Robustness in dynamical and control systems

We compile some results on robustness of dynamical and control systems. As control theory is preoccupied with stability problems, the robustness put forward in this paper is related to stability. We ask the question whether an asymptotically stable system remains asymptotically stable when perturbations are affecting it. We analyze robustness of control systems by examining vector fields in C^r topology, by studying associated Lyapunov functions, and by studying corresponding input-output maps. In the first case, we conclude that there is an open set of perturbations such that the system that is affected by them stays asymptotically stable. In the second case, we estimate the size of perturbations that do not destabilize the system. In the third and last case, we provide conditions on the gains of the interconnected systems such that the closed loop system has finite gain.