Controllability of k-Valued Fuzzy Cognitive Maps

Fuzzy cognitive maps (FCMs) as a kind of knowledge-based tools are widely applied to model complex dynamical systems using causal relations. Besides the representation and reasoning of systems behaviors, how to control the given systems into a desirable target by causal objects established by FCMs is also an open problem. Although, so far, there are some existing works about the applications of FCMs on the control-related problems, it is still a lack of the theoretical analysis in this domain. In this paper, the controllability of k-valued FCMs is studied. To improve the universality of models, a temporal extension of generalized FCMs is implemented. By means of semitensor product, the algebraic representation of k-valued FCMs with controls is established and a generalized formula of control-depending network transition matrices is achieved. A necessary and sufficient condition is proved to determine the control-depending fixed points of k-valued FCMs with temporalization. By utilizing three kinds of controls, the controllability of the discrete FCMs is discussed, respectively. The reachability condition of a specific target state from a given initial state at time s is studied, and the reachable set along with the corresponding reachable probability are also provided by analytic formula. Results provide a way to make FCMs evolving into the designed states by controls, which can further conduct the behaviors of the modeled systems in reality. Examples are shown to demonstrate the effectiveness and feasibility of the proposed scheme.