Stability conditions for non-autonomous linear differential equations in a Hilbert space via commutators

Abstract

In a Hilbert space $\mathcal{H}$ we consider the equation $dx(t)/dt=(A+B(t))x(t)$ $(t\ge 0),$ where $A$ is a constant bounded operator, and $B(t)$ is a piece-wise continuous function defined on $[0,\8)$ whose values are bounded operators in $\mathcal{H}$. Conditions for the exponential stability are derived in terms of the commutator $AB(t)-B(t)A$. Applications to integro-differential equations are also discussed. Our results are new even in the finite dimensional case.