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Article Contents

# Solving optimal control problem using Hermite wavelet

• * Corresponding author: Akram Kheirabadi
• 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.

Mathematics Subject Classification: Primary: 34H05, 37N35, 93C15; Secondary: 49N05.

 Citation:

• Figure 1.  Approximate (linestyle is -) and exact (linestyle is :) solution for x(t)

Figure 2.  Approximate (linestyle is -) and exact (linestyle :) solution for u(t)

Figure 3.  Approximate (linestyle -) and exact (linestyle :) solution for x(t)

Figure 4.  Approximate (linestyle -) and exact (linestyle :) solution for u(t)

Table 1.  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

Table 2.  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

Table 3.  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

Table 4.  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

Table 5.  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

Table 6.  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

Table 7.  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

Figures(4)

Tables(7)