Critical cases in the theory of absolute stability of control systems

Popov well-known criteria of absolute stability of controlled systems firstly were formulated for non-extremal cases of “direct” and “indirect” control. The possibility of the application of the criteria to special cases requires the careful investigation and till now the criterion is used only in the particular cases of “indirect” control.
The paper shows that when in the definition of the absolute stability the inequality $0\leq\varphi(\sigma)/\sigma\leq k$ (or $0<\varphi(\sigma)/\sigma\leq k$) is replaced by the inequality $\varepsilon<\varphi(\sigma)/\sigma\leq k$ ($\varepsilon>0$ is small quantity) Popov criterion is applicable to all possible special cases including the particular cases of “direct” control.