In the theory of linear time-invariant systems, 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 claim 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 infinite order of defect for an observability inequality, in terms of controllability properties.
