Alexei V. Pokrovskii

Expansivity of semi-hyperbolic Lipschitz mappings

Semi-hyperbolic dynamical systems generated by Lipschitz mappings are shown to be exponentially expansive, locally at least, and explicit rates of expansion are determined. The result is applicable to nonsmooth noninvertible systems such as those with hysteresis effects as well as to classical systems involving hyperbolic diffeomorphisms.

Transients in Quasi-Controllable Systems. Overshooting, Stability and Instability

Abstract Families of regimes for control systems are studied possessing the so called quasi-controllability property that is similar to the Kalman controllability property. A new approach is proposed to estimate the degree of transients overshooting in quasi-controllable systems. This approach is conceptually related with the principle of bowlded regimes absence in the absolute stability problem. Its essence is in obtaining of constructive a priori bOWlds for degree of overshooting in terms of the so called quasi-controllability measure. It is shown that relations between stability, asymptotic stability and instability for quasi-controllable systems are similar to those for systems described by linear differential or difference equations in the case when the leading eigenvalue of the corresponding matrix is simple. The results are applicable for analysis of transients, classical absolute stability problem, stability problem for desynchronized systems and so on.

On the fragmentary complexity of symbolic sequences

A measure of the ability of a symbolic sequence to be covered by initial fragments of another symbolic sequence is introduced and its basic properties are investigated. Applications to the characterization of symbolic sequences associated with shift mappings on a torus corresponding to a special partitioning of the torus and to multirate systems of coprocessors are considered.

Influence of Controllability-Type Properties on Reliability and Stability of Desynchronized Systems

Abstract Systems with high level of failures are considered that permanently operate due to this reason in a transient dynamical regime. In real systems amplitude restrictions for state variables are inevitable and also the so called overshooting effect is inwardly intrinsic. Therefore the operation in transient regime can lead to a great number of secondary failures. At the same time it is known that some classes of the so called desynchronized systems are insensitive to failures in data communication links. In the paper the notion of quasi-controllability for desynchronized systems is introduced that allow easy and efficient to estimate the overshooting measure. This presents the means to design the variety of highly reliable fault-tolerant control and manufacturing systems.

Functional Analysis and Dynamics of Nonlinear Control Systems

Abstract For the last decades the role of functional analysis in the development of investigation techniques for both the classical and new problems in the control theory is over growing. These techniques proved to be especially effective in studying various specific classes of systems with nonstandard nonlinear elements A number of workshops held at the Institute of Control Sciences (Moscow) are active in the field. Some new ideas developed at these workshops are presented below. This paper, of necessity, considers these ideas only in simplest situations. We shall mainly treat nonlocal problems, i.e systems described by nonlinear equations without small parameters.