Optimal Control Problems‎: ‎Convergence and Error Analysis in Reproducing Kernel Hilbert Spaces

[1] Aronszajn, N. (1950). “Theory of reproducing kernels”, Transactions of the American Mathematical Society, 68, 337-404.

[2] Betts, J. T., Biehn, N., and Campbell, S. L., (2002). “Convergence of non-convergent IRK discretization of optimal control problems with state inequality constraints”, SIAM Journal on Scientific Computing, 23( 6), 1981–2007.

[3] Bouboulis, P., and Mavroforakis, M. (2011). “Reproducing Kernel Hilbert spaces and fractal interpolation”, Journal of Computational and Applied Mathematics, 235(12), 3425-3434.

[4] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A. (1988). “Spectral methods in fluid dynamics”, Spinger-Verlag, Heidelberg, Germany.

[5] Chryssoverghi, I., Coletsos, J., and Kokkinis B., (2001). “Approximate relaxed descent method for optimal control problems”, Control and Cybernetics, 30(4), 385-404.

[6] Chryssoverghi, I., Coletsos I., and Kokkinis, B., (2006). “Discretization methods for optimal control problems with state constraints”, Journal of Computational and Applied Mathematics, 19(1), 1-31.

[7] Cialenco, I., Fasshauer, G. E., and Ye, Q. (2012). “Approximation of stochastic partial differential equations by a kernel-based collocation method”, International Journal of Computer Mathematics, 89(18), 2543–2561.

[8] Cui, M., and Geng, F. (2007). “A computational method for solving one-dimensional variable-coefficient Burgers equation. Applied Mathematics and Computation, 188(2), 1389–1401.

[9] Cui, M., and Lin, Y. (2009). “Nonlinear Numerical Analysis in Reproducing Kernel Space”, (UK ed.). Nova Science Publishers, Inc.

[10] EL-Gindy, T. M., EL-Hawary, H. M., Salim, M. S. and EL-Kady, M. (1995). “A Chebyshev approximation for solving optimal control problems”, Computers & Mathematical with Application, 29(6), 35-45.

[11] Garg, D., Patterson, M., Hager, W. W., Rao, A. V., Benson, D. A., and Huntington, G. T. (2010). “A unified framework for the numerical solution of optimal control problems using pseudo spectral methods”, Automatica, 46(11), 1843–1851.

[12] Garrard, W. L., and Jordan, J. M. (1977). “Design of nonlinear automatic flight control systems”, Automatica, 13(5), 497–505.

[13] Geng, F. (2009). “Solving singular second-order three-point boundary value problems using reproducing kernel Hilbert space method”, Applied Mathematics and Computation, 215(6), 2095–2102.

[14] Geng, F. (2011). “A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method”, Applied Mathematics and Computation, 218(8), 4211–4215.

[15] Geng, F., and Cui, M. (2007). “Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space”, Applied Mathematics and Computation, 192(2), 389–398.

[16] Geng, F., and Cui, M., (2007). “Solving a nonlinear system of second order boundary value problems”, Journal of Mathematical Analysis and Applications, 327, 1167-1181.

[17] Geng, F., and Cui, M. (2012). “A reproducing kernel method for solving nonlocal fractional boundary value problems”, Applied Mathematics Letters, 25(5), 818–823.

[18] Geng, F. Z., and Li, X. M. (2012). “A new method for Riccati differential equations based on reproducing kernel and quasi-linearization methods”, Abstract and Applied Analysis, 2012, 1–8.

[19] Ghasemi, M., Fardi, M., and Khoshsiar Ghaziani, R. (2015). “Numerical solution of non-linear delay differential equations of fractional order in reproducing kernel Hilbert space”, Applied Mathematics and Computation, 268, 815–831.

[20] Gholami Baladezaei, M., Ghachpazan, M., and Hashemi Borzabadi, A., (2020). “Extraction of approximate solution for a class of nonlinear optimal control problems using 1/G′- expansion technique”, Control and Optimization in Applied Mathematics, 5(2), 56-82.

[21] Itik, M., Salamci, M. U., and Banks, S. P. (2009). “Optimal control of drug therapy in cancer treatment”, Nonlinear Analysis: Theory, Methods and Applications, 71(12), e1473– e1486.

