A stability theorem for a class of linear evolution systems

Abstract

Dynamical systems of the form\((*)\tfrac{\partial }{{\partial t}}u = Bu, u(0) = u_0 \),u(0)=u0, where,B is a densely defined linear operator mapping its domainD (B⊂ e) into e — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC0 — are investigated, such that for each solutionu to\((*)\lim _{t \to + \infty } u(t) = Pu_0 \), whereP is the spectral eigenprojection onto the null space ofB.