A direct computational method for nonlinear variable‐order fractional delay optimal control problems

Abstract

This paper introduces a highly accurate operational matrix technique based upon the Chebyshev cardinal functions (CCFs) for solving a new category of nonlinear delay optimal control problems (OCPs) involving variable-order (VO) fractional dynamical systems. The VO fractional derivatives are defined in the Caputo type. The main aim is converting such OCPs into systems of algebraic equations. Thus, we first expand the state and control variables in terms of the CCFs with undetermined coefficients. Then, by utilizing the cardinal property of these basis functions, the delay terms in the problem under consideration are expanded in terms of the CCFs. Thereafter, these expansions are substituted into the cost functional, dynamical system and delay conditions. Next, the cardinality of the CCFs together with their operational matrix (OM) of VO fractional derivative are employed to extract a nonlinear algebraic equation from the cost functional, and several algebraic equations from the dynamical system and delay conditions. Eventually, the method of the constrained extremum is employed by coupling the algebraic constraints yielded from the dynamical system and delay conditions with the algebraic equation extracted from the cost functional using a set of Lagrange multipliers. The precision of the method is studied through different types of numerical examples. KEYWORDSChebyshev cardinal functions (CCFs), optimal control problems (OCPs), operational matrix (OM), variable-order (VO) fractional delay optimal control problems (OCPs)

Operator theoretic approach in fractional‐order delay optimal control problems

“…Moradi et al 5 compare the discrete and continuous approaches for solving time DFOCP by using Chebyshev polynomials. Heydari et al 6 proposed a new accurate operational matrix technique for solving variable‐order DFOCP by using Chebyshev cardinal functions. Bhrawy and Ezz‐Eldien 7 introduced another operational matrix technique that uses the Legendre orthonormal polynomial as a basis function to solve DFOCPs.…”

Numerical investigation of variable‐order fractional Benjamin–Bona–Mahony–Burgers equation using a pseudo‐spectral method

“…A suitable set of the polynomial basis functions which have been extensively applied for solving diverse categories of fractional functional equations is the set of the Chebyshev cardinal functions (CCFs). These polynomials have several useful properties, including the exponential (spectral) accuracy and cardinality 50 . The exponential accuracy allows us to obtain solutions with high precision for problems with smooth solution.…”

A numerical method based on the Chebyshev cardinal functions for variable‐order fractional version of the fourth‐order 2D Kuramoto‐Sivashinsky equation

“…In recent years, many researchers have used the cardinal approach to solve various equations. [11][12][13][14][15][16][17] Any smoothly differentiable function can be approximated by the cardinal functions because these are produced by the eigenfunctions of some singular Sturm-Liouville problems, such as the Chebyshev or Legendre equation. Using these polynomials results in the spectral accuracy.…”

“…The most significant privilege of approximation using cardinal functions is solving a less complex system of equations instead of the origin problem. In recent years, many researchers have used the cardinal approach to solve various equations 11‐17 . Any smoothly differentiable function can be approximated by the cardinal functions because these are produced by the eigenfunctions of some singular Sturm‐Liouville problems, such as the Chebyshev or Legendre equation.…”