In this paper we analyse the optimal infinite-horizon advertising policy of a monopolist firm in a market for durable goods, based on classic models by Vidale–Wolfe (Oper. Res. 1957; 5(3):370–381) and Nerlove–Arrow (Economica 1962; 29 (114):129–142). A set of necessary conditions for optimality generalizing previous results is provided for the resulting non-convex system. In addition, we establish local (and in some cases global) asymptotic convergence of an optimal trajectory towards the unique optimal steady state.