Strongly exponentially separated linear systems (Q2419923)

The paper is concerned with the linear differential systems $\dot x=A(t)x$, $x\in\mathbb{R}^n$, where $A(t)$ is a piecewise continuous matrix function. The authors present a developed theory of exponential separation which is applied to unbounded systems. An additional condition for this investigation is that the angle between two separated subspaces is bounded below, that leads to the notion of strong exponential separation (SES). They give a necessary and sufficient condition for SES, and then they use it to show that the condition of SES is preserved by the operation of taking the adjoint of the system. The robustness of SES under small perturbations of the coefficient matrix is also proved. The authors investigate the block upper triangular systems and give necessary and sufficient conditions for the strongly exponentially separated property of these systems. Finally, they prove that if a bounded linear Hamiltonian system is exponentially separated into two subspaces with the same dimension, then it must have an exponential dichotomy.