For a class of non-linear dynamic systems, a quadratic regulator problem is considered in which the cost functional contains a small parameter multiplied to the control term. In the analysis of the problem, an appropriate transformation of the state variables and an eigenvector technique are utilized, which were devised originally for analysing complex linear cheap control problems and for solving a Lyapunov-type system of linear equations, respectively. In so doing, the class of nonlinear systems is characterized and extended in a mathematically simple way, for which the optimal cost becomes zero as the small parameter goes to zero.