Constrained controllability in Banach spaces

The aim of this paper is to study null-controllability of the linear infinite dimensional control problem $\dot x = Ax + Bu$ where the control $u$ is constrained to lie in a convex, weakly compact subset $\Omega $ of the control space with $0 \in \Omega $. A necessary and sufficient condition for a particular initial state to be $\Omega $-null-controllable within a fixed, finite time $T$ is given. The result is extended to the case $\Omega = $ convex cone with vertex at 0. Applications to the one-dimensional heat- and wave-equation are given.