Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

Nik, Hassan Saberi; Effati, Sohrab; Motsa, S. S.; Shirazian, Mohammad

doi:10.1007/s11075-013-9700-4

Cited by 23 publications

(16 citation statements)

References 25 publications

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“…By applying the HAM and parametrization method (proposed method) the computed results for J k are given in table 1. The maximum absolute error of proposed method, HPM and SHAM [31] are given in table 2. Furthermore, in table 3, minimum of J for our approch is obtained, and shows comparison between proposed method, SHAM, HPM, OHPM [12] and MVIM [27] of J.…”

Section: Examplementioning

confidence: 99%

“…By applying the proposed method and considering = 0.0002, the computed results of applying our method for J k are given in table 4. The maximum absolute errors of HAM and parameterization method (proposed method), DT [9] and SHAM [31] are given in table 5.…”

Section: Examplementioning

confidence: 99%

“…In ([6], [33]) converts the problem into a non-linear programming by using the discretization or parametrization techniques. Recently solutions based on He's variational iteration method [32], boubaker polynomials expansion scheme [17], the control parameterization method [26], OHPM [12], SHAM [31], differential transform method [9,29], Galerkin approximation solutions [38], homotopy analysis method [2], optimal homotopy analysis method [14], modified homotopy perturbation method [3], RBF collocation method [28], Legendre approximations [16] have been used for solving NOCPs.…”

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confidence: 99%

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

Vali¹,

Borzabadi²

2019

Numerical Algebra, Control &Amp; Optimization

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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.

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Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

mentioning

confidence: 99%

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

Vali¹,

Borzabadi²

2019

Numerical Algebra, Control &Amp; Optimization

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“…In previous papers addressing the solution of various nonlinear problems with numerical analogues of the homotopy analysis method, the linear equations have been cast in terms of Chebyshev differentiation matrices, which results in a technique known as the spectral homotopy analysis method (sham) [7,9,10]. While this approach is well understood, it produces dense matrices that often are ill-conditioned for variable coefficient boundary value problem.…”

Section: Linear Solvermentioning

confidence: 99%

A fast, spectrally accurate solver for the Falkner--Skan equation

Cullen

Clarke²

2017

ANZIAMJ

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We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner-Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered.

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“…Theoretical results concerning, among others, convergence of the method in case of differential equations are included in papers [29,31,38,39,[44][45][46]52]. Various modifications of homotopy analysis method have been also elaborated, for example, the spectral homotopy analysis method [37], the optimal homotopy analysis method [19] and the optimal homotopy asymptotic method [22,34] (see also [33]). …”

Section: Introductionmentioning

confidence: 99%

Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

et al. 2013

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The paper presents an application of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind. In this method a series is created, sum of which (if the series is convergent) gives the solution of discussed equation. Conditions ensuring convergence of this series are presented in the paper. Error of approximate solution, obtained by considering only partial sum of the series, is also estimated. Examples illustrating usage of the investigated method are presented as well, including the example having practical application for calculating the charge in supply circuit of flash lamps used in cameras.

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Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

Cited by 23 publications

References 25 publications

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

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

A fast, spectrally accurate solver for the Falkner--Skan equation

Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

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