Delay-dependent stability for vector nonlinear stochastic systems with multiple state delays

Abstract

Global asymptotic stability conditions for vector nonlinear stochastic systems with multiple state delays are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov-Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.

Full text

Page 1

Delay-Dependent Stability for Vector Nonlinear Stochastic Systems with
Multiple State Delays
Michael Basin Dario Calderon-Alvarez
Abstract—Global asymptotic stability conditions for vector
nonlinear stochastic systems with multiple state delays are
obtained based on the convergence theorem for semimartingale
inequalities, without assuming the Lipschitz conditions for non-
linear drift functions. The Lyapunov-Krasovskii and degenerate
functionals techniques are used. The derived stability conditions
are directly expressed in terms of the system coefficients.
Nontrivial examples of nonlinear systems satisfying the obtained
stability conditions are given.
I. INTRODUCTION
The stability and stabilizabilty problems for time-delay
systems have been extensively studied in recent years due to
direct applicability of the obtained results to various technical
problems ([1], [2], [3]). Initiated in the background works
[4], [5], [6], the stability theory for linear time-delay systems
is now actively being developed. To prove stability results for
a selected class of linear time-delay systems, the Lyapunov-
Krasovskii or Lyapunov-Razumikhin functionals are applied
in the framework of the Lyapunov direct method. Two types
of stability conditions can be obtained: delay-independent,
establishing stability for all possible delay values, or delay-
dependent, corresponding to some restricted values of delay
shifts. While the first type of conditions is comprehensive
but conservative, the second one is more selective, flexible,
and, as a consequence, preferable. Some examples of delay-
dependent stability conditions can be found in ([7], [8], [9],
[10], [11], [12], [13], [14]) for various deterministic linear
time-delay systems and in [15], [16], [17], [18], [19] for
stochastic ones. Note that it is frequently needed to make
a special transformation of an original time-delay system to
obtain such stability conditions. Virtually all known results
involving delay-dependent stability conditions have been
obtained for linear time-delay systems, with certain or even
uncertain coefficients.
This paper concentrates on design of the stability condi-
tions for vector nonlinear stochastic time-delay systems gov-
erned by multidimensional nonlinear Ito differential equa-
tions with multiple state delays and a nontrivial diffusion
term. To obtain the results, a modified Lyapunov-Krasovskii
functional, known as degenerate functional, is employed,
which was introduced and described in details in [4], [5].
Applications of degenerate functionals for various classes of
deterministic functional-differential equations can be found
The authors thank the Mexican National Science and Technology Council
(CONACyT) for financial support under Grants no. 55584 and 52953.
M. Basin and D. Calderon-Alvarez are with Department of Physical
and Mathematical Sciences, Autonomous University of Nuevo Leon, San
Nicolas de los Garza, Nuevo Leon, Mexico mbasin@fcfm.uanl.mx
dcalal@hotmail.com
in [4], [5], [6]. In [20], the degenerate functionals are used
for obtaining delay-dependent stability conditions for deter-
ministic scalar delay-differential equations. This paper gen-
eralizes the result of [20] to vector stochastic nonlinear time-
delay systems. The convergence theorem for semimartingale
inequalities [21] serves as a key tool for obtaining stability
conditions in terms of stochastic system coefficients, without
any transformation of the original system itself. However,
some conditions in this paper are more restrictive than those
in [20], because of the nature of the solutions of stochastic
Ito equations. Even small initial values cannot guarantee that
solutions to stochastic Ito equation with nontrivial diffusion
will be almost surely (a.s.) bounded on a finite interval.
Therefore, a global linear growth condition for nonlinear drift
functions is used in this paper instead of a local one in [20].
Nonetheless, a significant advance reached in this paper in
comparison to [20] is elimination of the Lipschitz condition
for nonlinear drift functions. Similar delay-dependent sta-
bility conditions for discrete-time systems can be found in
[22]. Finally note that design of a stabilizing controller for a
class of nonlinear stochastic systems, based on the stability
conditions given in this paper, would be a direct application
of the obtained results (see [10] for a similar scheme of
stabilizing controller design for linear systems).
II. BASIC DEFINITIONS AND RESULTS
In this section, some basic definitions and results from
the theory of stochastic processes are briefly reviewed (see
([23], [24] for details). All stochastic variables or processes
are allowed to be multi-dimensional (stochastic vector or
vector processes), for which equalities and inequalities are
regarded component-wisely. The following notation is used:
xTmeans the transpose of a vector x, |x| =?(xTx) denotes
Euclidean 2-norm of a matrix σ, i.e., the sum of squares of
all matrix entries.
Let (Ω,F,P) be a complete probability space with a non-
decreasing right-continuous family of σ-algebras (filtrations)
F = {Ft}t≥t0. A stochastic process Mt is said to be an Ft-
martingale, if E|Mt| < ∞ and E?Mt
admits the representation Xt= Xt0+Mt+At, where Mt is a
martingale, Mt0= 0, and At is a process with almost surely
(a.s.) bounded variation, At0= 0, Xt0is a random variable.
The following lemma, originally proved in [21], presents
a modification of the martingale convergence theorems (cf.
[23]) in terms of inequalities, which plays a key role in
establishing the asymptotic stability conditions.
the Euclidean 2-norm of a vector x, and |σ| denotes the
??Fs
?= Ms for all t >
s ≥ t0. A stochastic process is called a semimartingale if it
2008 American Control Conference
Westin Seattle Hotel, Seattle, Washington, USA
June 11-13, 2008
ThAI01.4
978-1-4244-2079-7/08/$25.00 ©2008 AACC.1899

