Fractional-order Lagrange polynomials: An application for solving delay fractional optimal control problems

Abstract

The main purpose of this work is to provide an efficient method for solving delay fractional optimal control problems (DFOCPs). Our method is based on fractional-order Lagrange polynomials (FLPs) and the collocation method. The FLPs are used to achieve a new operational matrix of fractional derivative. Also, we present a delay operational matrix of FLPs. These operational matrices are driven without considering the nodes of Lagrange polynomials. The operational matrices and collocation method are applied to a constrained extremum in order to minimize the performance index. Then, the problem reduces to the solution of a system of algebraic equations. Convergence of the algorithm and approximation of FLPs are proposed. Furthermore, the upper bound of the error for the operational matrix of fractional derivatives is obtained. Numerical tests for demonstrating the efficiency and effectiveness of the method are included. Moreover, the method is used for numerical solution of a mathematical model of chemotherapy in breast cancer.

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

“…Consider a DFOCPs with two different delays in the form such that where . Table 3 shows the obtained values of J for with our scheme, Chelyshkov wavelets [ 27 ], Bernoulli polynomials [ 41 ], fractional-order Lagrange polynomials [ 42 ], Bernoulli wavelets basis [ 28 ], Müntz-Legendre polynomials [ 39 ], the least square method [ 40 ], and fractional-order Boubaker functions [ 29 ]. Again, the proposed algorithm also reported a very efficient performance.…”

“…Consider the following time-varying DFOCP, subject to: where . This example have been solved by Rahimkhani et al [ 28 ], Haddadi et al [ 41 ], Ordukhani et al [ 42 ], Moradi et al [ 27 , 38 ], and Rabiei et al [ 29 ], but any of them reached the initial condition . A comparison of the values of J obtained by these methods and that reported by the proposed scheme is presented in Table 5 .…”

Optimal control problems with Atangana‐Baleanu fractional derivative

“…Nonstandard finite difference method used to obtain numerical solutions of variable order FOCPs was presented in Reference 30. Finally, numerical schemes for solving FOCPs were carried out in References 31‐36.…”

A new operational matrix based on Boubaker wavelet for solving optimal control problems of arbitrary order

“…The application of fractional optimal control problems can be seen in engineering and physics and the aim of solving an optimal control problem is extremizing a cost function over an admissible set of control and state functions. Several numerical methods are applied to find an approximate solution to one-dimensional fractional optimal control problems, such as eigen functions method (Agrawal, 2008), rational approximation method (Tricaud and Chen, 2010), Legendre orthonormal basis method (Lotfi et al, 2013), Legendre operational technique (Bhrawy and Ezz-Eldien, 2016), Bernoulli polynomials method (Rabiei et al, 2018b), Hybrid of block-pulse functions and Bernoulli polynomials (Mashayekhi and Razzaghi, 2018), hybrid Chelyshkov functions (Mohammadi et al, 2018), fractional order Lagrange polynomials (Sabermahani et al, 2019), Adomian decomposition method (Alizadeh and Effati, 2018), Chebysheve collocation method (Rabiei and Parand, 2019), Grunwald-Letnikov, trapezoidal and Simpson fractional integral formulas (Salati et al, 2019), low dimensional approximations (Peng et al, 2019) and Spectral Galerkin approximation (Zhang and Zhou, 2019). But there are few researches devoted to two-dimensional problem especially in fractional area; for example, the authors in Nemati and Yousefi (2017) used the Ritz method to solve a class of these problems.…”