Lyapunov-like solutions for stability problems of the most general stationary Lurie-Postnikov systems

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Abstract
The most general class of Lurie-Postnikov stationary systems with multiple non-linearities is considered. No special requirement (e.g. controllability, observability or stability) is imposed on the linear part of the systems. For these systems Lyapunov-like conditions are shown to be both necessary and sufficient for absolute stability, or for validity of the Aizerman conjecture. Engineering requirements pose a particular problem of the practical application of the absolute stability concept; that is, the problem of the domain of asymptotic stability of the equilibrium state on the Lurie-Postnikov matrix set. A solution for this problem is established. Examples are aimed to illustrate the approach of the paper and the application of new algebraic criteria for absolute stability and domain estimation. They show applicability of the criteria in cases when the frequency domain criteria cannot provide the problem solution.

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