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Vol 53, No 13 (2017)
- Year: 2017
- Articles: 5
- URL: https://journals.rcsi.science/0012-2661/issue/view/9339
Control Theory
Global Problems for Differential Inclusions. Kalman and Vyshnegradskii Problems and Chua Circuits
1671-1702
Existence and Dimension Properties of a Global B-Pullback Attractor for a Cocycle Generated by a Discrete Control System
Abstract
We consider cocycles on finite-dimensional manifolds generated by discrete-time control systems. Frequency conditions for the existence of a global B-pullback attractor for such cocycles considered over a general base system on a metric space are given. Upper bounds for the Hausdorff dimension of the global B-pullback attractor of a discrete cocycle are obtained using the transfer function of the linear part of the cocycle and the discrete Kalman–Yakubovich–Popov frequency theorem.
1703-1714
Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold
Abstract
We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.
1715-1733
Boundary Value Problems with Free Surfaces in the Theory of Phase Transitions
Abstract
The aim of the paper is to show, using the one-dimensional problem as an example, what is to be expected and what should be pursued when studying the multidimensional case. The one-dimensional case has been chosen as a model, because here the problem admits an explicit solution permitting one to follow the phase transformation process.
1734-1763
Differential Equations with Hysteresis Operators. Existence of Solutions, Stability, and Oscillations
1764-1816