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

Authors : Michael GİL`

Pages : 6-11

View : 19 | Download : 10

Publication Date : 2018-06-30

Article Type : Research Paper

Abstract :In a Hilbert space $\mathcal{H}$ we consider the equation $dxinsert ignore into journalissuearticles values(t);/dt=insert ignore into journalissuearticles values(A+Binsert ignore into journalissuearticles values(t););xinsert ignore into journalissuearticles values(t);$ $insert ignore into journalissuearticles values(t\ge 0);,$ where $A$ is a constant bounded operator, and $Binsert ignore into journalissuearticles values(t);$ 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 $ABinsert ignore into journalissuearticles values(t);-Binsert ignore into journalissuearticles values(t);A$. 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