Legendre wavelet method for solving variable‐order nonlinear fractional optimal control problems with variable‐order fractional Bolza cost

Abstract

This paper deals with a numerical method for solving variable‐order fractional optimal control problem with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a variable‐order Riemann–Liouville fractional integral. Using the integration by part formula and the calculus of variations, the necessary optimality conditions are derived in terms of two‐point variable‐order boundary value problem. Operational matrices of variable‐order right and left Riemann–Liouville integration are derived, and by using them, the two‐point boundary value problem is reduced into the system of algebraic equations. Additionally, the convergence analysis of the proposed method has been considered. Moreover, illustrative examples are given to demonstrate the applicability of the proposed method.

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

“…Kumar and Mehra [3] have proposed a direct method to solve FOCPs with dynamic constraints as inequality by using Hermite wavelet. A FOCP with fractional Bolza has also been solved by deriving necessary optimality conditions [14,15]. Mehandiratta [16] has derived the necessary optimality conditions for a FOCP on a star graph and solved it by using the finite difference method.…”

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

“…Applying spectral analysis, Legendre polynomials [16] are used to develop different numerical approaches. The authors consider FOCPs with both integer-order and Caputo derivatives and propose an indirect numerical method in terms of truncated Bessel series in Tohidi and Nik [17]; other relevant results may be found in previous studies [18][19][20][21][22][23][24].…”

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

“…Kumar and Mehra (2021a) has used a Hermite wavelet collocation method to solve a FOCP with constraints as inequality. A Legendre wavelet collocation method has been used to solve FOCP with Bolza cost (Kumar and M. Mehra 2021) and variable-order (VO) FOCP with VO Bolza cost (Kumar and M. Mehra, 2022). An optimality system and numerical scheme for FOCP on a star graph have been given in (Mehandiratta et al, 2021).…”