Semitensor Product Approach to Controllability, Reachability, and Stabilizability of Probabilistic Finite Automata

Abstract

This paper proposes a matrix-based approach to investigate the controllability, reachability, and stabilizability of probabilistic finite automata (PFA). Firstly, the state transition probabilistic structure matrix is constructed for PFA, based on which a kind of controllability matrix is defined for PFA. Secondly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of PFA with positive probability by using the controllability matrix. Finally, an illustrate example is given to validate the obtained new results.

Matrix Expression of Shapley Value in Graphical Cooperative Games

“…In our recent work [10], a simple matrix formula is obtained to calculate the Shapley value and the basic mathematical method is the semitensor product of matrices (STP), which is a generalization of the traditional matrix product [11]. is method has been successfully applied in the analysis and control problems of logical networks [12][13][14][15][16][17][18][19][20][21][22][23][24][25], and then it has been used in finite game theory [26][27][28][29][30], evolutionary games [31][32][33][34][35][36][37][38], finite automata [39,40], graph theory [41], etc. Unlike the canonical cooperative games being considered in [10], this paper studies a new class of cooperative games, called graphical cooperative games.…”