Aizerman's problem in absolute stability theory for regulated systems

Abstract

A new method for investigating the absolute stability of regulated systems with limited resources is proposed. It is based on estimating improper integrals along the solution of the system. A nonsingular transformation is obtained which allows information about the nonlinearity properties to be taken into account. A class of regulated systems is distinguished for which Aizerman's problem is solvable. For this class a necessary and a sufficient condition for absolute stability are found. The proposed method for investigating absolute stability differs from the other available methods by the fact that conditions for absolute stability are derived without using the Lyapunov function and the frequency theorem. For systems with limited resources the phase variables are bounded, uniformly continuous functions. These properties were used in deriving a condition for stability and in estimating improper integrals. The estimate obtained allows the domain of absolute stability in the space of constructive parameters of the system to be greatly extended by comparison with the earlier known results, and in a number of cases a necessary and a sufficient condition for absolute stability can be obtained. Bibliography: 15 titles.