An Efficient Numerical Scheme for Solving Multi‐Dimensional Fractional Optimal Control Problems With a Quadratic Performance Index

Abstract

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

“…where (t) must be chosen such that the summation term in (17) satisfies the homogenous initial and boundary conditions, that is, x (i) (0) = 0, x (i) (1) = 0 and w(t) satisfies the given initial and boundary conditions of (16). Considering this fact we take…”

“…Fractional calculus provides a powerful mathematical tool for the description of physical systems in many fields, such as, among others, physics, engineering and bioengineering [3], mechanical systems [4], stochastic systems [5,6], mobile sensor [7], and control systems [8]. An interesting active research area of fractional calculus is that of fractional optimal control (FOC), in which the dynamical system as well as cost function involve not only integer order derivatives but also fractional order derivatives or integrals [15,16]. An interesting active research area of fractional calculus is that of fractional optimal control (FOC), in which the dynamical system as well as cost function involve not only integer order derivatives but also fractional order derivatives or integrals [15,16].…”

“…The first is to construct a Hamiltonian system and thereafter solve the arising two-point boundary value problem; and second is to solve the optimal control problem directly by discretizing or approximating the functions without constructing the Hamiltonian equations [16][17][18][19]. The first is to construct a Hamiltonian system and thereafter solve the arising two-point boundary value problem; and second is to solve the optimal control problem directly by discretizing or approximating the functions without constructing the Hamiltonian equations [16][17][18][19].…”

An efficient computational method based on the hat functions for solving fractional optimal control problems

“…For example in [3,4], the authors have achieved the necessary conditions for optimization of FOCPs with the Caputo fractional derivative. The interested reader can refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] for some recent works on FOCPs. In this paper, we propose a new efficient and accurate computational method based on HFs for solving the following FOCP [6]:…”

Approximate Technique for Solving Class of Fractional Variational Problems

“…We use the Legendre collocation method to discretize FDEs to get linear or non-linear system of algebraic equations, thus greatly simplifying the proposed problem. Recently, the shifted Legendre polynomials have been used as basis functions of numerical techniques for solving types of fractional optimal control problems, see [16] [17]. Legendre expansion method for solving fractional-order delay differential equations is given in [18].…”