In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping. To do so, we first characterize, in terms of Lyapunov functionals, several types of asymptotic stability for linear infinite-dimensional systems, namely the exponential and the polynomial stability. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term with compensates the nonlinearity. Our results are then applied to the linearized Korteweg-de Vries equation and the 1D wave equation. * 1 Swann Marx is with LAAS-Let H be a real Hilbert space equipped with the scalar product ·, · H . Let A : D(A) ⊂ H → H be a (possibly unbounded) linear operator whose domain D(A) is dense in H. We suppose that A generates a strongly continuous semigroup of contractions denoted by (e tA ) t≥0 . We use A ⋆ to denote the adjoint operator of A.In this section, we consider the linear system given by d dt z = Az, z(0) = z 0 .(1)Since A generates a strongly continuous semigroup of contractions, there exist both