Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations

Heydari, Mohammad Hossein; Mahmoudi, Mohammad Reza; Avazzadeh, Z.; Băleanu, Dumitru

doi:10.1177/1077546319889862

Cited by 7 publications

(7 citation statements)

References 39 publications

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“…Koleva et al presented a numerical method based on finite difference discretization for solving the Monge-Ampere equation that is used in fluid dynamics [8]. The Chebyshev cardinal functions are utilized to find an approximate solution of a class of nonlinear optimal control problems [9]. A computational scheme based on the Fibonacci wavelets and the Galerkin method was presented for finding solution of fractional optimal control problems [10].…”

Section: Introductionmentioning

confidence: 99%

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Rahimkhani¹,

Ordokhani

Sabermahani

2023

Math Methods in App Sciences

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In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

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Section: Introductionmentioning

confidence: 99%

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Rahimkhani¹,

Ordokhani

Sabermahani

2023

Math Methods in App Sciences

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“…Koleva et al present a numerical method based on finite difference discretization for solving the Monge-Ampere equation that is used in fluid dynamics [8]. Chebyshev cardinal functions are utilizing to find approximate solution of a class of nonlinear optimal control problem with dynamics system [9]. Muftu et.…”

Section: Introductionmentioning

confidence: 99%

Bernoulli wavelet least square support vector regression: Robust numerical method for systems of fractional differential equations

Rahimkhani

Ordokhani

Sabermahani

2022

Preprint

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In this study, a new hybrid method is developed to solve a system of linear or nonlinear fractional differential equations using Bernoulli wavelets (Bws) and the least square support vector regression(LS-SVR). For this purpose, we present Bw operator of Riemann-Liouville fractional integration for Bw and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, then with the help of Bw operator of fractional integration and LS-SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

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“…Combining nonlinear systems with VO calculus has become the main research object, and many scholars have studied the optimal control of these systems. From the viewpoint of polynomial approximation, Heydari (2020), Pho et al (2020), and Heydari et al (2020) proposed different methods based on Chebyshev cardinal functions for solving optimal control problems with VO derivatives. Heydari et al (2019) and Heydari and Avazzadeh (2019) solved nonlinear VO optimal control problems based on the classical Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

Variable fractional order sliding mode control for seismic vibration suppression of building structure

Wang

Zhou

Jin

et al. 2021

Journal of Vibration and Control

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Constant fractional order vibration control strategy has been one of research hotspots in recent decades. However, the variable fractional order control method is seldom concerned up to now. In this article, a novel variable fractional order sliding mode control (VOSMC) method is proposed to suppress the responses of building structure caused by seismic excitations, including El Centro, Hachinohe, Northridge, and Kobe earthquakes. Based on the proposed variable fractional order sliding mode surface, the control law of VOSMC is presented. The global asymptotic stability of the control system is analyzed and proved by utilizing variable fractional order Lyapunov stability theorem. Besides, the corresponding constant fractional order sliding mode control (COSMC) method is also given. The control effects of VOSMC and COSMC methods are discussed by four performance indices. Finally, the utilizability and reasonability of the proposed control method is verified by using two examples (include two-story and five-story shear buildings). Compared with the COSMC method, the proposed variable fractional order controller not only has a lesser control output, but also has a higher utilization of the output, which is conducive to energy saving.

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Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations

Cited by 7 publications

References 39 publications

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Bernoulli wavelet least square support vector regression: Robust numerical method for systems of fractional differential equations

Variable fractional order sliding mode control for seismic vibration suppression of building structure

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