Relay control of an evolution process in a Banach space in a neighborhood of an unstable stationary regime

We study a controlled evolution process whose states are characterized by points of a real Banach space and which is described by a family of semigroups in a neighborhood of an unstable stationary regime. The goal is to find control to keep the state of the process however long in the neighborhood. We find conditions under which this control is possible, and prove an existence theorem. Also, we apply the theorem to the processes described by systems of parabolic differential equations.