On he asymp o ic beha io o highly nonlinea hyb id
s ochas ic delay diffe en ial equa ions1
Tian Zhang†Huabin Chen†2Chenggui Yuan‡3and Tom´as Ca aballo♯4
†Depa men o Ma hema ics, Nanchang Uni e si y, Nanchang 330031, China
‡Depa men o Ma hema ics, Swansea Uni e si y, Swansea SA2 8PP, UK
♯The Dep o. Ecuaciones Di e enciales y An´alisis
Num´e ico Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain
Abs ac . In his pape , unde a local Lipschi z condi ion and a mono onici y condi ion,
he p oblems on he exis ence and uniqueness heo em as well as he almos su ely asymp-
o ic beha io o he global solu ion o highly nonlinea s ochas ic diffe en ial equa ions
wi h ime- a ying delay and Ma ko ian swi ching a e discussed by using he Lyapuno
unc ion and some s ochas ic analysis echniques. Two in eg al lemmas a e fi s ly es ab-
lished o o e come he difficul y s emming om he coexis ence o he s ochas ic pe u -
ba ion and he ime- a ying delay. Then, wi hou any edundan es ic i e condi ion
on he ime- a ying delay, by u ilizing he in eg al inequali y, he exponen ial s abili y
in p h(p≥1)-momen o such equa ions is in es iga ed. By employing he nonnega i e
semi-ma ingale con e gence heo em, he almos su e exponen ial s abili y is analyzed.
Finally, wo examples a e gi en o show he use ulness o he esul s ob ained.
AMS Subjec Classifica ion: 60H15
Key wo ds and ph ases: S ochas ic diffe en ial equa ions, ime- a ying delay, asymp o ic
beha io , s abili y, Ma ko swi ching.
1The esea ch o Huabin Chen is suppo ed by he Na ional Na u al Science Founda ion o
China (61364005, 11401292, 61773401), he Na u al Science Founda ion o Jiangxi P o ince o China
(20171BAB201007, 20171BCB23001), and he Founda ion o Jiangxi P o incial Educa ions o China
(GJJ160061, GJJ14155). The esea ch o Tom´as Ca aballo was pa ially suppo ed by he p ojec s
MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Jun a de Andaluc´ıa).
2E-mail add ess: chb [email p o ec ed] (H. Chen)
3E-mail add ess: [email protected](C. Yuan).
4E-mail add ess: [email p o ec ed] (T. Ca aballo).
1
1 In oduc ion
Many dynamical sys ems no only depend on he p esen s a e bu also he pas ones,
which a e desc ibed by diffe en ial delay equa ions (DDEs) [1]. Since DDEs ha e been
used in many fields, such as he popula ion ecology, s eam o wa e pipes, hea exchang-
e s, lossless ansmission lines, and he mass-sp ing-dampe model, e c, he dynamical
beha io o DDEs has been widely in es iga ed in [2, 3, 4]. When DDEs a e subjec o
he en i onmen al dis u bances, i can be cha ac e ized by s ochas ic delay diffe en ial
equa ions (SDDEs), see [5, 6, 7, 8, 9, 10, 11]. One o he impo an issues in he s udy
o SDDEs is au oma ic con ol, wi h consequen emphasis being placed on he s abili y
analysis. Some excellen wo ks on he s ochas ic s abili y analysis ha e been p esen ed in
[5, 12, 13, 14, 15, 16, 17] and he e e ences he ein. Fo ins ance, in [12], he dynamical
beha io o s ochas ic delay Lo ka-Vol e a model as a pa icula ly impo an applica ion
o SDDEs was analyzed. In [15], he exponen ial s abili y analysis o linea s ochas ic
delay diffe en ial equa ion has been in es iga ed by one use ul and ad anced me hod such
as he compa ison p inciple. In [16], by es ablishing he LaSalle heo em, he s abili y
analysis o SDDEs has been in es iga ed.
Hyb id sys ems d i en by con inuous- ime Ma ko chains ha e been used o desc ibe
many p ac ical sys ems, in which hey may expe ience ab up changes in hei s uc-
u e and pa ame e s, o example, elec ic powe sys ems, manu ac u ing sys ems, fi-
nancial sys ems [18, 19, 20], e c. Many excellen wo ks a e seen in [21, 22] and he
e e ences he ein. The hyb id sys ems comp ise wo pa s: one is ha he s a e akes
alues con inuously, and he o he is ha he s a e akes disc e e alues. Recen ly, he
s abili y analysis o SDDEs wi h Ma ko ian swi ching has been ex ensi ely s udied in
[18, 20, 23, 24, 25, 26, 27, 28, 29, 30] and he e e ences he ein. Fo ins ance, in [23],
he compa ison p inciple was used o s udy he s abili y o SDDEs wi h Ma ko ian
swi ching. In [25], by using he Lyapuno unc ional app oach, he exponen ial s abil-
i y in p h(p≥1)-momen and he almos su e exponen ial s abili y o SDDEs wi h
Ma ko ian swi ching ha e been in es iga ed unde one mono onici y condi ion, which
likes (2.6) (see Hypo hesis IV ). In [28], by u ilizing a linea ma ix inequali y app oach,
he delay-dependen exponen ial s abili y o s ochas ic sys ems wi h ime- a ying delays,
Ma ko ian swi ching and nonlinea i ies has been discussed. In [18], by using he Lyapuno
unc ional app oach, he delay eedback con ol was designed o achie e he s abiliza ion
o hyb id SDDEs. In [21], in o de o educe he con ol cos , he eedback con ol based
on disc e e- ime s a e obse a ions was designed o gua an ee he s abiliza ion o hyb id
SDEs.
