Constrained controllability of linear discrete nonstationary systems in Banach spaces

This paper studies local null-controllability of linear infinite-dimensional, nonstationary, discrete-time systems of the form $x_{k + 1} = A_k x_k + B_k u_k $, $u_k \in \Omega \subset U$, $x_k \in M_k \subset X$, where $X$, $U$ are Banach spaces; $A_k $, $B_k $ are linear bounded operators; $M_k $, $\Omega $ are given nonempty subsets. New necessary and sufficient conditions for local null-controllability are given. The main tool is the surjectivity theorem for convex multivalued mappings in Banach spaces.