    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:
  A. A. Abu Haya, Solving Optimal Control Problem Via Chebyshev Wavelet, Masters thesis, Islamic University of Gaza, 2011.  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.  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.  T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, California, 1995. 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.  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. 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.  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.  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.  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. 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.  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.  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.  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.  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.  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.  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.  A. Majdalawi, An Iterative Technique for Solving Nonlinear Quadratic Optimal Control Problem Using Orthogonal Functions, Ph. D thesis, Alquds University, 2010.  E. R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, 1995. 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. 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. 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.  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.  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.  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. 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.  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.  show all references

##### References:
  A. A. Abu Haya, Solving Optimal Control Problem Via Chebyshev Wavelet, Masters thesis, Islamic University of Gaza, 2011.  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.  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.  T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, California, 1995. 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.  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. 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.  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.  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.  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. 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.  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.  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.  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.  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.  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.  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.  A. Majdalawi, An Iterative Technique for Solving Nonlinear Quadratic Optimal Control Problem Using Orthogonal Functions, Ph. D thesis, Alquds University, 2010.  E. R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, 1995. 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. 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. 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.  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.  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.  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. 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.  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.   Approximate (linestyle is -) and exact (linestyle is :) solution for x(t)
Comparison of the optimal values of J (Example 4.1)
 Exact value of J Kafash et al.  Saberi Nik et al.  Approximated solution via HW 0.1929092981 0.192914197 0.193415452 0.1929092981
 Exact value of J Kafash et al.  Saberi Nik et al.  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  Hashemi Mehne and Hashemi Borzabadi Approximated solution via HW 6.1586 6.1748 6.1495
 Exact solution  Hashemi Mehne and Hashemi Borzabadi 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  Jaddu  Majdalawi  Our proposed method 0.06936094 0.0702 0.0693689 0.0693668896 0.0693688962
 Exact value Hsieh  Jaddu  Majdalawi  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  Jaddu  Abu Haya  Rafiei  Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
 Elnagar  Jaddu  Abu Haya  Rafiei  Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
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