| We study the control system x(t) = Ax(t) + bu(t), where x(t) belongs to a Hilbert Space h, b is an "admissible input element", and u is the control. A is an operator on H which has eigenvalues {lcub}(lamda)(,k){rcub} and eigenvectors which form a Riesz basis for H. We assume that A generates a holomorphic semigroup, so that {lcub}(lamda)(,k){rcub} lies in a sector in the left half plane. A canonical form is developed to study this system.; The canonical form is a functional equation ((eta) * z) (t) = u(t), which is shown to be equivalent to a state space equation. We can map the original equation to this state space equation. In the construction of these equations, we develop a Laplace Transform theory relevant to holomorphic semigroups. The Fourier transform of (eta) is a meromorphic function p with zeros at {lcub}(lamda)(,k){rcub} and certain growth properties. This function is used to represent (eta) in the Laplace Tansform space, and is in many ways analogous to the characteristic polynomial in the ordinary differential equation canonical form. We call such a function a "cardinal function".; We can use the canonical form to construct feedback controls. If q is a cardinal function with the same poles as p and zeros at {lcub}(alpha)(,k){rcub}, and if b satisfies certain conditions, we can construct a feedback control u(t) = h*x(t) such that A + bh* has eigenvalues at {lcub}(alpha)(,k){rcub} and a Riesz basis of eigenvectors. Here h* is an "admissible feedback element", and we obtain formulas for h* and the eigenvectors. If we consider only bounded inputs and feedbacks, we can prove Sun's sufficient condition for there to exist h* such that A + bh* has eigenvalues at {lcub}(alpha)(,k){rcub}. We also construct admissible controls using "almost cardinal functions". Finally, we briefly discuss a notion of "Relative Control", which is a generalization of feedback control. |
