Approximate Controllability and Stabilizability of Time Delay Systems: Functional Analytic and Algebraic Results

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 genrating a Co - semigroup in the state space. It is shown that spectral controllability is necessary for approximate coollability 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 controllabiity implies spectral controllability. Also, open loop stabilizability is necessary for approximate null controllability. We show that three large families of concrete function spaces, C^(k), M^p, and w^(k,p), satisfy all our geneal assumptions. We generalize the notions of a dual transposed system and dual observability concepts, developed in [1], corresponding to approximate controllability concepts. This enables completion of our general results by proving that spectral controllability is sufficient for approximate null controllability and by deriving testable criteria from the dual problems. Such criteria assume an especially simple form when the final time T is greater than nh and also in case of discrete delays where connections to controllability over the ring of polynomials in a delay operator are established