Page 2

Lemma 1. Let A1, A2, B1, B2
decreasing processes with B1≤A1, B2≥A2and A=B1−B2.
Let also Z =Zt0+A+M be a non-negative semimartingale.
Then, a.s.
be almost sure (a.s.) non-
?ω : A1
∞< ∞?⊆ {Z →}∩?ω : A2
where {Z →} denotes the set of all ω ∈ Ω for which Z∞=
lim
t→∞Ztexists and is finite.
Remark 1. In this paper, the notation x(t) is used for
a value of any stochastic process (other than a Wiener
process wt) at a moment t, although the notation xt is
commonly accepted for this purpose in the literature on
stochastic processes. This enables one to distinguish between
the pointwise value x(t) of a stochastic process and its history
up to the moment t, which is denoted by xt, xt(s) = x(s),
s ≤ t, as commonly accepted in the literature on time delay
systems.
The space C[a,b]of continuous vector functions u(s) ∈ Rn
in the interval [a,b] is defined as
∞< ∞?,
(1)
C[a,b]= {u : [a,b] → Rn| u(s) is continuous for any s ∈ [a,b],
with the norm ?u?C[a,b]= sup
1≤k≤n
sup
a≤s≤b|uk(s)| < ∞}.
III. STABILITY CONDITIONS
In this section, the asymptotic stability problem is consid-
ered for a state x(t)∈Rnsatisfying a system of Ito stochastic
nonlinear differential equations with several nonlinear func-
tions Ni(u) ∈ Rn, i = 1,...,m, and multiple discrete delays
hi> 0
m
∑
i=1
σ(t,xt)dwt, t ≥t0,
where φ(t) ∈ Rnis an initial continuous function defined
on [t0−h,t0], h = max(h1,...,hm), and wt∈ Rpis a vec-
tor Wiener process with independent components, i.e., a
stochastic process whose components are scalar independent
standard Wiener processes. The diffusion matrix σ(t,xt) is
of dimension n× p, and coefficients aiand bi, i = 1,...,m,
are scalar constants.
The equation (2) is equivalent to the following equation
dx(t) = −
aiNi(x(t))dt −
m
∑
i=1
biNi(x(t −hi))dt +
x(t) = φ(t), t ∈ [t0−h,t0], (2)
d[x(t)−
m
∑
i=1
bi
?t
t−hi
Ni(x(θ))dθ]
= −
x(t) = φ(t),
m
∑
i=1
(ai+bi)Ni(x(t))dt +σ(t,xt)dwt,
t ≥t0,
t ∈ [t0−h,t0].
(3)
Sometimes, the equation (3) is called a neutral type equation,
since it contains the unknown function evaluated in some
points θ ≤t under the sign of differential.
The functions Ni(u), i = 1,...,m, in the equations (2),(3)
are assumed to satisfy the sector-like condition
uTNi(u) > 0,
for any
u ∈ Rn, u ?= 0, Ni(0) = 0.
(4)
The first part of this condition, which should hold for any
real nonzero value u = x(t) or u = x(t −hi), i = 1,...,m,
rules out some unstable (for a nonzero initial function)
systems, such as dx(t) = x(t)dt, dx(t) = x(t − h)xT(t −
h)x(t −h)dt, or not asymptotically stable systems, such as
dx(t) = sin(x(t)xT(t)x(t))dt. Nonetheless, a modified sin
function would satisfy the condition (4), as shown further
in Example 5.
Next, assume that there exist such constants Ki,γ(i)
0,i = 1,...,m, and a function γ : [t0,∞) → R+that the
following conditions are satisfied
1,γ(i)
2>
|Ni(u)|2≤ KiuTNi(u),
tr(σ(t,xt)σT(t,xt)) ≤
for any
m
∑
k=1
u ∈ Rn;(5)
γ(k)
1|Nk(x(t))|2
+
m
∑
k=1
γ(k)
2|Nk(x(t −hk))|2+γ(t)
for any xt∈C[t0−h,t];
ai+bi> 0;
m
∑
i=1
|bk|hk
2(ak+bk)(
|bi|hi+γ(k)
?∞
Discussing the conditions (5)-(10), note that the deter-
ministic system obtained by setting σ(t,xt) = 0 would be
asymptotically stable in view of Theorem 1. The conditions
(6) and (10) imply that the diffusion term σ(t,xt) converges
to zero as time tends to infinity, i.e., the system (2) becomes
deterministic at the infinity, although remains stochastic for
any fixed large t. The condition (7) indicates that the ”total”
summarized coefficient of the current and delayed values for
each function Nishould have a strictly negative value, which
assures asymptotic stability of the corresponding system
mode. Note that ai and bi cannot be simultaneously equal
to zeros, since the corresponding mode could be stable but
not asymptotically stable, as for the system dx(t) = 0, or
even unstable, as for the system dx(t) = K(t)dw(t), where
K(t)=1 for t0≤t ≤T, T is a sufficiently large time moment,
and K(t) = 0 for t > T.
Remark 2. If Ki≡ K, γ(i)
it is not difficult to show that the condition (9) implies (8).
Indeed, the condition (9) gives
(6)
(7)
α2=|bi|hiKi< 1;(8)
βk= (
m
∑
i=1
(ai+bi))
+1
2
m
∑
i=1
1+γ(k)
2(ak+bk))Kk< 1;
2
(9)
t0
γ(s)ds < ∞.
(10)
1= γ(i)
2= 0 for all i = 1,...,m,
|bk|hkK
2(ak+bk)
?
m
∑
i=1
(ai+bi)
?
+1
2
m
∑
i=1
|bi|hiK < 1,
|bk|hkK
?
m
∑
i=1
(ai+bi)
?
+
?
m
∑
i=1
|bi|hiK
?
(ak+bk)
< 2(ak+bk), and
m
∑
k=1
|bk|hkK
?
m
∑
I=1
(ai+bi)
?
+
?
m
∑
i=1
|bi|hiK
?
m
∑
k=1
(ak+bk)
1900

