Optimal control of nonlinear fractional order delay systems governed by Fredholm integral equations based on a new fractional derivative operator.

We aim to develop a direct transcription approach for solving a notable category of optimal control problems governed by nonlinear fractional Fredholm integral equations having delays in both input and output signals. The foundation of the new methodology is based on a multi domains decomposition scheme by utilizing the fractional-order Legendre functions. A new fractional derivative operator associated with the fractional basis is introduced by using the Caputo fractional derivative operator. With the use of derivative and delay operators, one can transform the dynamical system related to the fractional control problem into a new system containing algebraic equations. A wide variety of challenging test problems are studied to provide a detailed explanation of the designed approach.

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.