Comparison of direct and perturbation approaches to analysis of infinite–dimensional feedback control systems

For general boundary control systems in factor form some necessary and sufficient conditions for generation of an analytic exponentially stable semigroup are proposed in both direct and perturbation forms for comparison. The direct approach is applicable to operators with the numerical range satisfying certain additional conditions. In particular, it applies to operators similar to convexoids and therefore it generalizes previous results known for hyponormal operators. The perturbation result (indirect approach) is derived and formulated as an exponential stability robustness result using the frequence-domain considerations. It is expressed in terms of some estimates of the resolvent growth over the open right complex half-plane and compared with some recent results. The analysis is illustrated in detail with examples of an unloaded and loaded electric RC-transmission line with proportional negative feedback.