Page 3

< 2
m
∑
k=1
(ak+bk).
Dividing the latter by ∑m
i=1(ai+bi) yields
m
∑
i=1
m
∑
k=1
|bk|hkK+|bi|hiK < 2.
For all t ≥t0and xt∈C[t0−h,t], x(s) = φ(s), s ∈ [t0−h,t0],
the degenerate functional is defined
¯Y(t,xt) = x(t)−
m
∑
i=1
bi
?t
t−hi
Ni(x(θ))dθ.
(11)
The functional Y(t,xt) is not negative but also is not positive
definite. However, for every t ≥t0, the norm ?Y(•,x•)?C[t0,t]
can be bounded from the below by the norm ?x?C[t0−h,t], as
shown in the following Lemma 2.
The next lemma and following theorem establish asymp-
totic stability conditions for solutions of the equation (3).
Lemma 2. Let conditions (5), (9) be satisfied and α2=
∑m
i=1|bi|hiKi< 1. Then, for t ≥ t0 and xt∈ C[t0−h,t], x(s) =
φ(s), s ∈ [t0−h,t0], the state trajectory norm satisfies the
inequality
?x?C[t0−h,t]≤
Proof.
1
1−α2
?
?¯Y(•,x•)?C[t0,t]+?φ?C[t0−h,t0]
?
.
(12)
?x(s)?C[t0,t]= ?x(s)−
m
∑
i=1
bi
?t
t−hi
Ni(x(θ))dθ
+
m
∑
i=1
bi
?t
t−hi
Ni(x(θ))dθ?C[t0,t]
≤ ?¯Y(s,xs)?C[t0,t]+
m
∑
i=1
m
∑
i=1
|bi|?
?t
t−hi
N(x(θ))dθ?C[t0,t]
≤ ?¯Y(•,x•)?C[t0,t]+
|bi|hiKi?x?C[t0−hi,t].
Then,
?x?C[t0−h,t]≤ ?x?C[t0,t]+?φ?C[t0−h,t0]
≤ ?¯Y(•,x•)?C[t0,t]+α2?x?C[t0−h,t]+?φ?C[t0−h,t0],
which implies (12).
Theorem 1. Let conditions (4)-(10) be satisfied. Then,
lim
t→+∞x(t) = 0 holds a.s. for all solutions x of the equation
(2).
Proof. Define functional V =V(x(t),t) by the formula
V = |¯Y|2+V1+V2,
?
∑
i=1
m
∑
k=1
where¯Y is given by (11).
Applying Ito formula to theV(x(t),t) along the trajectories
of solutions of the equation (3) yields
V1=
m
(ai+bi)
?
m
∑
k=1
|bk|
?t
t−hk
ds
?t
s
|Nk(x(θ))|2dθ,
V2=
γ(k)
2
?t
t−hk
|Nk(x(θ))|2dθ,
V(x(t),t) =V(0,t0)+
?t
t0
¯Fsds+¯ Mt,
with V(0,t0) = 0,
¯Ft= −2[xT(t)−
m
∑
i=1
bi
?t
t−hi
NT
i(x(θ))dθ]
×
m
∑
k=1
m
∑
i=1
?t
m
∑
k=1
(ak+bk)Nk(x(t))+tr(σ(t,xt)σT(t,xt))
+(ai+bi)(
m
∑
k=1
|bk|[hk|Nk(x(t))|2
−
t−hk
|Nk(x(θ))|2dθ])
+
γ(k)
2[|Nk(x(t))|2−|Nk(x(t −hk))|2]
(13)
and
d¯ M(t) = 2
?
xT(t)−
m
∑
i=1
bi
?t
t−hi
NT
i(x(θ))dθ
?
σ(t,xt)dwt.
Applying the Minkowski’s inequality yields
2(
m
∑
i=1
m
∑
i=1
?t
|bi|
?t
t−hi
NT
i(x(θ))dθ)(
m
∑
k=1
(ak+bk)Nk(x(t)))
≤
m
∑
k=1
(ak+bk)|bi|[hi|Nk(x(t))|2
+
t−hi
|Ni(x(θ))|2dθ].
(14)
Using (14) in (13) leads to
Ft≤ −2
m
∑
k=1
(ak+bk)xT(t)Nk(x(t))
+
m
∑
i=1
m
∑
i=1
|bi|hi
m
∑
k=1
(ak+bk)|Nk(x(t))|2
+
m
∑
k=1
(ak+bk)|bi|
?t
t−hi
|Ni(x(θ))|2dθ
+(
m
∑
i=1
m
∑
i=1
m
∑
k=1
(ai+bi))
m
∑
k=1
m
∑
k=1
|bk|hk|Nk(x(t))|2
−
(ai+bi)|bk|
?t
t−hk
m
∑
k=1
|Nk(x(θ))|2dθ
+
γ(k)
1|Nk(x(t))|2+
γ(k)
2|Nk(x(t −hk))|2
+γ(t)+
m
∑
k=1
γ(k)
2[|Nk(x(t))|2−|Nk(x(t −hk))|2]
≤ −2
m
∑
k=1
(ak+bk)xT(t)Nk(x(t))
+(
m
∑
i=1
m
∑
i=1
m
∑
k=1
(ai+bi))
m
∑
k=1
|bk|hk|Nk(x(t))|2
+|bi|hi
m
∑
k=1
(ak+bk)|Nk(x(t))|2
+
(γ(k)
1+γ(k)
2)|Nk(x(t))|2+γ(t)
≤ −2
m
∑
k=1
(ak+bk)[1−(
|bk|hk
2(ak+bk)
m
∑
i=1
(ai+bi)
1901

