An accurate method for fractional optimal control problems governed by nonlinear multi-delay systems

Abstract

This study introduces a novel framework based on a combination of block-pulse and fractional-order Chebyshev functions. The new framework is a generalization of the fractional-order Chebyshev functions called the fractional hybrid functions. An accurate method is designed for solving nonlinear fractional optimal control problems with fractional multi-delay systems. Two essential linear operators, specifically, the fractional derivative operator and the fractional integral operator are introduced by implementing the Caputo and the Riemann–Liouville fractional operators. The two mentioned operators have a fundamental impact on reducing the computational complexity of the problem under study. Furthermore, these operators enable us to simply transform the principal problem into a new optimization one. Due to the structure of the fractional framework, we can construct an accurate solution for an extensive family of fractional multi-delay systems. By using a generalization of Taylor’s theorem, we prove that the proposed framework is convergent. The reliability, feasibility and accuracy of the new fractional framework are validated through examining a wide range of nontrivial examples.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

“…An extension of the fractional-order Chebyshev functions for constant order α , 0 < α ≤ 1, has been initially introduced in (Marzban 2022). Here, we provide a new generalization for Chebyshev functions of piecewise constant order α ( t ) where 0 < α ( t ) ≤ 1.…”

“…For determining the product operator normalΥ˜, we proceed the same way as carried out in (Marzban 2022).…”