# American Institute of Mathematical Sciences

March  2019, 9(1): 101-112. doi: 10.3934/naco.2019008

## Solving optimal control problem using Hermite wavelet

 1 Department of Mathematics, Faculty of Mathematical Science and Statistics, University of Birjand, Birjand, Iran 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author: Akram Kheirabadi

Received  May 2018 Revised  July 2018 Published  October 2018

In this paper, we derive the operational matrices of integration, derivative and production of Hermite wavelets and use a direct numerical method based on Hermite wavelet, for solving optimal control problems. The properties of Hermite polynomials are used for finding these matrices. First, we approximate the state and control variables by Hermite wavelets basis; then, the operational matrices is used to transfer the given problem into a linear system of algebraic equations. In fact, operational matrices of Hermite wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system gives us the solution of the original problem. Numerical examples with time varying and time invariant coefficient are given to demonstrate the applicability of these matrices.

Citation: Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008
##### References:
 [1] A. A. Abu Haya, Solving Optimal Control Problem Via Chebyshev Wavelet, Masters thesis, Islamic University of Gaza, 2011. Google Scholar [2] A. Ali, M. A. Iqbal and S. T. Mohyud-Din, Hermite wavelets method for boundary value problems, International Journal of Modern Applied Physics, 3 (2013), 38-47.   Google Scholar [3] E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188 (2007), 417-426.  doi: 10.1016/j.amc.2006.10.008.  Google Scholar [4] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, California, 1995.  Google Scholar [5] M. Behroozifar and S. A. Yousefi, Numerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials, Computational Methods for Differential Equations, 1 (2013), 78-95.   Google Scholar [6] C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proceedings - Control Theory and Applications, 144 (1997), 87-94.  doi: 10.1049/ip-cta:19970702.  Google Scholar [7] G. Elnagar, State-control spectral Chebyshev parameterization for linearly constrained quadratic optimal control problems, Journal of Computational and Applied Mathematics, 79 (1997), 19-40.  doi: 10.1016/S0377-0427(96)00134-3.  Google Scholar [8] M. Ghasemi, E. Babolian and M. Tavassoli Kajani, Hybrid Fourier and block-pulse functions for applications in the calculus of variations, International Journal of Computer Mathematics, 83 (2006), 695-702.  doi: 10.1080/00207160601056016.  Google Scholar [9] M. Ghasemi and M. Tavassoli Kajani, Numerical solution of time-varying delay systems by Chebyshev wavelets, Applied Mathematical Modelling, 35 (2011), 5235-5244.  doi: 10.1016/j.apm.2011.03.025.  Google Scholar [10] J. S. Gu and W. S. Jiang, The Haar wavelets operational matrix of integration, International Journal of Systems Science, 27 (1996), 623-628.  doi: 10.1080/00207729608929258.  Google Scholar [11] N. Haddadi, Y. Ordokhani and M. Razzaghi, Optimal control of delay systems by using a hybrid functions approximation, Journal of Optimization Theory and Applications, 153 (2012), 338-356.  doi: 10.1007/s10957-011-9932-1.  