Page 4

+1
2
m
∑
i=1
|bi|hi+γ(k)
1+γ(k)
2(ak+bk))]KkxT(t)Nk(x(t))+γ(t)
|bk|hk
2(ak+bk)(
|bi|hi+γ(k)
2
≤ −2
m
∑
k=1
(ak+bk)[1−(
m
∑
i=1
(ai+bi))
+1
2
m
∑
i=1
1+γ(k)
2(ak+bk))]KkxT(t)Nk(x(t))+γ(t)
2
≤ −2
m
∑
k=1
(ak+bk)[1−βk]xT(t)Nk(x(t))+γ(t).
(15)
Then,
V(x(t),t) =V(0,t0)+
?t
t0
?¯Fs+2
m
∑
k=1
m
∑
k=1
(ak+bk)
×[1−βk]xT(s)Nk(x(s))?ds−2
×
t0
(ak+bk)[1−βk]
?t
xT(s)Nk(x(s))ds+¯ Mt=V(0,t0)+¯B1
t−¯B2
t+¯ Mt,
where
¯B1
t=
?t
t0
max{0,¯Fs+2
m
∑
k=1
(ak+bk)[1−βk]xT(s)Nk(x(s))}ds,
¯B2
t
=
?t
t0
[2
m
∑
k=1
(ak+bk)[1−βk]xT(s)Nk(x(s))
(16)
+max{0,−¯Fs−2
m
∑
k=1
(ak+bk)[1−βk]xT(s)Nk(x(s))}]ds.
It is easy to see from (15) and (16) that
¯B1
t≤¯A1
t=
?t
t0
γ(s)ds,
¯B2
t≥¯A2
t= 2
m
∑
k=1
(ak+bk)[1−βk]
?t
t0
xT(t)Nk(x(t))dt.
Now, applying Lemma 1 and the inequalities (1) and (10)
implies that P{V →} = 1 and, therefore, P{supt≥t0Vt<
H} = 1 almost surely for a certain random variable H =
H(ω) < ∞. Next, the definition (13) of the functional V
yields P{supt≥t0|Yt|2<H}=1, P{supt≥t0(V1)t<H}=1 and
P{supt≥t0(V2)t< H} = 1. Finally, applying Lemma 2 (see
(12)) and the inequality (5) implies that a.s.
?xt? ≤
1
1−α2[?Y(•,x•)?C[t0,t]+?φ?C[t0−h,t0]]
1−α2[H +?φ?C[t0−h,t0]] = H1,
??Ni(x(t))??2≤ KixT(t)Ni(x(t)) ≤ Ki?x(t)??Ni(x(t))?,
≤
1
therefore, ?Ni(x(t))? ≤ Ki?x(t)? ≤ KiH1.
Thus,
P{sup
t≥t0
From (13), V = |Y|2+V∗, where V∗=V1+V2. Almost sure
convergence of Vt=V(xt,t), as t →∞, implies, in particular,
that Vt is a.s. uniformly continuous on [t0,∞). Let us show
