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We consider the weak regularity problem of  linear systems  in Banach state spaces when its associated unbounded controllers take values in the extrapolated Favard class. We prove that this type of  linear systems are weakly regular and this regularity is output operators independent up to the well-posedness, provided the associated state spaces are non-reflexive and that the adjoint of its associated semigroups are strongly continuous on the dual of its state space. As applications, we consider a class of boundary control systems and a class of well-posed bilinear systems introduced in [2]. 

well-posed linear system, well-posed bilinear system, regular linear systems, regular bilinear systems, admissible operators, Favard spaces.