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2 Krumov A. V.

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Model of time-varying linear systems and Kolmogorov equations

Krumov A. V.

Archives of Control Sciences

2015

Vol. 25, no. 2

201--214

In the paper an approximate model of time-varying linear systems using a sequence of timeinvariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with timevarying coefficients, namely analytical solution of the Kolmogorov equations is shown.

Lagrange and practical stability criteria for dynamical systems with nonlinear perturbations

Krumov A. V.

Archives of Control Sciences

2012

Vol. 22, no. 1

43-58

In the paper two classes of nonlinear dynamical system with perturbations are considered. The sufficient conditions for robust Lagrange and practical stability are proven with theorems, applying the theory of nonlinear operators of the functional analysis. The presented criteria give also the bounds of the analyzed dynamical processes. Three examples comparing the numerical computer solutions and the analytical investigation of the stability of the systems are given. The method can be applied to analytical and computer modeling of nonlinear dynamical systems, synthesis of computer control and optimization.