On the robust stability of implicit linear systems containing a small parameter in the leading term

Abstract

This paper deals with the robust stability of implicit linear systems containing a small parameter in the leading term. Based on possible changes in the algebraic structure of the matrix pencils, a classification of such systems is given. The main attention is paid to the cases when the appearance of the small parameter causes some structure change in the matrix pencil. First, we give sufficient conditions providing the asymptotic stability of the parameterized system. Then, we give a formula for the complex stability radius and characterize its asymptotic behaviour as the parameter tends to zero. The structure-invariant cases are discussed, too. A conclusion concerning the parameter dependence of the robust stability is obtained.