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International Conference: 2004 - Dynamical Systems and Applications
July 5-10, 2004
Antalya, Turkey |
|
Organizers
Akca Haydar, Basem S. Attili, Boucherif Abdelkader, Cho Yeol Je, Covachev Valery, Gyori Istvan, Maksimov Vyacheslav, Stavroulakis Ioannis P.
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The majorizing evaluation of convergence of solutions of the linear stationary system with delay
by
Denys Khusainov
Professor Faculty of Cybernetics Kyiv National University, 01033, UKRAINE, Kyiv, Vladimyrskaya Str., 64
The majorizing evaluation of convergence of solutions of the
linear stationary system with aftereffect.
Denys Khusainov
(Kyiv national university, Kyiv, Ukraine)
The linear stationary aftereffect systems with lag
|
|
d
dt
|
\x(t)=Ax(t)+Bx(t-\tau) |
| (1) |
and of neutral type
|
|
d
dt
|
[x(t)- Dx(t-\tau)]=Ax(t)+Bx(t-\tau) |
| (2) |
are being considered.
The simple statements about stability (asymptotic stabili-ty) of
systems (1), (2) are frequently insufficient to the
solving of the real-world problems. Transient processes
characteristics may greatly differ even for asym-ptotically stable
systems with the equal exponent factors. For some asymptotically
stable systems the ratio of maximal deviation to the original
state can be so high that the solutions convergence for real-world
problems is not important anymore. Therefore it is considered
important to obtain N > 0, gamma > 0 so that the relation
|
|x(t)| <= N||x(0)||\tau exp{ - |
1
2
|
\gammat}, t >= 0 |
| (3) |
holds
true. The relation (3) is obtained by using 2nd Liapunov's
method in two modifications.
1. Method of Liapunov's finite-dimensional quadratic functions V(x, t)=e\gammatxTHx is applied with Ruzumahin's additional condition V(x(s), s) < V(x(t), t), s < l. Symmetric positive defined matrix H is chosen as the solution of Liapunov's matrix equation
2. Method of Liapunov-Krasovski's quadratic functionals in the form
|
V[x(t)]=x(t)THx(t)+ |
ó
õ
|
t
t-\tau
|
e-\betaxT(s)Gx(s)ds |
|
is applied to systems with lag (1) and in the form
|
V[x(t)]=[x(t)-Dx(t-\tau)]TH[x(t)-Dx(t-\tau)]+ |
ó
õ
|
t
t-\tau
|
e-\betasxT(s)Gx(s)ds |
|
to systems of neutral type (2). Positively defined matrices
H and G are chosen so that matrices (for systems (1),
(2) respectively)
Date received: December 9, 2003
Copyright © 2003 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # calu-31.