On the control of LTI systems with rough control laws

The theory of linear time invariant systems is well established and allows, among other things, to formulate and solve control problems in finite time. In this context the control laws are typically taken in a space of the form L^p(0,T;U). In this paper we consider the possibility of taking control laws in (H^1(0,T;U))* , which induces non-trivial issues. We overcome these difficulties by adapting the functional setting, notably by considering a generalized final state for the systems under consideration. In addition we collect time regularity properties and we pretend that in general it is not possible to consider control laws in H^{-1}(0,T;U). Then, we apply our results to propose an interpretation of the inifinite order of defect for an observability inequality, in terms of controllability properties.