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

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.