The present work is devoted to the study of the asymptotic equilibrium state for different classes of linear and nonlinear systems. The second method of Lyapunov is used as a basic research method. On the basis of Zubov Theorem sufficient conditions for the existence of an asymptotic equilibrium state are obtained.

Analysis of asymptotic equilibrium state of differential systems using Lyapunov function method

In the paper the analysis of asymptotic behavior of the solutions of differential systems is considered. Using integral inequalities sufficient conditions for the existence of an asymptotic equilibrium state are presented for various classes of differential systems.

On the subject of asymptotic behavior of differential systems

Controllable systems are considered. Sufficient conditions for global null-controllability are obtained for a given set of admissible controls. The Lyapunov functions are used for analyzing the presence of an asymptotic stationary point in the system.

On the question of zero-controllability of nonstationary bilinear systems

This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov’s results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.

/pdf/lyapunov-s-first-method-estimates-of-characteristic-numbers-2244fjjfkz.pdf

Lyapunov’s first method: estimates of characteristic numbers of functional matrices

Analysis of the global null-controllability of linear-in-control systems

Analysis of the attainability set and construction of its estimates greatly facilitates the solution of a variety of problems in mathematical control theory. In the paper, the problem of boundedness of the integral funnel of nonlinear controlled system is considered. Some estimates of the attainability sets for a nonlinear controlled system are presented. Theorems on the boundedness of considered integral funnels are proved.

On the estimation of the attainability set of nonlinear control systems

Bilinear controlled systems are considered. For some sets of admissible controls sufficient conditions for the boundedness of the integral funnel are obtained. Estimations of bounded attainability set are constructed using the method of Lyapunov functions.

Yu. E. Balykina

Papers

On the estimation of the attainability set of nonlinear control systems

On the subject of asymptotic behavior of differential systems

Analysis of attainability sets of bilinear control systems

Analysis of the global null-controllability of linear-in-control systems