|x|2(t) ≤ H2
1} = 1,
P{sup
t≥t0
?Ni(x(t))? ≤ KiH1} = 1.
now that V∗
Indeed,
t
is also a.s. uniformly continuous on [t0,∞).
(V1)t−(V1)θ= (V1)′
κ(t −θ),
where κ is a point between t and θ, and
??(V1)′
−
κ
??= (
|Nk(x(θ))|2dθ| ≤ (
?κ
m
∑
k=1
m
∑
i=1
m
∑
i=1
(ai+bi))
m
∑
k=1
|bk||hk||Nk(x(κ))|2
?κ
m
∑
k=1
κ−hk
m
∑
i=1
(ai+bi))
×|bk||
κ−hk
(|Nk(x(κ))|2−|Nk(x(θ))|2)dθ|
≤ (
m
∑
i=1
(ai+bi))
|bk||
?κ
κ−hk
K2
k
??x(κ)−x(θ)??2dθ|
|bk|hkK2
≤ 4H2
1((ai+bi))(
m
∑
k=1
k).
Then, a.s.
??(V1)t−(V1)θ
It means that (V1)tis a.s. uniformly continuous on [t0,∞). To
prove that (V2)tis also a.s. uniformly continuous on [t0,∞),
note that
??≤ 4H2
1
?
m
∑
i=1
(ai+bi)
??
m
∑
k=1
|bk|hkK2
k
?
|t −θ|.
|(V2)t−(V2)θ| =
m
∑
k=1
γ(k)
2
×|
?t
m
∑
k=1
?t−hk
t−hk
γ(k)
|Nk(x(θ))|2dθ −
?t
?θ
θ−hk
|Nk(x(τ))|2dτ|
?θ
?θ−hk
θ−hk
?
t−hk
≤
2|
t0
|Nk(x(θ))|2dθ −
t0
|Nk(x(τ))|2dτ
−
t0
|Nk(x(θ))|2dθ +
t0
|Nk(x(τ))|2dτ|
=
m
∑
k=1
γ(k)
2[
θ
?
t
|Nk(x(τ))|2dτ +
|Nk(x(τ))|2dτ]
≤ 2H2
1
m
∑
k=1
γ(k)
2K2
k|t −θ|.
Thus, Y2
on [t0,∞). The a.s. uniform continuity of x(t) on [t0,∞) can
be obtained from the following inequalities
t=Vt−V∗
thas also to be a.s. uniformly continuous
|x(t)−x(s)| =
???x(t)−
Nk(x(θ))dθ ]+
m
∑
k=1
[bk
?t
t−hk
m
∑
k=1
Nk(x(τ))dτ]−x(s)
+
m
∑
k=1
m
∑
k=1
m
∑
k=1
[bk
?s
?s
s−hk
[bk
?t
t−hk
Nk(x(τ))dτ]
−
[bk
s−hk
?t
Nk(x(θ))dθ]
??? ≤ |Y(x(t),t)−Y(x(s),s)|
∑
k=1
s−hk
m
∑
k=1
+|bk||
s
Nk(x(τ))dτ|+
m
|bk||
?t−hk
N(x(τ))dτ|
≤ |Y(x(t),t)−Y(x(s),s)|+2H1
[|bk|Kk]|t −s|.
1902

