The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

Nemati, Ali; Mamehrashi, Kamal

doi:10.1002/asjc.1791

Cited by 10 publications

(4 citation statements)

References 40 publications

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“…During last years, various numerical methods have been used for numerically solving nonlinear fractional OCPs (for instance, see previous studies [2][3][4][5][6][7] ). The interested reader can also consult on previous studies [8][9][10][11][12][13][14] to see more details about the latest published papers about the fractional OCPs.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Heydari

Atangana

Avazzadeh

2020

Math Methods in App Sciences

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This article is devoted to developing an accurate operational matrix (OM) method for the solution of a new category of nonlinear optimal control problems (OCPs) explained by fractal‐fractional dynamical systems. The fractal‐fractional differentiation is considered in the sense of Atangana‐Riemann‐Liouville. The method is based on the shifted Legendre polynomials (LPs). Through the way, a new OM of fractal‐fractional differentiation is extracted for the shifted LPs. The optimality conditions are converted to an algebraic system of equations by using the shifted LP expansions of the state and control variables and applying the method of constrained extrema. More precisely, these variables are approximated by the shifted LPs with undetermined coefficients. Then, these expansions are replaced in the performance index, and the Gauss‐Legendre integration method is used to approximate the performance index for extracting a nonlinear algebraic equation. In the sequel, the mentioned OM and the collocation method are used to generate some algebraic equations from the dynamical system. Finally, the method of the constrained extrema is used by coupling the algebraic constraints generated from the dynamical system and the initial condition with the algebraic equation elicited from the performance index by a set of unknown Lagrange multipliers. The accuracy of the method is studied by solving some numerical examples.

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Section: Introductionmentioning

confidence: 99%

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Heydari

Atangana

Avazzadeh

2020

Math Methods in App Sciences

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“…The computational method based on Legendre wavelets (LWs), the Ritz spectral, and the shifted Legendre orthogonal polynomials for fractional optimal control problems (FOCPs) are developed in [9][10][11] respectively. [10,[12][13][14][15][16] present several methods to solve fractional optimal control problems, including the Ritz method, Laguerre functions, and the Bernstein polynomial basis. Complex-order fractional derivatives have also attracted attention.…”

Section: Introductionmentioning

confidence: 99%

Variational problem containing psi‐RL complex‐order fractional derivatives

Zhao

Qin

Wang

et al. 2020

Asian Journal of Control

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The main purpose of this paper is to solve the variational problem containing real-and complex-order fractional derivatives. We define a new version of the complex-order derivative based on the ψ-Riemann-Liouville fractional derivative, and get the Euler-Lagrange equation for the variational problem. By introducing the approximated expansion formula of the complex-order fractional derivative to the variational problem, we derive the corresponding approximated Euler-Lagrange equation. It is proved that the approximated Euler-Lagrange equation converges to the original one in the weak sense. At the same time, the minimal value of the approximated action integral tends to the minimal value of the original one. We also conduct a stress relaxation experiment and discuss the feasibility of the complex-order derivative in real problem modeling.

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“…Fractional calculus is a natural generalization of classical integer order calculus, whose inception can be traced back to 300 years ago. It is well known that fractional calculus has been widely applied to system modelling [1,2], stability analysis [3,4], controller synthesis [5,6], and optimization algorithm [7,8], etc. Due to the great efforts devoted by researchers, a large number of valuable results have been reported on fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

Wei

Wang

Tse

et al. 2019

Asian Journal of Control

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The paper focuses on the numerical approximation of nabla fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed models of discrete fractional order system are introduced. Then, resorting the nabla Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.

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The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

Cited by 10 publications

References 40 publications

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Variational problem containing psi‐RL complex‐order fractional derivatives

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

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