On a continuity of characteristic exponents of linear discrete time-varying systems

Abstract

<jats:title>On a continuity of characteristic exponents of linear discrete time-varying systems</jats:title><jats:p>In this paper we present a sufficient condition for continuity of Lyapunov exponents of discrete time-varying linear system. Basing on this result we show that Lyapunov exponents of time-invariant systems depend continuously on the time-varying perturbations.</jats:p>

Stability and controllability of switched systems

“…There are several approaches to defining the exponential growth rate, all of which turn out to be equivalent. A trajectory based definition considers Lyapunov exponents [36][37][38][39][40][41][42][43][44][45][46][47][48] of individual trajectories, which are defined in the discrete-time case as…”

“…The quantity ln ρ(d) is known in the control theory literature as the greatest Lyapunov exponent. Properties of Lyapunov exponents were studied in [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], where sufficient conditions so that Lyapunov exponents characterizes exponential stability for switched systems are shown. In particular, it was shown that the system (1) is absolutely stable if and only if sup…”

On the number of upper Bohl exponents for diagonal discrete time-varying linear system

“…Perron exponents, which are closely related to them, appeared for the first time in [61] and [62]. Both of them were introduced in the context of linear time-varying differential equations and since then, they are an object of intensive research both for continuous-time and discrete-time, have many generalizations and variants (see [1][2][3][4][5][6][9][10][11][12][16][17][18]20,21,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][40][41][42][43][44][45]49,52,53,69]). Moreover, the problem of sensitivity to parameter variation has been widely investigated (see for example [31] and the references therein).…”

Properties of the Lower Bohl Exponents of Diagonal Discrete Linear Time-Varying Systems

“…The exponential [37], sigma [38] and central [8,9,39,40,41] exponents describe possible changes of Lyapunov exponents caused by different types parametrical disturbances. It's well-known that Lyapunov exponents are not continuous functions of system coefficients [42,43]. In the effect, it is possible that a finite shift of the characteristic exponents of the original systems corresponds to an arbitrarily small parametrical disturbances.…”