[22] Jajarmi, A., Pariz, N., S. Effati, S., and Vahidian Kamyad A., (2011), “Solving infinite horizon nonlinear optimal control problems using an extended modal series method”, Journal of Zheijang University Science B., 12( 8), 667-677.

[23] Jiang, W., and Chen, Z. (2013). “A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation”, Numerical Methods for Partial Differential Equations, 30(1), 289–300.

[24] Kafash, B., Delavarkhalafi, A., and Karbassi, S. M., (2012). “Application of Chebyshev polynomials to derive eficient algorithms for the solution of optimal control problems”, Scientia Iranica, 19(3), 795-805.

[25] Kameswaran, S., and Biegler, L. T., (2008). “Convergence rates for direct transcription of optimal control problems using collocation at Radau points”, Computational Optimization and Applications, 41(1), 81–126.

[26] Mehne, H. H., and Borzabadi, A. H. (2006). A numerical method for solving optimal control problems using state parametrization. Numerical Algorithms, 42(2), 165–169.

[27] Mohammadi, M., and Mokhtari, R. (2011). “Solving the generalized regularized long wave equation on the basis of a reproducing kernel space”, Journal of Computational and Applied Mathematics, 235(14), 4003–4014.

[28] Momani, S., Abu Arqub, O., Hayat, T., and Al-Sulami, H. (2014). A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm Volterra type. Applied Mathematics and Computation, 240, 229–239.

[29] Nemati, A., Alizadeh, A. and F. Soltanian, F., (2020). “Solution of fractional optimal control problems with noise function using the Bernstein functions”, Control and Optimization in Applied Mathematics, 4(1), 37-51.

[30] Nezhadhosein, S., (2017). “Haar matrix equations for solving time-variant linear-quadratic optimal control problems”, Control and Optimization in Applied Mathematics, 2(2), 1-14.

[31] Pinch, E.R., (1993). “Optimal conntrol and calculus of variations”, Oxford University Press.

[32] Saberi Nik, H., and Effati, S. (2014). An approximate method for solving a class of non-linear optimal control problems, Optimal Control Applications and Methods, 35, 324-339.

[33] Saberi Nik, S. Effati, H., Mosta, S. S., and Shirazian, M., (2014). Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems, Numerical Algorithm, 65, 171-194.

[34] Saberi Nik, H., Effati, S., and Yildirim, A. (2012). “Solution of linear optimal control systems by differential transform method”, Neural Computing and Applications, 23(5), 1311–1317.

[35] Salim, M. S. (1994). “Numerical studies for optimal control problems and its applications”, Ph.D Thesis, Assiut University, Assiut, Egypt.

[36] Shirazian, M., and Effati, S. (2012). Solving a class of nonlinear optimal control problems via he’s variational iteration method. International Journal of Control, Automation and Systems, 10(2), 249–256. https://doi.org/10.1007/s12555-012-0205-z

[37] Vahdati, S., Fardi, M., and Ghasemi, M. (2018). “Option pricing using a computational method based on reproducing kernel”, Journal of Computational and Applied Mathematics, 328, 252–266.

[38] Vlassenbroeck, J., van Dooren, R. (1988). “A Chebyshev technique for solving nonlinear optimal control problems”, IEEE Transactions on Automatic Control, 33(4), 333–340.

[39] Wang, Y., Du, M., Tan, F., Li, Z., and Nie, T. (2013). “Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions”, Applied Mathematics and Computation, 219(11), 5918–5925.

[40] Wei, S. M., Zefran, S. M., and DeCarlo, R. A. (2008). “Optimal control of robotic system with logical constraints: application to UAV path planning”, IEEE International Conference on Robotic and Automation, Pasadena, 176-181.

[41] Yousefi, S. A., Dehghan, M., and Lotfi, A., (2010). “Finding the optimal control of linear systems via He’s variational iteration method”, International Journal of Computer Mathematics, 87(5), 1042-1050.

[42] Zaremba, S. (1907). “L’équation biharmonique et une classe remarquable de fonctions fondamentales harmoniques”. Bulletin International de l’Academie des Sciences de Cracovie, 147-196.