Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Abstract

In this paper, a method for finding an approximate solution of a class of 2D fractional optimal control problems with fractional-order dynamical system is discussed. In the proposed method, the fractional derivative is expressed in the Caputo sense. The method consists of expanding unknown functions as the elements of two-dimensional (2D) Müntz-Legendre wavelets. The 2DMüntz-Legendre wavelets are constructed and their properties are presented.The operational matrix of fractional-order integration for these wavelets is utilized to reduce the solution of 2D fractional optimal control problem to an optimization problem, which can then be solved easily. Some results concerning the error analysis are obtained. Finally, two illustrative test problems are included to demonstrate the validity and applicability of the technique. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. KEYWORDSCaputo derivative, fractional optimal control problem, numerical solution, Riemann-Liouville fractional integral operator, wavelet method INTRODUCTIONNowadays, fractional calculus have a wide range of applications in fields of fluid-dynamic traffic, 1 frequency dependent damping behavior of various viscoelastic materials, 2 colored noise, 3 solid mechanics, 4 economics, 5 signal processing, 6 and control theory. 7 Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, 8 finite difference method, 9 wavelet method, 10 least square approximation method, 11 finite element method, 12 spline collocation method, 13 fractional wavelet method, 14 shifted fractional-order Jacobi orthogonal functions, 15 Jacobi Tau method, 16,17 and Legendre Tau method. 18 An optimal control problem (OCP) is defined to minimize the functional over an admissible set of control functions subject to dynamic constraints on the state and control variables. A fractional OCP (FOCP) is an optimal control problem in which the performance index or the differential equations governing the dynamics of the system or both contains at least one fractional-order derivative term. In recent the years, several numerical methods have been devoted to solve of one-dimensional FOCPs. We list here some of these numerical methods, as follows: • Eigen functions method (Agrawal 19 ); • Quadratic numerical scheme (Agrawal 20 ); • Rational approximation method (Tricaud and Chen 21 ); • Legendre multiwavelet collocation method (Yousefi et al 22 ); 1916

Numerical solution of nonlinear fractional optimal control problems using generalized Bernoulli polynomials

“…The mixed conditions Λ j , (j = n + 1, n + 2, … , 2n), using Eq. (30). * Minimize the objective function  * [A, B, H, Q, ] subject to the constraints ( 29) and (30).…”

“…Tohidi and Nik 29 applied the truncated Bessel series approximation by using a collocation scheme, for solving linear and nonlinear FOCP. Rahimkhani and Ordokhani 30 introduced a numerical technique based on the Müntz-Legendre wavelets for 2-dim FOCP. Readers can refer to References 31-40 for other works on FOCP.…”

Numerical investigation of distributed‐order fractional optimal control problems via Bernstein wavelets

“…Heydari 25 presented a new direct method based on the Chebyshev cardinal functions to solve a class of variable‐order FOCPs. Also, two‐dimensional FOCPs are considered by the spectral method combined with Bernstein operational matrix, 26 generalized fractional‐order Bernoulli‐Legendre functions method, 27 and Müntz‐Legendre wavelets 28 …”

Optimal control problems with Atangana‐Baleanu fractional derivative

“…In recent years, fractional calculus (FC) is used to advance mathematical models of real‐world phenomena in science and engineering. Some interesting results obtained in FC including analysis and computations results were recently studied in References 1‐17. Fractional differential equations (FDEs) with singular kernels have difficulties in handling in many physical phenomena.…”