On the Admissible Control operators for Linear and Bilinear Systems and the Favard Spaces

The objective of this work is to give some relationship between the Favard spaces and the p-admissibility (resp. (p, q)-admissibility) of unbounded control operators for linear (resp; bilinear) systems in Banach spaces. For linear case, this enables to give a simple identification of the space of the 1 admissible control operators in Banach space and it enables us to extend the result of Weiss [29] (for p = 1) on reflexive Banach spaces to a general situation. This result is applied to boundary control systems. The results obtained for bilinear systems generalize those given in Idrissi [16] and Berrahmoune [2] and are applied to diffusion equations of fractional order time distributed order.