Page 5

Moreover, xT(t)Nk(x(t)) is also a.s. uniformly continuous on
[t0,∞) for any k = 1,...,m. Indeed, for any t,θ ≥t0, a.s.
|xT(t)Nk(x(t))−xT(θ)Nk(x(θ))| ≤ |Nk(x(t))||x(t)−x(θ)|
+|x(θ)||Nk(x(t))−Nk(x(θ))| ≤ 2H1Kk|x(t)−x(θ)|.
?
0. In view of continuity of a function Ni, i = 1,...,m,
and the condition (4), there exist a.s. such a finite random
variable ζ1(ω) > 0 and a sequence of random moments
tk=tk(ω)→∞, as k →∞, that P(Ω1)= p0>0, where Ω1=
{ω:|xT(tk)Ni(x(tk))|(ω)>ζ1(ω)>0}. Since xT(t)Ni(x(t)) is
a.s. uniformly continuous on [t0,∞), for ε =ε(ω)=ζ1(ω)/2,
there exists δ = δ(ω) such that
Suppose now that P
limsup
t→∞|x(t)| = ζ0(ω) > 0
?
= p0>
|xT(tk)Ni(x(tk))−xT(s)Ni(x(s))| ≤ ε = ζ1(ω)/2,
for ω ∈ Ω1and |s−tk| ≤ δ. Then, for ω ∈ Ω1and s ∈ [tk−
δ,tk+δ], the following inequality is obtained
|xT(s)Ni(x(s))| ≥ |xT(tk)Ni(x(tk))|
−|xT(tk)Ni(x(tk))−xT(s)Ni(x(s))| ≥ ζ1(ω)/2.
Without loss of generality, suppose that tk+1(ω)−tk(ω) >
2δ(ω) for any ω ∈ Ω1. Let k(n) be the number of elements
in the sequence {tk} belonging to the interval [t0,n]. Applying
inequality (17) implies, for ω ∈ Ω1, that
?n
k:t0≤tk+δ≤n
≥ζ1
2
∑
k:t0≤tk+δ≤n
as n → ∞, since k(n) → ∞ as n → ∞. Hence, P{A2
p0> 0, which contradicts (1). Theorem 1 is proved.
The following examples illustrate applicability of the
Theorem 1. The state equation in Example 1 contains two
different nonlinear functions and two delays, as well as
the linear term with a positive coefficient. Examples 2–4
present various nonlinear vector functions N(x), satisfying
the condition (5). This illustrates viability of the condition
(5) in the multi-dimensional case.
Example 1. Consider the nonlinear system (t0= 0)
t0
xT(s)Ni(x(s))ds ≥
∑
?tk+δ
tk−δ
xT(s)Ni(x(s))ds
?tk+δ
tk−δ
ds = δζ1 ∑
k≤k(n)
1 = k(n)δζ1→ ∞,
∞= ∞} ≥
dx(t) = −b1
x5(t −h1)
1+x4(t −h1)dt −b2x(t −h2)dt
+0.3x(t)dt +
1+t
?(1+x2(t))
dwt.
(17)
For a sufficiently large T and all t > T, the conditions of
Theorem 1 can be satisfied. Indeed, the conditions (4)-(5)
