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

Authors : Michael Gil'

Pages : 6-11

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Publication Date : 2018-06-30

Article Type : Research

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.
Keywords : Hilbert space, Differential equation, Stability, Integro-differential equation, Barbashin type equation