Some modifications of the Weiss-Staffans perturbation theorem
P. Grabowski · International Journal of Robust and Nonlinear Control · 2016
For boundary control systems in factor form, a version of the Weiss perturbation result is derived and formulated as an exponential stability robustness result (Theorem 3.1). An example of a heating rod control, illustrating its application, is given. Next, a generalization of the Weiss perturbation to a class of retarded systems of the neutral type is presented (Theorem 4.1). The characteristic feature of this generalization is that it allows to deal with a dynamic perturbation rather than a static one. Using this result, we get a new derivation of an exponential stability criterion. We also show that some parabolic systems without the admissibility of control operator still admit a weakened version of the Weiss perturbation result (Theorem 5.1). This result is a consequence of the necessary and sufficient conditions for generation of an analytic exponentially stable semigroup and some frequency-domain estimates and is illustrated in details by an example of an unloaded electric -transmission line. Copyright © 2016 John Wiley & Sons, Ltd.