  The paper deals with a problem of control of a system characterized by the
fact that the influence of controls on the dynamics of certain functions of
state variables (called observables) is relatively weak and the rates of change
of these observables are much slower than the rates of change of the state
variables themselves. The contributions of the paper are twofold. Firstly, the
averaged system whose solutions approximate the trajectories of the slow
observables is introduced, and it is shown that the optimal value of the
problem of optimal control with time discounting criterion considered on the
solutions of the system with slow observables (this problem is referred to as
perturbed) converges to the optimal value of the corresponding problem of
optimal control of the averaged system. Secondly, a way how an asymptotically
optimal control of the perturbed problem can be constructed on the basis of an
optimal solution of the averaged problem is indicated, sufficient and necessary
optimality conditions for the averaged problem are stated, and a way how a near
optimal solution of the latter can be constructed numerically is outlined (the
construction being illustrated with an example).
