Non-well-posed infinite-dimensional linear systems

We provide an abstract framework for infinite-dimensional linear systems that are not necessarily well-posed in the standard L2 sense. One example of a system that is not well-posed but does fit into the new abstract framework is the heat equation with Dirichlet control and Neumann observation. The new abstract framework presented is based on distribution theory. We discuss problems such as: the linear quadratic regulator problem, coprime factorization, robustly stabilizing controllers with respect to factor perturbations and LQG-balanced model reduction. All these problems admit a satisfactory solution at the level of generality of the new abstract framework