Necessary conditions for exact controllability with a finite-dimensional input space

We derive necessary conditions for exact controllability of infinite-dimensional systems described by ẋ = Ax + Bu, where the generator A has a Riesz basis of eigenvectors and the input space is finite-dimensional. These conditions are in terms of the eigenvalues of A and the degree of unboundedness of B, but no further information about B is needed. Our results easily imply lack of exact controllability for a wide range of distributed parameter systems. It is known from partial differential equation (PDE) examples that the addition of damping into an exactly controllable second order system can destroy exact controllability. We discuss the effect of damping on such systems using our general conditions. We also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension.