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

AN ANALYTICAL APPROCH FOR SOLVING FRACTIONAL FUZZY OPTIMAL CONTROL PROBLEM WITH FUZZY INITIAL CONDITIONS

Abstract. A fractional – fuzzy optimal control problem is an optimal control problem in which it is governedby a fuzzy system of fractional differential equation. The aim of this paper is to introduce an analytically solution for such Bolza problems when the initial state is also fuzzy. For this purpose, first the problem is turned to two fractional optimal control problems by concept of 훽-cut and complex numbers. Then, we apply a new method to solve these ractional optimal control problems, analytically by applying a new Riccati differential equation determined from PMP. Indeed this Riccati equation transfer each mentioned fractional optimal control problem to a fractional differential system. We show that if the new system has close solution, one is able to obtain the analytical solution of the fractional – fuzzy optimal control problems. A numerical simulation based on the new method is presented for different values of 훽 and fractional order and the results are compered. In the last section, a numerical example of fractional-fuzzy optima control problem is solved by the new method for different 훽 and 훾; and compared with the exact state; also, they are shown in figures for each cases.Key words: Fractional differential equation, optimal control, fuzzy

Numerical Solution of Fuzzy Arbitrary Order Predator-Prey Equations

This paper seeks to investigate the numerical solution of fuzzy arbitrary order predator-prey equations using the Homotopy Perturbation Method (HPM). Fuzziness in the initial conditions is taken to mean convex normalised fuzzy sets viz. triangular fuzzy number. Comparisons are made between crisp solution given by others and fuzzy solution in special cases. The results obtained are depicted in plots and tables to demonstrate the efficacy and powerfulness of the methodology

Well-posedness and stability for fuzzy fractional differential equations

In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo–Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach fixed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results

SOLVING HYBRID FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS BY IMPROVED EULER METHOD

In this paper we study numerical methods for hybrid fuzzy fractional differential equations and the iteration method is used to solve the hybrid fuzzy fractional differential equations with a fuzzy initial condition. We consider a differential equation of fractional order  and we compared the results with their exact solutions in order to demonstrate the validity and applicability of the method. We further give the definition of the Degree of Sub element hood of hybrid fuzzy fractional differential equations with examples.

A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201