  Many problems in systems and control theory can be formulated in terms of
robust D-stability analysis, which aims at verifying if all the eigenvalues of
an uncertain matrix lie in a given region D of the complex plane. Robust
D-stability analysis is an NP-hard problem and many polynomial-time algorithms
providing either sufficient or necessary conditions for an uncertain matrix to
be robustly D-stable have been developed in the past decades. Despite the vast
literature on the subject, most of the contributions consider specific families
of uncertain matrices, mainly with interval or polytopic uncertainty. In this
work, we present a novel approach providing sufficient conditions to verify if
a family of matrices, whose entries depend polynomially on some uncertain
parameters, is robustly D-stable. The only assumption on the stability region D
is that its complement is a semialgebraic set described by polynomial
constraints, which comprises the main important cases in stability analysis.
Furthermore, the D-stability analysis problem is formulated in a probabilistic
framework. In this context, the uncertain parameters characterizing the
considered family of matrices are described by a set of non a priori specified
probability measures. Only the support and some of the moments (e.g., expected
values) are assumed to be known and, among all possible probability measures,
we seek the one which provides the minimum probability of D-stability. The
robust and the probabilistic D-stability analysis problems are formulated in a
unified framework, and relaxations based on the theory of moments are used to
solve the D-stability analysis problem through convex optimization. Application
to robustness and probabilistic analysis of dynamical systems is discussed.
