A direct method of Lyapunov for analyzing the stability of functional-differential equations with discontinuous right parts and controlled systems with piecewise-continuous controls with lagging feedback is developed. The problem of stabilization of program movements is investigated, accounting for the delay in the feedback structure in a class of piecewise-continuous controlling effects, discontinuous on a surface of a certain kind. The problem is analyzed on the basis of the method of limiting equations, using constant-sign Lyapunov functional. As an example, a mechanical system with non-stationary, holonomous and ideal connections is analyzed.
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