An application of Generalized Fuzzy Hyperbolic Model for solving fuzzy optimal control problems under granular differentiability

Abstract

The nature of real-world phenomena are often imprecision and vagueness, i.e., there is always a need to take into consideration the uncertainty factors when modeling real-world phenomena. In this paper, a generalized fuzzy hyperbolic model is employed for solving fuzzy optimal control problems, under the granular differentiability concept. Due to the characteristics of fewer identification parameters, GFHM can simplify the complexity of traditional ship fuzzy models. At the first step, we consider the granular Euler–Lagrange conditions for fuzzy variational problems and Pontryagin’s maximum principle for fixed and free final states of fuzzy optimal control problems, based on the ideas of horizontal membership function and granular differentiability via the calculus of variations. The necessary optimality conditions for these problems are derived in the form of two-point boundary value problems. Here, for the first time, generalized fuzzy hyperbolic models are used to approximate the solutions of the related two-point boundary value problems. This fuzzy hyperbolic models uses of the number of sample points as the training dataset, and the Levenberg–Marquardt algorithm is selected as the optimizer. By relying on the ability of the generalized fuzzy hyperbolic models as function approximator, the fuzzy solutions of variables are substituted in the related two-point boundary value problem. The obtained algebraic nonlinear equations system is then reduced into an error function minimization problem. A learning scheme based on the Levenberg–Marquardt algorithm is employed as the optimizer to derive the adjustable parameters of fuzzy solutions. In order to clarify the effectiveness of the studied approach, some numerical results are supplied.