Optimal control of nonlinear time-delay fractional differential equations with Dickson polynomials

Fractals ◽  
2020 ◽  
Author(s):  
Shu-Bo Chen ◽  
Samaneh Soradi-Zeid ◽  
Maryam Alipour ◽  
Yu-Ming Chu ◽  
J. F. Gomez-Aguilar ◽  
...  
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Dickson Polynomials ◽  
Fractional Differential

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

2021 ◽  
pp. 107754632110310
Author(s):  
Ahmed S Hendy ◽  
Mahmoud A Zaky ◽  
José A Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Heat Equation ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Variable Order ◽  
Integer Order ◽  
Order Variable ◽  
Fractional Differential ◽  
Distributed Order

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

Application of Fractional Optimal Control Problems on Some Mathematical Bioscience

2020 ◽  
pp. 41-56
Author(s):  
Ismail Gad Ameen ◽  
Hegagi Mohamed Ali
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Short Review ◽  
Decision Makers ◽  
Fractional Optimal Control ◽  
Fractional Models ◽  
Fractional Differential ◽  
Fractional Optimal Control Problems

In this chapter, the authors present a short review of a fractional-order control models, which are described by a system of fractional differential equations. Fractional derivatives describe the behaviour of dynamical systems better than classical calculus, where it can reflect the effect of memory. This survey shows the effect of the control on the fractional models, which represents epidemiological and biomedicine problems. So, the solution to such models is necessary for decision makers in health organizations.

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