Fuzzy Bang-Bang control problem under granular differentiability

In this paper, a class of fuzzy optimal control problem called fuzzy Bang-Bang (FBB) control problem is revisited. The FBB control problem aims at finding control inputs which transfer the states of an uncertain dynamical system to the origin in a minimum time. Since the conditions and/or parameters of the dynamical system are uncertain, the FBB control problem is a challenge task. To address the problem, we first introduce the concepts of the granular integral and derivative of fuzzy functions whose domain is uncertain. In addition, the notion of granular partial derivative of a multivariable fuzzy function whose variables are fuzzy functions with uncertain domains is presented. Then, we propose a theorem which is proved to be applicable to the FBB control problem. Moreover, taking the relative-distance-measure fuzzy interval arithmetic and horizontal membership functions into consideration, we further give complementary theorems to ensure that if the problem has a solution, then the controller assumes its boundary values. The simulated results confirm this theoretical conclusion. These findings may enrich our insight of the behavior of the uncertain dynamical system subjected to the FBB control problem, and guide to predict the uncertain trajectories.