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Stability of nonautonomous differential equations in Hilbert spaces. (English) Zbl 1088.34053

This is a very nice and carefully written article about the problem whether asymptotic stability of the zero solution of the linear system \[ \dot v = A(t)v \] persists under small nonlinear perturbations \[ \dot v = A(t)v + f(t,v) . \] In fact, it is known that, even if all Lyapunov exponents of the linear system are negative, still the nonlinear perturbation might be unstable. However, if the linear system is Lyapunov regular, then asymptotic stability persists under small perturbations. The authors introduce Lyapunov regularity in Hilbert spaces and establish the persistence of the asymptotic stability of the zero solution under sufficiently small perturbations of Lyapunov regular nonautonomous differential equations, in the infinite-dimensional setting of Hilbert spaces.

MSC:

34G10 Linear differential equations in abstract spaces
34D20 Stability of solutions to ordinary differential equations
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