Approximate Controllability of Linear Functional-Differential Systems: A State Space Independent Approach

Approximate versions of controllability, reachability, and null controllability for linear functional-differential systems of retarded type are considered in relation to spectral controllability and stabilizability in general function spaces not necessarily generating a C_o - semigroup in the state space. It is shown that spectral controllability is necessary for approximate controllability and reachability in a large variety of state spaces. Spectral controllability is also necessary for approximate null controllability on a fixed time interval. On the other hand, if the space of control functions is closed under right shifts, uniform approximate controllability implies spectral controllability. Also, open loop stabilizability is necessary for approximate null controllability. Finally, we show that three large families of concrete function spaces, C^(k) M^p, and W^(k,p), satisfy all our general assumptions.