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

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
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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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