[Studies in Computational Intelligence 481] Artur Babiarz, Robert Bieda, Karol Jędrasiak, Aleksander Nawrat (auth.), Aleksander Nawrat, Zygmunt Kuś (eds.) - Vision Based Systemsfor UAV Applications (2013, Sprin

312 A. Czornik, A. Nawrat, and M. Niezabitowski
x(n,x 0 )=A (n,0)x 0 .
The main result of this paper is to present an upper, and in some cases, a lower
bound for norm of solution of system (1) by elements of matrices A(n). Estimation
problem of the solutions of continuous linear systems in terms of the coefficients
is discussed in literature (for example see [1], Chapter IV, §6). The author shows
on examples that is impossible to judge objectively, which of the presented there
estimates is in general better. This problem is also discussed in [3], [15] and [16].
Estimation problem for the solution of dynamical systems is closely related to
the concept of Lyapunov exponents. Now we present the definitions.
Let a =(a(n)) n∈N be a sequence of real numbers. The numbers (or the symbol
±∞)definedas
1
λ (a)=limsup
n→∞ n ln|a(n)|
is called the characteristic exponent of sequence (a(n)) n∈N .
For x 0 ∈ R s ,x 0 ≠ 0 the Lyapunov exponent λ A (x 0 ) of (1) is defined as characteristic
exponent of (‖x(n,x 0 )‖) n∈N , therefore
1
λ A (x 0 )=limsup
n→∞ n ln‖x(n,x 0)‖.
It is well known [4] that the set of all Lyapunov exponents of system (1) contains
at most s elements, say −∞ < λ 1 (A) < λ 2 (A) < ... < λ r (A) < ∞, r ≤ s and the set
{λ 1 (A),λ 2 (A),...,λ r (A)} is called the spectrum of (1).
For exhaustive presentation of theory of Lyapunov exponents we recommend the
following literature positions: [2], [4]. Problem of estimation of Lyapunov exponents
is discussed in [5]-[9], [12]-[14] and [18].
In the paper we will also use the following conclusion from publication [10] and
[11].
Theorem 1. Suppose, that for system (1) we have
lim A(n)=A. (2)
n→∞
Then, Lyapunov exponents of system (1) coincides with logarithms of moduli’s
eigenvalues of matrix A.
2 MainResults
Let introduce the following notations:
{
aij (p) for i ≠ j
a ij (p)=
a ii (p) − 1fori = j , (3)