A discrete-time stochastic control problem is considered for a dynamical model with piecewise linear coefficients and not necessarily Gaussian disturbances. The cost criteria and the class of admissible controls include piecewise polynomial costs and piecewise linear controls respectively. It is shown that relevant asymptotic (for vanishing noise) properties of this problem coincide with the corresponding properties of a suitably chosen adaptive control problem with linear dynamics. In particular, it turns out that the optimal values of the two problems tend to coincide and that almost optimal controls for one problem are almost optimal also for the other
