moghadam, Kh. (2022). “A numerical solution for fractional linear quadratic optimal control problems via shifted Legendre polynomials”, International Journal of Applied and Computational Mathematics, 8.

[28] Rastegar, M., Bazrafshan Moghaddam, A., Erfanian, M., Bazrafshan Moghaddam, B. (2019). “Using matrix-based rationalized Haar wavelet method for solving consolidation equation”, Asian-European Journal of Mathematics, 12, 1-9.

[29] Razzaghi, M., Tahai, A., Arabshahi, A. (1989). “Solution of linear two-point boundary value problems via Fourier series and application to optimal control of linear systems”, Journal of the Franklin Institute, 326, 523-533.

[30] Senthil Arumugam, M., Rao, M.V.C. (2008). “On the improved performances of the particle swarm optimization algorithms with adaptive parameters, cross-over operators and root mean square (RMS) variants for computing optimal control of a class of hybrid systems”, Application Soft Computing, 8, 324-336.

[31] Senthil Arumugam, M., Ramana Murthy, G., Rao, M.V.C., Loo, C.K. (2009). “On the optimal control of the steel annealing processes as a two stage hybrid systems via PSO algorithms”, International Journal Bio-Inspired Computing, 1, 198-209.

[32] Shi, X.H., Wan, L.M., Lee, H.P., Yang, XW., Wang, L.M., Liang, Y.C. (2003). “An improved genetic algorithm with variable population-size and a PSO-GA based hybrid evolutionary algorithm, in: Machine learning and cybernetics”, International Conference.

[33] Sim, Y.C., Leng, S.B., Subramaniam, V. (2000). “A combined genetic algorithms-shooting method approach to solving optimal control problems”, International Journal of Systems Science, 31, 83-89.

[34] Srinivasan, B., Palanki, S., Bonvin, D. (2003). “Dynamic optimization of batch processes: I. Characterization of the nominal solution”, Computers and Chemical Engineering, 27, 1-26.

[35] Vincent Antony Kumar, A., Balasubramaniam, P. (2009). “Optimal control for linear system using genetic programming”, Optimal Control Applications and Methods, 30, 47-60.

[36] Wang, X.T. (2007). “Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials”, Applied Mathematics and Computation, 184, 849-856.

[37] Wang, F.S., Chiou, J.P. (1997). “Optimal control and optimal time location problems of differential-algebraic systems by differential evolution”, Industrial and Engineering Chemistry Research, 36, 5348-5357.

[38] Wang, L., Ma, Y., Meng, Z. (2014). “Haar wavelet method for solving fractional partial differential equations numerically”, Applied Mathematics and Computation, 227, 66-76.