Adam Czornik

Lyapunov exponents for systems with unbounded coefficients

In this article we propose a generalization of the concept of Lyapunov exponents for discrete linear systems which may be used in the case of unbounded coefficients. We show some simplest properties of this generalization and apply it to define a generalization of regular system. Finally, we discuss the problem of stability by linear approximation. † This article was originally published with an error. This version has been corrected. Please see Corrigendum (http://dx.doi.org/10.1080/14689367.2012.756700)

Controllability and minimum energy control of linear fractional discrete-time infinite-dimensional systems

The main purpose of the paper is to discuss controllability problem for infinite-dimensional fractional linear discrete-time control systems. Moreover, the minimum energy control problem of infinite-dimensional fractional-discrete time linear systems is also addressed. Using theory of linear operators necessary and sufficient conditions for the exact controllability of the system are established. Sufficient conditions for the solvability of the minimum energy control of the infinite-dimensional fractional discrete-time systems are given. A procedure for computation of the optimal sequence of inputs minimizing the quadratic performance index is proposed.

On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations

In this note we describe the maximal and the minimal values of Lyapunov exponents for second-order discrete time-invariant linear system perturbed by time-varying bounded perturbations. An interpretation of the results in terms of generalized spectral radius is given. An application of obtained formulas to the robust stability problem is demonstrated on a numerical example.

Technical communique: Set of possible values of maximal Lyapunov exponents of discrete time-varying linear system

In this paper we consider discrete time varying linear systems with coefficients in fixed set of invertible matrices and we describe the set of all possible maximal Lyapunov exponents for the system. We show that the set includes an interval bounded by the generalized spectral subradius and the generalized spectral radius.

On direct controllability of discrete time jump linear system

In this paper we consider a problem of controllability of discrete time linear system with randomly jumping parameters which can be described by finite state Markov chain. Necessary and sufficient conditions for existence of control which governs the system from any initial conditions to zero, from zero initial condition to any final value and from any initial condition to any final value in given time and with probability one are given. The relations between such kinds of controllability and stochastic stabilizability are discussed.

On controllability with respect to the expectation of discrete time jump linear systems

Abstract In this paper we consider a problem of controllability of discrete time linear systems endowed with randomly jumping parameters which can be described by a finite state Markov chain. Necessary and sufficient conditions for existence of a control which governs the expectation of the state of the system from any initial condition to a given target value at a given time are presented. Comparison with other definitions of controllability for such systems is also done.

Controllability and stability of switched systems

The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aim to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability.

Technical communique: On new estimates for Lyapunov exponents of discrete time varying linear systems

In this paper, we propose certain new bounds for the Lyapunov exponents of discrete time varying linear systems. The bounds are expressed in terms of spectral radii of matrix coefficients and therefore may be used to establish the exponential stability of time varying system on the basis of eigenvalues of individual coefficient. This approach is known in the literature as frozen time method.

Alternative formulae for lower general exponent of discrete linear time-varying systems

Abstract The Bohl exponents, similarly as Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. Properties of the Lyapunov characteristics are well described in the literature. Properties of the Bohl exponents are much less investigated. In this paper we consider the so-called junior lower general exponent of discrete linear time-varying system and present some alternative formulae for it. We also discuss relations between lower Bohl exponents of the perturbed system and junior lower general exponent of the unperturbed system.

On the Set of Perron Exponents of Discrete Linear Systems

Abstract The Lyapunov, Bohl and Perron exponents belong to the most important numerical characteristics of dynamical systems used in control theory. Properties of the first two characteristics are well described in the literature. Properties of the Perron exponents are much less investigated. In this paper we show an example of two-dimensional discrete-time linear system with bounded coefficients for which the set of Perron exponents constitutes an interval.