Reduced stability of parameter-dependent matrices

The stability properties of parameter-dependent linear systems , with A(0) = block-diag(0, F), F a stable matrix, and 0 of order n × n, are analyzed near = 0 by a reduction principle which amounts to considering the Schur complement G( ) of A( ). Some sufficient conditions are given for A( ) and G( ) to have the same stability properties (the so-called principle of reduced stability) in terms of the asymptotic expansion of G( ). Explicit necessary and sufficient conditions are given in the case F = −I, n = 2 in terms of the location of the spectrum of G( ), allowing for geometrical interpretation.