Google Scholar [12] H. Hashemi Mehne and A. Hashemi Borzabadi, A numerical method for solving optimal control problems using state parametrization, Numerical Algorithms, 42 (2006), 165-169.  doi: 10.1007/s11075-006-9035-5.  Google Scholar [13] H. C. Hsieh, Synthesis of adaptive control systems by function space methods, Advances in control systems, 2 (1965), 117-208.  doi: 10.1016/B978-1-4831-6712-1.50008-1.  Google Scholar [14] C. Hwang and Y. P. Shih, Laguerre series direct method for variational problems, Journal of Optimization Theory and Applications, 39 (1983), 143-149.  doi: 10.1007/BF00934611.  Google Scholar [15] C. Hwang and Y. P. Shih, Optimal control of delay systems via block pulse functions, Journal of Optimization Theory and Applications, 45 (1985), 101-112.  doi: 10.1007/BF00940816.  Google Scholar [16] H. M. Jaddu, Numerical Methods for Solving Optimal Control Problems Using Chebyshev Polynomials, Ph. D thesis, School of Information Science, Japan Advanced Institute of Science and Technology, 1998. Google Scholar [17] B. Kafash, A. Delavarkhalafi and S. M. Karbassi, Application of variational iteration method for hamilton-jacobi-bellman equations, Applied Mathematical Modelling, 37 (2013), 3917-3928.  doi: 10.1016/j.apm.2012.08.013.  Google Scholar [18] A. Majdalawi, An Iterative Technique for Solving Nonlinear Quadratic Optimal Control Problem Using Orthogonal Functions, Ph. D thesis, Alquds University, 2010. Google Scholar [19] E. R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, 1995.  Google Scholar [20] Z. Rafiei, B. Kafash and S. M. Karbassi, A new approach based on using Chebyshev wavelets for solving various optimal control problems, Computational and Applied Mathematics, 36 (2017), 1-14.  doi: 10.1007/s40314-017-0419-z.  Google Scholar [21] M. Razzaghi, Solution of multi-delay systems via combined block-pulse functions and Legendre polynomials, Analele Stiintifice ale Universitatii Ovidius Constanta, 17 (2009), 223-232.   Google Scholar [22] M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32 (2001), 495-502.  doi: 10.1080/00207720120227.  Google Scholar [23] V. Rehbockt, K. L. Teo, L. S. Jenning and H. W. J. Lee, A survey of the control parametrization and control parametrization enhancing methods for constrained optimal control problem, in Progress in Optimization, Springer, Boston, (1999), 247-275. doi: 10.1007/978-1-4613-3285-5_13.  Google Scholar [24] H. Saberi Nik, S. Effati and M. Shirazian, An approximate-analytical solution for the hamilton-jacobi-bellman equation via homotopy perturbation method, Mathematical and Computer Modelling, 36 (2012), 5614-5623.  doi: 10.1016/j.apm.2012.01.013.  Google Scholar [25] H. R. Sharif, M. A. Vali, M. Samavat and A. A. Gharavizi, A new algorithm for optimal control of time-delay systems, Applied Mathematical Sciences, 5 (2011), 595-606.   Google Scholar [26] X. T. Wang, Numerical solution of time-varying systems with a stretch by general Legendre wavelets, Applied Mathematics and Computation, 198 (2008), 613-620.  doi: 10.1016/j.amc.2007.08.058.  Google Scholar [27] S. Yousefi and M. Razzaghi, Legendre wavelets method for the nonlinear volterra-fredholm integral equations, Mathematics and Computers in Simulation, 70 (2005), 1-8.  doi: 10.1016/j.matcom.2005.02.035.  Google Scholar

