On finite energy solutions of fractional order equations of the Choquard type

Abstract Fractional circuit elements become increasingly popular due to their versatility in various applications. However, the bottleneck in deploying these tools in practice is related to an open problem, i.e, infinite energy problem. On this topic, many valuable achievements have been made. Some scholars don’t dare to use fractional circuit elements because of the infinite energy problem while some scholars believe that there is no paradox compared with classical finite energy or even some scholars think that this problem has been successfully solved. However, there is still no consensus on this topic and confusion remains widespread. Consequently, a comprehensive review on infinite energy problem is needed imperatively. At this point, this paper reviews the consequences, root causes, and potential mitigation approaches through the modeling analysis and literature survey. This review starts with the fractional capacitors. Subsequently, other fractional circuit elements and fractional order operators/systems are considered. Finally, the main technical challenges as well as future researches on this topic are highlighted carefully.

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Finite energy Lyapunov function candidate for fractional order general nonlinear systems

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2019.104886 ◽

2019 ◽

Vol 78 ◽

pp. 104886 ◽

Cited By ~ 7

Author(s):

Yan Li ◽

Daduan Zhao ◽

YangQuan Chen ◽

Igor Podlubny ◽

Chenghui Zhang

Keyword(s):

Nonlinear Systems ◽

Lyapunov Function ◽

Fractional Order ◽

Finite Energy

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Modeling of Atmospheric Turbulence as Disturbances for Control Design and Evaluation of High Speed Propulsion Systems

Volume 3: Controls, Diagnostics and Instrumentation; Cycle Innovations; Marine ◽

10.1115/gt2010-22851 ◽

2010 ◽

Author(s):

George Kopasakis

Keyword(s):

Atmospheric Turbulence ◽

Fractional Order ◽

High Speed ◽

Transfer Functions ◽

Turbulence Models ◽

Finite Energy ◽

Atmospheric Disturbances ◽

Flight Controls ◽

Propulsion Control ◽

Supersonic Vehicles

Atmospheric turbulence models are necessary for the design of both inlet/engine and flight controls, as well as for studying integrated couplings between the propulsion and the vehicle structural dynamics for supersonic vehicles. Models based on the Kolmogorov spectrum have been previously utilized to model atmospheric turbulence. In this paper, a more accurate model is developed in its representative fractional order form, typical of atmospheric disturbances. This is accomplished by first scaling the Kolmogorov spectral to convert them into finite energy von Karman forms. Then a generalized formulation is developed in frequency domain for these scale models that approximates the fractional order with the products of first order transfer functions. Given the parameters describing the conditions of atmospheric disturbances and utilizing the derived formulations, the objective is to directly compute the transfer functions that describe these disturbances for acoustic velocity, temperature, pressure and density. Utilizing these computed transfer functions and choosing the disturbance frequencies of interest, time domain simulations of these representative atmospheric turbulences can be developed. These disturbance representations are then used to first develop considerations for disturbance rejection specifications for the design of the propulsion control system, and then to evaluate the closed-loop performance.

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Modeling of Atmospheric Turbulence as Disturbances for Control Design and Evaluation of High Speed Propulsion Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4005368 ◽

2011 ◽

Vol 134 (2) ◽

Cited By ~ 5

Author(s):

George Kopasakis

Keyword(s):

Atmospheric Turbulence ◽

Fractional Order ◽

High Speed ◽

Transfer Functions ◽

Turbulence Models ◽

Finite Energy ◽

Atmospheric Disturbances ◽

Flight Controls ◽

Propulsion Control ◽

Supersonic Vehicles

Atmospheric turbulence models are necessary for the design of both inlet/engine and flight controls, as well as for studying integrated couplings between the propulsion and the vehicle structural dynamics for supersonic vehicles. Models based on the Kolmogorov spectrum have been previously utilized to model atmospheric turbulence. In this paper, a more accurate model is developed in its representative fractional order form, typical of atmospheric disturbances. This is accomplished by first scaling the Kolmogorov spectral to convert them into finite energy von Karman forms. Then a generalized formulation is developed in frequency domain for these scale models that approximates the fractional order with the products of first order transfer functions. Given the parameters describing the conditions of atmospheric disturbances and utilizing the derived formulations, the objective is to directly compute the transfer functions that describe these disturbances for acoustic velocity, temperature, pressure, and density. Utilizing these computed transfer functions and choosing the disturbance frequencies of interest, time domain simulations of these representative atmospheric turbulences can be developed. These disturbance representations are then used to first develop considerations for disturbance rejection specifications for the design of the propulsion control system and then to evaluate the closed-loop performance.

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