A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems

This paper aims to devise a novel fractional orthogonal basis to solve a certain class of nonlinear fractional optimal control problems with delay whose system dynamics is governed by a nonlinear fractional differential equation of the Caputo type. The foundation of the new framework is based on a hybrid of block-pulse and fractional-order Legendre functions. A new integral operator associated with the proposed orthogonal basis is constructed by using the Riemann-Liouville integral operator. This operator enables one to immensely reduce the complexity of computations related to the Riemann-Liouville integral operator. Some significant theoretical results concerning the new fractional basis are provided. Several problems are tested for the validation and verification of our numerical findings. It is demonstrated that the new fractional basis produces an exact solution for a specific class of nonlinear delay fractional optimal control problems. Generally, the developed fractional basis is a promising mathematical tool for investigating fractional-order systems.