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##### References:
 [1] A. A. Abu Haya, Solving Optimal Control Problem Via Chebyshev Wavelet, Masters thesis, Islamic University of Gaza, 2011. Google Scholar [2] A. Ali, M. A. Iqbal and S. T. Mohyud-Din, Hermite wavelets method for boundary value problems, International Journal of Modern Applied Physics, 3 (2013), 38-47.   Google Scholar [3] E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188 (2007), 417-426.  doi: 10.1016/j.amc.2006.10.008.  Google Scholar [4] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, California, 1995.  Google Scholar [5] M. Behroozifar and S. A. Yousefi, Numerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials, Computational Methods for Differential Equations, 1 (2013), 78-95.   Google Scholar [6] C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proceedings - Control Theory and Applications, 144 (1997), 87-94.  doi: 10.1049/ip-cta:19970702.  Google Scholar [7] G. Elnagar, State-control spectral Chebyshev parameterization for linearly constrained quadratic optimal control problems, Journal of Computational and Applied Mathematics, 79 (1997), 19-40.  doi: 10.1016/S0377-0427(96)00134-3.  Google Scholar [8] M. Ghasemi, E. Babolian and M. Tavassoli Kajani, Hybrid Fourier and block-pulse functions for applications in the calculus of variations, International Journal of Computer Mathematics, 83 (2006), 695-702.  doi: 10.1080/00207160601056016.  Google Scholar [9] M. Ghasemi and M. Tavassoli Kajani, Numerical solution of time-varying delay systems by Chebyshev wavelets, Applied Mathematical Modelling, 35 (2011), 5235-5244.  doi: 10.1016/j.apm.2011.03.025.  Google Scholar [10] J. S. Gu and W. S. Jiang, The Haar wavelets operational matrix of integration, International Journal of Systems Science, 27 (1996), 623-628.  doi: 10.1080/00207729608929258.  Google Scholar [11] N. Haddadi, Y. Ordokhani and M. Razzaghi, Optimal control of delay systems by using a hybrid functions approximation, Journal of Optimization Theory and Applications, 153 (2012), 338-356.  doi: 10.1007/s10957-011-9932-1.  Google Scholar [12] H. Hashemi Mehne and A. Hashemi Borzabadi, A numerical method for solving optimal control problems using state parametrization, Numerical Algorithms, 42 (2006), 165-169.  doi: 10.1007/s11075-006-9035-5.  Google Scholar [13] H. C. Hsieh, Synthesis of adaptive control systems by function space methods, Advances in control systems, 2 (1965), 117-208.  doi: 10.1016/B978-1-4831-6712-1.50008-1.  Google Scholar [14] C. Hwang and Y. P. Shih, Laguerre series direct method for variational problems, Journal of Optimization Theory and Applications, 39 (1983), 143-149.  doi: 10.1007/BF00934611.  Google Scholar [15] C. Hwang and Y. P. Shih, Optimal control of delay systems via block pulse functions, Journal of Optimization Theory and Applications, 45 (1985), 101-112.  doi: 10.1007/BF00940816.  Google Scholar [16] H. M. Jaddu, Numerical Methods for Solving Optimal Control Problems Using Chebyshev Polynomials, Ph. D thesis, School of Information Science, Japan Advanced Institute of Science and Technology, 1998. Google Scholar [17] B. Kafash, A. Delavarkhalafi and S. M. Karbassi, Application of variational iteration method for hamilton-jacobi-bellman equations, Applied Mathematical Modelling, 37 (2013), 3917-3928.  doi: 10.1016/j.apm.2012.08.013.  Google Scholar [18] A. Majdalawi, An Iterative Technique for Solving Nonlinear Quadratic Optimal Control Problem Using Orthogonal Functions, Ph. D thesis, Alquds University, 2010. Google Scholar [19] E. R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, 1995.  Google Scholar [20] Z. Rafiei, B. Kafash and S. M. Karbassi, A new approach based on using Chebyshev wavelets for solving various optimal control problems, Computational and Applied Mathematics, 36 (2017), 1-14.  doi: 10.1007/s40314-017-0419-z.  Google Scholar [21] M. Razzaghi, Solution of multi-delay systems via combined block-pulse functions and Legendre polynomials, Analele Stiintifice ale Universitatii Ovidius Constanta, 17 (2009), 223-232.   Google Scholar [22] M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32 (2001), 495-502.  