The time average reward for some dynamic fuzzy systems

In this paper, by using a fuzzy relation, we define a dynamic fuzzy system with a bounded convex fuzzy reward on the positive orthant Rn+ of an n-dimensional Euclidean space. As a measure of the systemʹs performance, we introduce the time average fuzzy reward, which is characterized by the limiting fuzzy state under the contractive properties of the fuzzy relation. In the one-dimensional case, the average fuzzy reward is expressed explicitly by the functional equations concerning the extreme points of its α-cuts. Also, a numerical example is given to illustrate the theoretical results.