A numerical method for approximating the solution of fuzzy fractional optimal control problems in caputo sense using legendre functions

• Published in: Journal of intelligent & fuzzy systems Vol. 43; no. 3; pp. 3827 - 3858
• Main Authors: Mirvakili, M., Allahviranloo, T., Soltanian, F.
• Format: Journal Article
• Language: English
• Published: Amsterdam IOS Press BV 01.01.2022
• Subjects: Ambiguity; Approximation; Basis functions; Differential equations; Dynamical systems; Euler-Lagrange equation; Fractional calculus; Fuzzy control; Fuzzy sets; Initial conditions; Lagrange multiplier; Legendre functions; Numerical analysis; Numerical methods; Optimal control; Performance indices; Polynomials; Time invariant systems; Time varying control systems
• DOI: 10.3233/JIFS-210583
• ISSN: 1064-1246

Fractional order differential equations accurately model dynamic systems and processes. In some of the fractional optimal control problems (FOCPs), due to the ambiguity in the initial conditions and the transfer of ambiguity to the solution, it is necessary to use fuzzy mathematics. In this paper, a numerical method is presented to approximate the solution for a class of Fuzzy Fractional Optimal Control Problems (FFOCPs) using the Legendre basis functions. The fuzzy fractional derivative is described in the Caputo sense. The performance index of an FFOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fuzzy Fractional Differential Equations (FFDEs). After obtaining Euler–Lagrange equations for FFOCPs and the necessary and sufficient conditions for the existence of solutions, using the definition of generalized Hukuhara differentiability (types I, II), the problem is considered in two cases. Then the distance function and an approach similar to the variational type along with the Lagrange multiplier method are used to formulate and solve the equations in a system. Time-invariant and time-varying examples are provided to assess the presented method. Numerical results show a similar trend for the state and control variables for various numbers of Legendre polynomials. Also, the convergence of state and control variables for the time-invariant system can be seen, and the same is true for control variables for the time-varying system.