Approximation methods for solving fractional optimal control problems

Zeid, Samaneh Soradi; Effati, Sohrab; Kamyad, Ali Vahidian

doi:10.1007/s40314-017-0424-2

Cited by 17 publications

(6 citation statements)

References 113 publications

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“…The approximate value of the cost function for different choices of α, is compared with other methods in Table 9. Probably, the results are in agreement with the results of Zeid et al (2018).…”

Section: Illustrative Test Problemssupporting

confidence: 91%

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Soradi-Zeid

2019

Comp. Appl. Math.

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The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods. Keywords Optimal control problems • Radial basis functions • Direct and indirect methods • Fractional two-point boundary value problem Mathematics Subject Classification 49M37 • 49M05 • 49L99 • 65K05

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Section: Illustrative Test Problemssupporting

confidence: 91%

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Soradi-Zeid

2019

Comp. Appl. Math.

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“…Such problems appear in various fields of knowledge including, e.g., chemistry [10], biology [50], electrical engineering [19], and medicine [21]. Main directions of research here are related to necessary optimality conditions (see, e.g., [4,30] and the references therein) and numerical methods for constructing optimal controls (see, e.g., [29,45,51] and the references therein). In addition, note that several problems for linear systems are considered and studied in detail in, e.g., [2,13,18,20,27,40].…”

Section: Introductionmentioning

confidence: 99%

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

Gomoyunov

2022

ESAIM: COCV

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We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order $\alpha$. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov--Krasovskii functional.

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“…Because nearly all FOCPs do not chiefly have exact solutions, this highlights the importance of utilizing approximated and numerical methods. One can find analytical solution for solving FOCPs in Salati et al (2019), Zeid et al (2018), Chen et al (2020), Ameen et al (2020), Kumar and Mehra (2020), Mohammadi et al (2020), Song et al (2020), Soradi-Zeid (2021), and Sweilam et al (2020).…”

Section: Introductionmentioning

confidence: 99%

Research on a collocation approach and three metaheuristic techniques based on MVO, MFO, and WOA for optimal control of fractional differential equation

Ebrahimzadeh

Khanduzi

Beik³

et al. 2021

Journal of Vibration and Control

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Exploiting a comprehensive mathematical model for a class of systems governed by fractional optimal control problems is the significant focal point of the current paper. The efficiency index is a function of both control and state variables and the dynamic control system relies on Caputo fractional derivatives. The attributes of Bernoulli polynomials and their operational matrices of fractional Riemann–Liouville integrations are applied to convert the optimization problem to the nonlinear programing problem. Executing multi-verse optimizer, moth-flame optimization, and whale optimization algorithm terminate to the most excellent solution of fractional optimal control problems. A study on the advantage and performance between these approaches is analyzed by some examples. Comprehensive analysis ascertains that moth-flame optimization significantly solves the example. Furthermore, the privilege and advantage of preference with its accuracy are numerically indicated. Finally, results demonstrate that the objective function value gained by moth-flame optimization in comparison with other algorithms effectively decreased.

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Approximation methods for solving fractional optimal control problems

Cited by 17 publications

References 113 publications

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

Research on a collocation approach and three metaheuristic techniques based on MVO, MFO, and WOA for optimal control of fractional differential equation

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