Our objective in this paper is to extend as much as possible the dissipativity approach for the study of robustness of stability in the presence of known/unknown but ignored input dynamics. This leads us to: give a new characterization of control Lyapunov functions (CLF) where LfV is upper-bounded by a function of LgV, define the dissipativity approach as assuming the ignored dynamics are dissipative with storage function W and (known) supply rate w, analyzing closed-loop stability with the sum of a CLF for the nominal part and the storage function W. Stability margin are given in terms of an inequality the supply should satisfy. Nevertheless, in spite of this extension, we show that the dissipativity approach cannot cope with ignored dynamics which have nonzero relative degree or are nonminimum phase.