Estimates by Lozinsky’s functional improved in the linear autonomous case

Abstract

Using a real-valued functional (denoted by $\mu$) introduced by Lozinsky and defined on matrices, lower and upper bounds can be given for the norm of solutions of linear differential equations. This functional also depends on the norm applied. It is pointed out that the best possible bounds can be obtained applying an appropriate linear transformation of the differential equation. In the real and autonomous case this appropriate linear transformation is real and does not depend on the time if $\mu$ is induced by the Euclidean norm. Moreover, close correspondence between $\mu$ and quadratic Liapunov functions is shown. It is proved that in the general case (not Euclidean norms) the best possible bounds can, generally, not be obtained if the linear transformation does not depend on the time.