Riccati differential inequalities and dichotomies for linear systems

Abstract A general notion of dichotomy for linear differential systems is investigated. It is well known that a system x ′= A ( t ) x in which the matrix A ( t ) is bounded and diagonally dominant by rows or columns has an invariant splitting of its solution space into two subspaces each uniformly asymptotically stable, one for increasing time and the other for decreasing time. Similar results are obtained here where the concept of diagonal dominance is weakened using Riccati inequalities.