No e ha he e a e some esul s on he s abili y analysis o SDDEs wi h Ma ko ian
swi ching, see [18, 20, 23, 24, 25, 27, 28] and he e e ences he ein, in which he diffusion
e m and he d i e m o he SDDEs obey he local Lipschi z condi ion and he linea
g ow h condi ion. Usually, o many nonlinea SDDEs, hese wo e ms o en do no
sa is y he linea g ow h condi ion, bu he local Lipschi z condi ion. When he linea
g ow h condi ion is eplaced wi h he mono onici y condi ion, one o he mos powe ul
echnique used in he s udy o s abili y o SDDEs wi h Ma ko ian swi ching is based
on a s ochas ic e sion o he Lyapuno di ec me hod, and he e a e some ep esen i e
2
wo ks on he s abili y analysis o highly nonlinea SDDEs wi h Ma ko ian swi ching, see
[31, 32, 33, 34, 35]. Fo example, In [31], he delay-dependen s abili y c i e ia o highly
nonlinea SDDEs wi h Ma ko ian swi ching ha e been discussed by using he Lyapuno
unc ion app oach. Wi hou he linea g ow h condi ion, he exis ence and uniqueness,
he s abili y analysis and boundedness o he global solu ion o highly nonlinea SDDEs
wi h Ma ko ian swi ching we e conside ed in [32, 33].
Howe e , he ob ained esul s in [31, 32, 33, 34, 35] a e sui able o he cons an delay
o he ime- a ying delay wi h i s de i a i e alue being less han one. I is well known
ha in mos indus ial p ocess in ol ing anspo a ion o ma e ials, delay a ia ion is
one among he well-known s uc u al ime a ia ions in he p ocess plan s. Since he
anspo a ion ime a ies equen ly acco ding o a ying flow a es, ime- a ying delay
is an inhe en cha ac e is ics o hese p ocesses, which a ies a ound a cons an alue
and depends on he equency o he ex e nal exci a ion [36]. Thus, we will analyze
he exis ence and uniqueness o solu ions as well as hei s abili y p ope ies when he
es ic i e condi ions imposed on he ime-delay is emo ed, he local Lipschi z condi ion
is sa isfied o he d i e m and he diffusion e m, and he linea g ow h condi ion is
eplaced by he mono onici y condi ion.
In his pape , he exis ence and uniqueness heo em o highly nonlinea SDDEs wi h
Ma ko ian swi ching is p ima ily conside ed unde a local Lipschi z condi ion and a mono-
onici y condi ion. Wi hou any edundan es ic i e condi ion on he ime- a ying delay,
he exponen ial s abili y in p h(p≥1)-momen o such equa ions is discussed by using
he in eg al inequali y, and he almos su e exponen ial s abili y is analyzed by employ-
ing he nonnega i e semi-ma ingale con e gence heo em. The almos su e asymp o ical
s abili y o he global solu ion o highly nonlinea SDDEs wi h Ma ko ian swi ching is
also in es iga ed by i ue o some s ochas ic analysis echnique. Finally, wo examples
including one coupled sys ems consis ing o a mass-sp ing-dampe wi h he nonlinea ex-
e nal andom o ces a e p o ided o alida e he effec i eness o he heo e ical esul s
ob ained.
No a ions: Th oughou his pape , unless o he wise specified, we use he ollowing
no a ion. Le |·| deno e he Euclidean no m in Rn. I Ais a ec o o ma ix, i s anspose
is deno ed by AT. I Ais a ma ix, i s ace no m is deno ed by |A|=√ ace(ATA).
