Analysis of nonlinear fractional optimal control systems described by delay Volterra–Fredholm integral equations via a new spectral collocation method

Abstract

We aim to introduce a new spectral collocation method for investigating and analyzing nonlinear delay control systems governed by the fractional mixed Volterra–Fredholm integral equations (MVFIEs). The generalized fractional Legendre basis (GFLB) is used as a complete orthogonal basis, and the fractional Legendre–Gaussian nodes are introduced and employed as the fractional collocation points. These nodes correspond to the zeros of the fractional-order Legendre function of degree M. The convergence of the generalized orthogonal basis is discussed in detail based on the Sobolev and L2 norms. Additionally, new fractional integral and delay operators are implemented to reduce the primary control system to a static optimization system. Two benchmark fractional control problems, including delay, are considered to demonstrate the powerfulness and superiority of the new methodology. The introduced collocation approach can be successfully performed for solving even those fractional control problems having any irregularities in the control function, including jump discontinuities and bang–bang behavior.