The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects

Abstract

We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known particular cases of unbounded LMI regions (namely, conic sectors and shifted halfplanes) are considered. A new Dstability criterion as well as sufficient conditions for generalized D-stability are analyzed. Several applications of the developed theory to dynamical systems are shown.

Some bounds for determinants of relatively $D$-stable matrices

“…Corollary 1 Let a matrix A ∈ M n×n be diagonally stable. Then A is relatively D-stable and relatively additive D-stable for any value of θ which satisfies Inequality (11).…”

“…To prove that A is diagonally C 0 θ -stable for any value of θ satisfying (11), we consider the matrix…”

“…We need to show that W(A, D 0 ) is negative definite for any value of θ satisfying (11). Taking into account that Sym(D 0 A) is negative definite for any Lyapunov scaling D 0 ∈ D(A), it will be sufficient to show that…”

Generalized D-stability and diagonal dominance with applications to stability and transient response properties of systems of ODE

“…In [21], the following generalizations of the concept of D-stability to the case of an unbounded LMI region D were considered (separately for the cases when 0 ∈ D and when 0 ∈ C \ D). Definition 5.…”