The method of recursive aim inequalities has been used in the theory of adaptive control since the late 1960s by a group of Leningrad mathematicians. The problem is reduced to an infinite system of recursive inequalities on the vectors of unknown parameters. These inequalities can be solved by using specially developed, finitely convergent algorithms. The method is illustrated by the solution of adaptive control problems for linear discrete systems under different assumptions on the plant and disturbances. KEY WORDS XI+ 1 = Wxb, ub, t , 0, L7f = S(x& ud-', t , 6)

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We consider the problem of asymptotic analysis of the zeros of a sampled system of a linear time-invariant continuous system as the sampling period decreases. We show that for a continuous prototype system with delay the limits of a part of the model's zeros are roots of certain polynomials whose coefficients are determined by the relative order of the prototype system and delay. In the special case of zero delay these polynomials coincide with Euler polynomials. Zeros of these generalized Euler polynomials are localized: we show that they are all simple and negative, and that they move monotonically between the zeros of classical Euler polynomials as the fractional part of the delay divided by the sampling period grows. Our results lead to sufficient and "almost necessary" conditions for the stable invertible of the discrete model for all sufficiently small values of the sampling period.