# American Institute of Mathematical Sciences

December  2019, 9(4): 493-506. doi: 10.3934/naco.2019037

## A hybrid parametrization approach for a class of nonlinear optimal control problems

 1 Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran 2 Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran 3 Department of Applied Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

Received  November 2018 Revised  April 2019 Published  May 2019

In this paper, a suitable hybrid iterative scheme for solving a class of non-linear optimal control problems (NOCPs) is proposed. The technique is based upon homotopy analysis and parametrization methods. Actually an appropriate parametrization of control is applied and state variables are computed using homotopy analysis method (HAM). Then performance index is transformed by replacing new control and state variables. The results obtained from the given method are compared with the results which are obtained using the spectral homotopy analysis method (SHAM), homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM), modified variational iteration method (MVIM) and differential transformations. The existence and uniqueness of the solution are presented. The comparison and ability of the given approach is illustrated via two examples.

Citation: M. Alipour, M. A. Vali, A. H. Borzabadi. A hybrid parametrization approach for a class of nonlinear optimal control problems. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 493-506. doi: 10.3934/naco.2019037
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##### References:
Approximate solution of $x_1(t)$ and $u_1(t)$ for (m = 4, k = 2)
Approximate solution of x2(t) and u2(t) for (m = 4, k = 2)
Approximate solution of x3(t) and u3(t) for (m = 4, k = 2)
h-curve at 4-order of approximation of $x_1(t)$ and $x_2(t)$
h-curve at 4-order of approximation of $x_3(t)$
Approximate solution of $x_1(t)$ and $x_2(t)$ for (m = 7, k = 3)
Approximate solution of $u(t)$
h-curve at 7-order of approximation of x1(t) and x2(t)
Minimum of performance index value $J_k$ of the proposed method
 Itr CPU time (sec.) HAM and parametrization approaches m=4, k=1 $0.109$ $0.00468778$ m=4, k=2 $0.121$ $0.00468778$
 Itr CPU time (sec.) HAM and parametrization approaches m=4, k=1 $0.109$ $0.00468778$ m=4, k=2 $0.121$ $0.00468778$
The Max error of the proposed method for $x_1(t)$ that $k = 2$ and $h = -1$ in comparison to SHAM and HPM
 Method CPU time (sec.) Max error proposed method (m=4, k=2) $0.155$ $2.93152 * 10^{-17}$ SHAM (Legendre) (m=6, N=50, h=-1.2) $0.224$ $1.0589* 10^{-9}$ SHAM (Chebyshev) (m=6, N=50, h=-1.2) $0.224$ $1.0586* 10^{-9}$ HPM (m=6) $46.401$ $3.1420* 10^{-8}$
 Method CPU time (sec.) Max error proposed method (m=4, k=2) $0.155$ $2.93152 * 10^{-17}$ SHAM (Legendre) (m=6, N=50, h=-1.2) $0.224$ $1.0589* 10^{-9}$ SHAM (Chebyshev) (m=6, N=50, h=-1.2) $0.224$ $1.0586* 10^{-9}$ HPM (m=6) $46.401$ $3.1420* 10^{-8}$
Minimum of performance index value $J$ of the proposed method and other methods
 Method Cost function CPU time (sec.) Proposed Method (m=4, k=2, h=-1) $0.00468778$ $0.141$ SHAM Chebyshev (m=6, N=50, h=-1.2) $0.0046877944625923$ $0.226$ SHAM Legendre (m=6, N=50, h=-1.2) $0.0046877944625906$ $0.227$ HPM (m=3) $0.004687795533$ $10.821$ OHPM (m=1) $0.004688009428$ $-$ MVIM (m=3) $0.004687986656$ $-$
 Method Cost function CPU time (sec.) Proposed Method (m=4, k=2, h=-1) $0.00468778$ $0.141$ SHAM Chebyshev (m=6, N=50, h=-1.2) $0.0046877944625923$ $0.226$ SHAM Legendre (m=6, N=50, h=-1.2) $0.0046877944625906$ $0.227$ HPM (m=3) $0.004687795533$ $10.821$ OHPM (m=1) $0.004688009428$ $-$ MVIM (m=3) $0.004687986656$ $-$
Minimum of performance index value $J_k$ of the proposed method
 Itr CPU time (sec.) HAM and parametrization approaches m=7, k=1 $0.016$ $1.07504$ m=7, k=2 $0.031$ $1.0136$ m= 7, k=3 $0.032$ $1.01184$
 Itr CPU time (sec.) HAM and parametrization approaches m=7, k=1 $0.016$ $1.07504$ m=7, k=2 $0.031$ $1.0136$ m= 7, k=3 $0.032$ $1.01184$
The Max error of our method of $x_1(t)$ in comparison to SHAM and HPM
 Itr Max error Proposed Method (m=7, k=3, h=-0.9) $3.16673\times10^{-5}$ SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ DT (m=15) $4.4380\times10^{-4}$
 Itr Max error Proposed Method (m=7, k=3, h=-0.9) $3.16673\times10^{-5}$ SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ DT (m=15) $4.4380\times10^{-4}$
Minimum of performance index value J of the proposed method and other methods
 Method Cost function CPU time (sec.) Proposed Method (m=7, k=3, h=-0.9) $1.01184$ $0.032$ SHAM Chebyshev (m=15, N=50, h=-0.5) $1.0472$ $0.200$ SHAM Legendre (m=15, N=50, h=-0.5) $1.0472$ $0.188$ DT (m=15) $1.0478$ $87.74$
 Method Cost function CPU time (sec.) Proposed Method (m=7, k=3, h=-0.9) $1.01184$ $0.032$ SHAM Chebyshev (m=15, N=50, h=-0.5) $1.0472$ $0.200$ SHAM Legendre (m=15, N=50, h=-0.5) $1.0472$ $0.188$ DT (m=15) $1.0478$ $87.74$
Minimum of performance index value Jk of the proposed method
 Itr CPU time (sec.) HAM and parametrization approaches m=7, k=1 0.016 1.07504 m=7, k=2 0.031 1.0136 m= 7, k=3 0.032 1.01184
 Itr CPU time (sec.) HAM and parametrization approaches m=7, k=1 0.016 1.07504 m=7, k=2 0.031 1.0136 m= 7, k=3 0.032 1.01184
The Max error of our method of x1(t) in comparison to SHAM and HPM
 Itr Max error Proposed Method (m=7, k=3, h=-0.9) $5.36319\times10^{-5}$ SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ DT (m=15) $4.4380\times10^{-4}$
 Itr Max error Proposed Method (m=7, k=3, h=-0.9) $5.36319\times10^{-5}$ SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$ DT (m=15) $4.4380\times10^{-4}$
Minimum of performance index value J of the proposed method and other methods
 Method Cost function Proposed Method (m=4, k=2) 1.04483 SHAM Chebyshev (m=15, N=50, h=-0.5) 1.0472 SHAM Legendre (m=15, N=50, h=-0.5) 1.0472 DT (m=15) 1.0478
 Method Cost function Proposed Method (m=4, k=2) 1.04483 SHAM Chebyshev (m=15, N=50, h=-0.5) 1.0472 SHAM Legendre (m=15, N=50, h=-0.5) 1.0472 DT (m=15) 1.0478
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