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In an infinite horizon optimal control problem the Hamiltonian vanishes at infinity when the differential equation is autonomous and the integrand in the criterion satisfies some weak integrability conditions. A generalization of Michel's result (in Econometrica 50:975-985, 1982) is obtained.Keywords Optimal control · Infinite horizon ∞ 0 e rt g(x, u) dt, r a given real number. In problems with one state variable, this condition, together with the other conditions of the maximum principle, often determine a unique solution candidate. The main purpose of this note is to generalize 1 There is a misprint in the theorem. Please replace the inequality b > k by b > (n − m)k.

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