Complete Description of the Exponential Stability Index for Linear Parametric Systems as a Function of the Parameter

Abstract

For parametric families of n-dimensional linear differential systems on the time semiaxis with parameter varying in a metric space, we consider two functions of the parameter defined as the dimension of the subspace of solutions that have the characteristic exponent that, respectively, is less than or does not exceed a given real number. A complete description is derived both for the functions themselves and for the vector function composed of them, for the families of systems continuous in one of the two topologies: uniform or compact-open. In addition, the Lebesgue sets and the sets of points of upper and lower semicontinuity are described for the indicated functions.