A sufficient condition for a triple of linear systems to be simultaneously stabilizable in a behavioral framework

In this paper, we address simultaneous stabilization problem for a triple of linear systems within a behavioral framework. First, we provide a necessary and sufficient condition for a given pair of linear systems to be simultaneously stabilizable in a behavioral framework. Next, we show that a simultaneous stabilizer has a symmetric structure, which is also a self-standing interesting result from the theoretical points of view. By using these results, we give a new necessary and sufficient condition for a given pair of linear systems to be simultaneously stabilizable. This new condition is described in terms of the solvability of polynomial matrix equations consisting of polynomial matrices inducing kernel or image representations of the behaviors of given two behaviors. Finally, by using this new condition, we give a sufficient condition for a triple of linear systems to be simultaneously stabilizable in terms of the behaviors.