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

We derive necessary conditions for exact controllability of infinite-dimens ional systems described by (x)over dot = Ax + BEI, where the generator A ha s 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 unb oundedness 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) exa mples that the addition of damping into an exactly controllable second orde r system can destroy exact controllability. We discuss the effect of dampin g on such systems using our general conditions. We also give simple new pro ofs for results about the lack of exact controllability of hyperbolic syste ms in more than one space dimension. (C) 2000 Elsevier Science B.V. All rig hts reserved.