Absolute stability of multidimensional regulated systems. Aizerman problem

Abstract

We consider one class of ordinary differential equations describing the dynamics of multidimensional controlled systems with a single equilibrium state with nonlinearities from a given set. Such uncertainty of the nonlinear function generates a non-uniqueness of the solution, which leads to the study of the properties of solutions of equations with differential inclusions. A new method for studying the absolute stability of the equilibrium state of controlled systems with many
continuous nonlinearities with incomplete information about them is proposed. By non-singular transformation, the original system is reduced to a special form, which allows using information about the properties of nonlinearities. We study properties of the solutions, obtain estimates for the solutions of the original system and the transformed system, and prove their boundedness. The identities with respect to the components of the nonlinear function are obtained and their
connection with the phase variables is established. Estimates of improper integrals along the solution of the system are obtained and they are used to obtain conditions for absolute stability.
The class of multidimensional nonlinear controlled systems for which the problem of Aizerman has a solution is highlighted. For this class of regulated systems, necessary and sufficient conditions for absolute stability are obtained.