In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler-Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM).

Overlap functions, as one kind of particular binary aggregation functions, have been continuously studied in the literature for their vast preponderance in some real applications. Meanwhile, after Lucca et al. introduced the notion of preaggregation functions recently, the investigation related to such “aggregation” functions becomes an interesting, and natural research area. In this article, we continue to focus on this topic for overlap functions. First, we introduce the concept of $\vec{r}$ -(quasi-)overlap functions by replacing the monotonicity of (quasi-)overlap functions with directional monotonicity, and the notions of 0- $\vec{r}$ -(quasi-)overlap functions, 1- $\vec{r}$ -(quasi-)overlap function, and 0,1- $\vec{r}$ -(quasi-)overlap functions associated with them. And then, we propose some vital properties of $\vec{r}$ -(quasi-)overlap functions. Finally, we show several construction methods of $\vec{r}$ -(quasi-)overlap functions, 0- $\vec{r}$ -(quasi-)overlap functions, 1- $\vec{r}$ -(quasi-)overlap function, and 0,1- $\vec{r}$ -(quasi-)overlap functions.

Communication between information systems is a basic problem in granular computing. In current research, a homomorphism mapping between information systems is required. In this paper, the concept of the general relation mapping between two universes is proposed in order to construct the equivalence relation on one universe according to the given equivalence relation on the other universe based on the including degrees. The main properties of the mapping are studied, and it is proved that attribute reductions in the original system and image system are equivalent to each other under the given conditions. Finally, we also define the concept of fuzzy general relation mapping between two fuzzy information systems and give some properties.