doi: 10.1080/00207720120227.  Google Scholar [23] V. Rehbockt, K. L. Teo, L. S. Jenning and H. W. J. Lee, A survey of the control parametrization and control parametrization enhancing methods for constrained optimal control problem, in Progress in Optimization, Springer, Boston, (1999), 247-275. doi: 10.1007/978-1-4613-3285-5_13.  Google Scholar [24] H. Saberi Nik, S. Effati and M. Shirazian, An approximate-analytical solution for the hamilton-jacobi-bellman equation via homotopy perturbation method, Mathematical and Computer Modelling, 36 (2012), 5614-5623.  doi: 10.1016/j.apm.2012.01.013.  Google Scholar [25] H. R. Sharif, M. A. Vali, M. Samavat and A. A. Gharavizi, A new algorithm for optimal control of time-delay systems, Applied Mathematical Sciences, 5 (2011), 595-606.   Google Scholar [26] X. T. Wang, Numerical solution of time-varying systems with a stretch by general Legendre wavelets, Applied Mathematics and Computation, 198 (2008), 613-620.  doi: 10.1016/j.amc.2007.08.058.  Google Scholar [27] S. Yousefi and M. Razzaghi, Legendre wavelets method for the nonlinear volterra-fredholm integral equations, Mathematics and Computers in Simulation, 70 (2005), 1-8.  doi: 10.1016/j.matcom.2005.02.035.  Google Scholar
Approximate (linestyle is -) and exact (linestyle is :) solution for x(t)
Approximate (linestyle is -) and exact (linestyle :) solution for u(t)
Approximate (linestyle -) and exact (linestyle :) solution for x(t)
Approximate (linestyle -) and exact (linestyle :) solution for u(t)
Comparison of the optimal values of J (Example 4.1)
 Exact value of J Kafash et al. [17] Saberi Nik et al. [24] Approximated solution via HW 0.1929092981 0.192914197 0.193415452 0.1929092981
 Exact value of J Kafash et al. [17] Saberi Nik et al. [24] Approximated solution via HW 0.1929092981 0.192914197 0.193415452 0.1929092981
The exact and approximated values of x(t) and u(t) for Example 4.1
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 1.0000 1.0000 -0.3859 -0.3858 0.2 0.7594 0.7594 -0.2769 -0.2769 0.4 0.5799 0.5799 -0.1902 -0.1902 0.6 0.4472 0.4472 -0.1189 -0.1189 0.8 0.3505 0.3505 -0.0571 -0.0571 1 0.2820 0.2820 0.0000 0.0000
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 1.0000 1.0000 -0.3859 -0.3858 0.2 0.7594 0.7594 -0.2769 -0.2769 0.4 0.5799 0.5799 -0.1902 -0.1902 0.6 0.4472 0.4472 -0.1189 -0.1189 0.8 0.3505 0.3505 -0.0571 -0.0571 1 0.2820 0.2820 0.0000 0.0000
Comparison of the optimal values of J (Example 4.2)
 Exact solution [19] Hashemi Mehne and Hashemi Borzabadi[12] Approximated solution via HW 6.1586 6.1748 6.1495
 Exact solution [19] Hashemi Mehne and Hashemi Borzabadi[12] Approximated solution via HW 6.1586 6.1748 6.1495
The exact and approximated values of x(t) and u(t) for Example 4.2
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 0.0000 0.0000 1.1028 1.1029 0.2 0.2264 0.2265 1.4185 1.4188 0.4 0.4896 04897 1.9646 1.9648 0.6 0.8321 0.8324 2.8293 2.8293 0.8 1.3097 1.3100 4.1515 4.1526 1 2.0000 2.0000 6.1300 6.1493
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 0.0000 0.0000 1.1028 1.1029 0.2 0.2264 0.2265 1.4185 1.4188 0.4 0.4896 04897 1.9646 1.9648 0.6 0.8321 0.8324 2.8293 2.8293 0.8 1.3097 1.3100 4.1515 4.1526 1 2.0000 2.0000 6.1300 6.1493
Comparison between different methods for optimal value of J (Example 4.3)
 Exact value Hsieh [13] Jaddu [16] Majdalawi [18] Our proposed method 0.06936094 0.0702 0.0693689 0.0693668896 0.0693688962
 Exact value Hsieh [13] Jaddu [16] Majdalawi [18] Our proposed method 0.06936094 0.0702 0.0693689 0.0693668896 0.0693688962
The approximate and exact values of J (Example 4.4)
 Exact value Approximated value via HW Error 0.16666666666 0.1666666666 0.4×10−14
 Exact value Approximated value via HW Error 0.16666666666 0.1666666666 0.4×10−14
Comparison between different methods for optimal value of J (Example 4.5)
 Elnagar [7] Jaddu [16] Abu Haya [1] Rafiei [20] Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
 Elnagar [7] Jaddu [16] Abu Haya [1] Rafiei [20] Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
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