Optimal control based on the polynomial least squares method. (English) Zbl 1427.49030
Summary: In this paper an approach for computing an optimal control law based on the Polynomial Least Squares Method (PLSM) is presented. The initial optimal control problem is reformulated as a variational problem whose corresponding Euler-Lagrange equation is solved by using PLSM. A couple of examples emphasize the accuracy of the method.
MSC:
| 49M20 | Numerical methods of relaxation type |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
References:
| [1] | Sargent, R. W., Optimal control, Journal of Computational and Applied Mathematics, 124, 1-2, 361-371 (2000) · Zbl 0970.49003 · doi:10.1016/S0377-0427(00)00418-0 |
| [2] | Sussmann, H. J.; Willems, J. C., 300 years of optimal control: from the brachystochrone to the maximum principle, IEEE Control Systems Magazine, 17, 3, 32-44 (1997) · Zbl 1014.49001 · doi:10.1109/37.588098 |
| [3] | Berkani, S.; Manseur, F.; Maidi, A., Optimal control based on the variational iteration method, Computers & Mathematics with Applications. An International Journal, 64, 4, 604-610 (2012) · Zbl 1252.49051 · doi:10.1016/j.camwa.2011.12.066 |
| [4] | Luus, R., On the application of iterative dynamic programming to singular optimal control problems, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 37, 11, 1802-1806 (1992) · Zbl 0773.49017 · doi:10.1109/9.173155 |
| [5] | Sewell, M. J., Maximum and Minimum Principles. A Unified Approach, with Applications (1987), New York, NY, USA: Cambridge University Press, New York, NY, USA · Zbl 0652.49001 |
| [6] | Tatari, M.; Dehghan, M., Solution of problems in calculus of variations via He’s variational iteration method, Physics Letters A, 362, 5-6, 401-406 (2007) · Zbl 1197.65112 · doi:10.1016/j.physleta.2006.09.101 |
| [7] | van Brunt, B., The Calculus of Variations. The Calculus of Variations, Springer (2004), New York, NY, USA · Zbl 1039.49001 |
| [8] | He, J.-H., Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114, 2-3, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6 |
| [9] | Kucuk, I., Active Optimal Control of the KdV Equation Using the Variational Iteration Method, Mathematical Problems in Engineering, 2010 (2010) · Zbl 1207.35258 · doi:10.1155/2010/929103 |
| [10] | Bota, C.; Căruntu, B., Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method, Advances in Mathematical Physics, 2015 (2015) · Zbl 1338.34044 · doi:10.1155/2015/450235 |
| [11] | Bota, C.; Caruntu, B., Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method, Fractional Calculus and Applied Analysis, 20, 4, 1043-1050 (2017) · Zbl 1377.34007 · doi:10.1515/fca-2017-0054 |
| [12] | Bota, C.; Caruntu, B., Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the polynomial least squares method, Chaos, Solitons & Fractals, 102, 339-345 (2017) · Zbl 1374.34307 · doi:10.1016/j.chaos.2017.05.002 |
| [13] | Caruntu, B.; Bota, C., Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method, The Scientific World Journal, 201 (2014) · doi:10.1155/2014/721865 |
| [14] | Zahedi, M. S.; Nik, H. S., On homotopy analysis method applied to linear optimal control problems, Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, 37, 23, 9617-9629 (2013) · Zbl 1427.49035 · doi:10.1016/j.apm.2013.05.009 |
| [15] | Jia, W.; He, X.; Guo, L., The optimal homotopy analysis method for solving linear optimal control problems, Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, 45, 865-880 (2017) · Zbl 1446.49026 · doi:10.1016/j.apm.2017.01.024 |
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