Arch. Rational Mech. Anal. 128 (1994) 105-125. 9 Springer-Verlag 1994 Asymptotic Stabilityfor Perturbed Hamiltonian Systems GIOVANNI LEON1 Communicated by J. SERRIN w Introduction The purpose of this paper is to study the asymptotic stability of the trivial solution of the perturbed Hamiltonian system (1.1) x' = YJC~x(t, x) + R(t, x), where x : J ~/R 2s, with J a half-open interval of the form I-T, oc ), where Y -= - - I N ' and where j~l.Ca : J • IR2N -+ JR, R:J• are given functions, with ~t ~ of class C 1 and R continuous. The most important assumptions on system (1.1) are (1.2) (J/~x(t, x), x) > 0 for x 4= 0, Jug~(t, 0) = 0, (1.3) (~x(t, x), R(t, x)) < O, R(t, O) = O. Here (.,-) denotes the inner product in Euclidian space and = ,..., , x=(x, ..... xe~). Our approach depends on the construction of an appropriate Liapunov function for the system (1.1), based on the general theory of variational identities introduced by Pucc~ & SERRINin [9]. Our setting includes the usual special case treated in the literature (of., for example, [5]) in which (1.4) (~x(t,x),R(t,x))<-cr(t)q~(x), a(t)qS(x)>O forx+O.