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The estimation and adaptive control problems with the new class of disturbances are considered. It is assumed that disturbances are bounded in mean square and are arbitrary in other respects. It is shown that least squares estimates may diverge in closed-loop adaptive control systems under the action of disturbances from this class. The recursive estimation algorithm is proposed which converges under the action of mean square bounded disturbances. This algorithm is used to construct adaptive optimal control for nonlinear and non-stationary discrete-time plant.