and (7) are satisfied for N1(u) =
K2= 1, b1= b2= 0.5, a1= 0, a2= −0.3, h1= h2= 0.4.
The condition (8) is satisfied, since K1b1h1+K2b2h2= 0.4.
Finally, taking into account that
u5
1+u4, N2(u) = u, K1=
β1=
?
|b1|h1
2(a1+b1)
?
2
∑
i=1
(ai+bi)
?
+1
2
2
∑
i=1
|bi|hi
?
K1= 0.55
and
β2=
?
|b2|h2
2(a2+b2)
?
2
∑
i=1
(ai+bi)
?
+1
2
2
∑
i=1
|bi|hi
?
K2= 0.34,
the conditions (6) and (9) are also satisfied for γ(1)
γ(1)
1
= 0, γ(t) =
(17) has a unique solution on [−h,∞), all the conditions, in
particular, (6) and (9), can be verified only for sufficiently
large t >T. Then, the coefficient γ(2)
small.
Example 2. Let constants C,A > 0 be such that for all
x ∈ Rn, N(x) ∈ Rn, n ≥ 2,
|N(x)|2≤C|x|2and xTN(x) ≥ A|x|2.
Then, the condition (5) holds for K =C
Example 3. Let (Ni)2
xk?= 0, x ∈ Rn, k = 1,2,...,n, for any i = 1,2,...,m, where
Ki> 0 are positive constants. Then, for k = 1,2...n and i =
1,2,...,m,
2
= γ(2)
2
=
1
(1+t)2and γ(2)
1
=
1
(1+t)2. Since the equation
1
can be taken arbitrarily
(18)
A.
k(x) ≤ Kikx2
k, and xkNk(x) > 0 for all
(Ni)2
k(x) = (Ni)k(Ni)k≤
n
∑
k=1
where Ki= maxk=1,2,...,n{√Kik}. Thus, the condition (5)
holds.
Example 4. Let fi:R→R be scalar continuous functions,
i = 1,2, and
?Kikxk(Ni)k(x) and
n
∑
k=1
|Ni(x)|2≤
?Kikxk(Ni)k(x)≤Ki
xk(Ni)k(x)=KxTNi(x),
1
2|u| ≤ |fi(u)| ≤ |u|,
ufi(u) > 0,
(19)
for all
u ?= 0,
u ∈ R,
i = 1,2.
Let K = 3 and −0.36 < c < 0.224. For all x = (x1,x2) ∈ R2,
define
?
Since
N(x) =
f1(x1)+cx2
cx1+ f2(x2)
?
.
(20)
|N(x)|2
=
f2
f2
?1+c2??x2
?+2cx1x2, then
KxTN(x)−|N(x)|2≥ (K
+2c?Kx1x2−x2f1(x1)−x1f2(x2)?
?
For K = 3, the last inequality takes the form
1(x1)+c2x2
2(x2)+c2x2
2+2cx2f1(x1)
1+2cx1f2(x2)
1+x2
2
?+2c?x2f1(x1)+x1f2(x2)?
+
≤
and xTN(x) = x1f1(x1)+2cx1x2+x2f2(x2)
≥1
2
?x2
1+x2
2
2−1−c2)?x2
?+2cx1x2(K−2), if x1x2> 0;
1+x2
2
?
≥
(K
(K
2−1−c2)?x2
1+x2
1+x2
2
2−1−c2)?x2
2
?+2cx1x2(K−1), if x1x2≤ 0.
KxTN(x)−|N(x)|2
≥
(1
?
(1
2−c2)?x2
2−c2)?x2
1+x2
1+x2
2
?+2cx1x2, if x1x2> 0;
2
?+4cx1x2, if x1x2≤ 0.
(21)
1903