Strong D-stability
The concepts of strong D-stability and strong block D-stability are introduced. A matrix \(F\in R^{m\times m}\) is said to be D-stable [block D-stable with respect to a multiindex \((m_ 1,...,m_ M)]\) if DF is stable for all \(D=diag\cdot \{d_ 1,...,d_ m\}\) [for all \(D=block\) \(diag\cdot \{d,I_{m_ 1},...,d_ MI_{m_ M}\}]\) with \(d_ i>0\). Strong versions of D- and block D-stability require existence of a \(\mu >0\) such that \(F+G\) is D- or block D-stable for all G with \(| G| <\mu\). The paper first shows that a class of matrices known to be block D-stable are also strongly block D-stable. The concept of strong block D- stability is then applied to multiparameter singularly perturbed systems of the form \(\dot x=Ax+By,\) \(\epsilon_ i\dot y_ i=C_ ix+D_ iy,\) \(i=1,2,...,M\), to obtain a stability condition, which, unlike previous results, does not require that \(\epsilon_ i| \epsilon_ j\) be bounded. Finally, the results are stated for strongly block D-hyperbolic matrices and multiparameter singularly perturbed systems involving such matrices.
scientific article; zbMATH DE number 3962219
scientific article