c Allerton Press, Inc., 2012.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2012, Vol. 56, No. 2, pp. 29–36. c B.G. Grebenshchikov, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 2, pp. 34–42.
Original Russian Text The First-Approximation Stability
of a Nonstationary System with Delay
B. G. Grebenshchikov1
1
Ural Federal University, ul. Mira 19, Ekaterinburg, 620002 Russia
Received February 28, 2011
Abstract—We study the exponential stability of a nonlinear system of differential equations with
constant delay such that the right-hand side of one of its subsystems contains the multiplier et . We
obtain a sufficient condition for the first-approximation stability of this system.
DOI: 10.3103/S1066369X12020041
Keywords and phrases: instability, asymptotic stability, exponential estimation, first approximation.
We consider the following nonlinear system with constant delay:
dx(t)/dt = A1 x(t) + A2 x(t − τ ) + B1 y(t) + B2 y(t − τ )
+ f 1 (t, x(t), x(t − τ )) + f 2 (t, y(t), y(t − τ )),
dy(t)/dt = et [A3 x(t) + A4 x(t − τ )x(t − τ ) + B3 y(t) + B4 y(t − τ )
(1)
+ f 3 (et , x(t), x(t − τ )) + f 4 (et , y(t), y(t − τ ))], t ≥ t0 , τ = const, τ > 0.
Here Ak , Bk (k = 1, 2, 3, 4) are positive m × m matrices, x(t) and y(t) are m-dimensional vector
functions with respect to time (the argument) t, f j (t, x(t), x(t − τ )) and f l (t, y(t), y(t − τ )) (j = 1, 3;
l = 2, 4) are nonlinear vector functions such that f j (t, 0, 0) = 0 and f l (et , 0, 0) = 0, and the following
bounds are valid in a neighborhood of the origin of coordinates:
t
j
δ1 (s)ds ≤ ε1 ,
f (t, x(t), x(t − τ )) ≤ δ1 (t)[x(t) + x(t − τ )],
t−1
(2)
t
l t
t
δ2 (s)ds ≤ ε2 ,
f (e , y(t), y(t − τ )) ≤ δ2 (e )[y(t) + y(t − τ )],
t−1
where ε1 and ε2 are sufficiently small positive values. We define the norm of a vector w = {wj }()
m
(here wj are components of the vector w) by the equality w =
|wj |. We define the norm of a matrix
j=1
D = {dij } (i, j = 1, . . . , m) correspondingly to the norm of a vector ([1], P. 12), namely,
|dij |.
D = max
j
i
Consider the following linear homogeneous “unperturbed” system (the first approximation system):
dx0 (t)/dt = A1 x0 (t) + A2 x0 (t − τ ) + B1 y 0 (t) + B2 y 0 (t − τ ),
dy 0 (t)/dt = et [A3 x0 (t) + A4 x0 (t − τ ) + B3 y 0 (t) + B4 y 0 (t − τ )], t ≥ t0 > 0.
(3)
We assume that roots λ of the characteristic equation
|A1 + A2 e−λτ − λE| = 0
(4)
Re λ < −β1 , β1 = const, β1 > 0.
(5)
have negative real parts, i.e.,
29