Distribution semigroups and control systems

We introduce the new concept of a distributional control system. This class of systems is the natural generalization of distribution semigroups to input/state/output systems. We show that, under the Laplace transform, this new class of systems is equivalent to the class of distributional resolvent linear systems which we introduced in an earlier article. There we showed that this latter class of systems is the correct abstract setting in which to study many non-well-posed control systems such as the heat equation with Dirichlet control and Neumann observation. In this article we further show that any holomorphic function defined and polynomially bounded on some right half-plane can be realized as the transfer function of some exponentially bounded distributional resolvent linear system.