A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

A survey on fractional order control techniques for unmanned aerial and ground vehicles

In recent years, numerous applications of science and engineering for modeling and control of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) systems based on fractional calculus have been realized. The extra fractional order derivative terms allow to optimizing the performance of the systems. The review presented in this paper focuses on the control problems of the UAVs and UGVs that have been addressed by the fractional order techniques over the last decade

Enlarged Controllability of Riemann-Liouville Fractional Differential Equations

We investigate exact enlarged controllability for time fractional diffusion systems of Riemann-Liouville type. The Hilbert uniqueness method is used to prove exact enlarged controllability for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN 1555-1423, CODEN JCNDDM, available at [http://computationalnonlinear.asmedigitalcollection.asme.org]. Submitted 10-Aug-2017; Revised 28-Sept-2017 and 24-Oct-2017; Accepted 05-Nov-201

Chaotic multi-objective optimization based design of fractional order PI{\lambda}D{\mu} controller in AVR system

In this paper, a fractional order (FO) PI{\lambda}D\mu controller is designed to take care of various contradictory objective functions for an Automatic Voltage Regulator (AVR) system. An improved evolutionary Non-dominated Sorting Genetic Algorithm II (NSGA II), which is augmented with a chaotic map for greater effectiveness, is used for the multi-objective optimization problem. The Pareto fronts showing the trade-off between different design criteria are obtained for the PI{\lambda}D\mu and PID controller. A comparative analysis is done with respect to the standard PID controller to demonstrate the merits and demerits of the fractional order PI{\lambda}D\mu controller.Comment: 30 pages, 14 figure

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

Study on Viral Transmission Impact on Human Population Using Fractional Order Zika Virus Model

This work comprises the spread of Zika virus between humans and mosquitoes as a mathematical simulation under fractional order, which also incorporates the asymptotically infected human population. For determining the solution of the model the fuzzy Laplace transform technique is utilized. By combining fuzzy logic with the Laplace transform, we can analyze systems even when we lack precise information. Further, the sensitivity analysis is performed to validate the model. On top of that the population dynamics of both human and mosquito populations are discussed using numerical data and the graphical result of the model is presented. The main objective of this work is to study the dynamics of the Zika virus and to examine the effect of virus on humans when the transmission occurs between humans and from mosquitoes, under fractional order. The outcome of these comparisons suggests that even by reducing a minute fractional part of transmission through mosquitoes results in a greater reduction of Zika exposed population. The comparisons improve the understanding of fractional level transmission resulting in more effective drug administration to patients. The Hyers-Ulam stability method is a mathematical technique used to study the stability of functional equations. Eventually, Ulam Hyers and Ulam Hyers Rassias stability are employed to assess the stability of the proposed model

New variational problems with an action depending on generalized fractional derivatives, the free endpoint conditions, and a real parameter

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.publishe