well-posedness and stability analysis of hybrid feedback systems using shkalikov's theory

The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [Grabowski P., Well–posedness and stability analysis of hybrid feedback systems, Journal of Mathematical Systems, Estimation and Control 6 (1996), 121–124 (summary), full electronic manuscript – retrieval code 15844, Grabowski P., Spectral approach to well–posedness and stability analysis of hybrid feedback systems, In: Wajs W., Grabowski P. (Eds.), Studies in Automatics, 1996, Kraków, Wydawnictwa AGH, 104–139] two groups of similarity criteria for a given hybrid closed-lop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.