A new class of orthonormal basis functions: application for fractional optimal control problems

Heydari, M. H.; Razzaghi, Mohsen

doi:10.1080/00207721.2021.1947411

Cited by 10 publications

(3 citation statements)

References 27 publications

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“…Applying spectral analysis, Legendre polynomials [16] are used to develop different numerical approaches. The authors consider FOCPs with both integer-order and Caputo derivatives and propose an indirect numerical method in terms of truncated Bessel series in Tohidi and Nik [17]; other relevant results may be found in previous studies [18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Liu

et al. 2023

Asian Journal of Control

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This article considers fractional optimal control problems (FOCPs) including both integer‐order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method.

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Section: Introductionmentioning

confidence: 99%

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Liu

et al. 2023

Asian Journal of Control

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“…If the problem solution is piecewise, wavelet and piecewise functions are appropriate choices in the expressed method. Some of the basis functions with this property that have been used in recent years for FOCPs are hat functions [13], Legendre wavelets [14], Chebyshev wavelets [15], piecewise Chebyshev cardinal functions [16], piecewise Chelyshkov functions [17], and Chebyshev cardinal wavelets [18]. If the problem solution includes fractional and piecewise expressions at the same time, fractional piecewise functions are a suitable choice for the stated method.…”

Section: Introductionmentioning

confidence: 99%

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non‐smooth solutions

Zhagharian,

Heydari,

Razzaghi

2023

Asian Journal of Control

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In this study, to solve fractional problems with non‐smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non‐smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.

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“…In Heydari and Razzaghi (2021a), the piecewise Chebyshev cardinal functions are used for solving constrained fractional OCPs. The orthonormal piecewise Chelyshkov functions are used in Heydari and Razzaghi (2021b) for solving fractional OCPs. The Genocchi polynomials are used in Tajadodi (2020) for a category of variable-order fractional OCPs.…”

Section: Introductionmentioning

confidence: 99%

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

Heydari

Razzaghi

Avazzadeh

2022

Journal of Vibration and Control

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In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

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A new class of orthonormal basis functions: application for fractional optimal control problems

Cited by 10 publications

References 27 publications

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non‐smooth solutions

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

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