Le (Ω,F,{F } ≥0,P) ep esen s a comple e p obabili y space wi h a fil a ion {F } ≥0
sa is ying he usual condi ions (i.e., i is inc easing and igh -con inuous while F0con ains
all P-null se s). Le B( ) = col[B1( ), B2( ), . . . , Bm( )] be an m-dimensional B ownian
mo ion on (Ω,F,{F } ≥0,P). Fo τ > 0, le C([−τ, 0]; Rn) ep esen he amily o all
con inuous Rn- alued unc ions on [−τ, 0] wi h no m ∥φ∥C= sup{|φ(θ)|:−τ≤θ≤0}
o any φ∈ C([−τ, 0]; Rn). CF ([−τ, 0]; Rn) deno es he amily o all F -measu able and
C([−τ, 0]; Rn)- alued andom a iables ξ={ξ(θ) : −τ≤θ≤0}. Le E{·} s and o he
expec a ion ope a o . Fo any wo numbe s a,b,a∨band a∧bdeno e he maximum
alue and he minimum alue be ween aand b, espec i ely. H(a−) deno es he le -hand
limi o he unc ion H(·) a a,i. e. H(a−) = limu→0−H(a+u).
3
2 P oblem s a emen and p elimina ies
Le ( )( ≥0) be a igh -con inuous Ma ko chain on he p obabili y space aking
alues in a fini e s a e space S={1,2, ..., N}wi h gene a o Γ = (γij)N×Ngi en by
P{ ( +△) = j| ( ) = i}={γij△+o(△),i i=j,
1 + γij△+o(△),i i=j,
whe e lim△↓0o(△)
△= 0. He e, γij ≥0 is he ansi ion a e om i o j, i i=jwhile
γii =−∑j=iγij.
Fo a con inuous- ime Ma ko chain ( ) wi h i s gene a o Γ, i can be gi en as one
s ochas ic in eg al wi h espec o a Poisson andom measu e
d ( ) = ∫R
¯
h( ( −), y)ν(d , dy), ≥0
wi h he ini ial alue (0) = i0∈ S, whe e ν(d , dy) is a Poisson andom measu e wi h
in ensi y d ×m(dy) in which mis he Lebesgue measu e on R, while he explici defini ion
o ¯
h:S × R→Rcan be ounded in [12].
Conside he ollowing highly nonlinea hyb id s ochas ic delay diffe en ial equa ions:
dx( ) = ( , x( ), x( −τ( )), ( ))d +g( , x( ), x( −τ( )), ( ))dB( ), ≥0,(2.1)
wi h he ini ial alue {x(θ) : −τ≤θ≤0}=φ∈ CF0([−τ, 0]; Rn) and (0) = i0∈ S, whe e
x( ) = col[x1( ), x2( ), . . . , xn( )] ∈Rnis he s a e ec o . The ime- a ying delay τ(·) :
[0,∞)→[0, τ] is a bounded measu able unc ion. (·,·,·,·) : [0,∞)×Rn×Rn×S → Rn
is he d i coefficien ec o , and g(·,·,·,·) : [0,∞)×Rn×Rn× S → Rn×mis he
diffusion coefficien ma ix. In his pape , i is also assumed ha he Ma ko chain (·) is
independen o he B ownian mo ion B(·). Le x( , 0, φ, i0) be he solu ion o Eq. (2.1).
Fo simplici y, x( ) = x( , 0, φ, (0)).
In his pape , he exis ence-uniqueness heo em, and he asymp o ic beha io o Eq.
(2.1) will be checked. In gene al, he ollowing assump ions a e gi en o he exis ence
and uniqueness o he solu ion o Eq. (2.1), see [18].
Hypo hesis I (Local Lipschi z condi ion): Fo each k= 1,2, . . ., he e exis s a posi i e
cons an cksuch ha
| ( , x, y, i)− ( , ¯x, ¯y, i)| ∨ |g( , x, y, i)−g( , ¯x, ¯y, i)| ≤ ck(|x−¯x|+|y−¯y|)
o any ( , i)∈[0, T]× S (T > 0), x, y, ¯x, ¯y∈Rnwi h |x|∨|y|∨|¯x|∨|¯y| ≤ k.In addi ion,
( , 0,0, i) = 0 and g( , 0,0, i) = 0.
Hypo hesis II (Linea g ow h condi ion): The e is a posi i e cons an Lsuch ha
| ( , x, y, i)| ∨ |g( , x, y, i)| ≤ L(1 + |x|+|y|)
o any ( , x, y, i)∈[0, T]×Rn×Rn× S.
4
No e ha Hypo hesis II is a conse a i e condi ion o check he exis ence o he global
solu ion. Fo example, when S={1,2}, ( , x, y, 1) = −0.15x−2x3+ 0.4y, ( , x, y, 2) =
−2x−0.5xy4+0.82y,g( , x, y, 1) = 2x2, and g( , x, y, 2) = xy2, o any ≥0, Hypo hesis II
does no hold o (·,·,·,·) and g(·,·,·,·). He e, we shall pe sis Hypo hesis I bu eplace
Hypo hesis II by a mo e gene al condi ion o gua an ee he exis ence o he unique global
solu ion o Eq. (2.1). To s a e a gene al condi ion, we need a ew no a ions. Le C1,2=
C1,2([0,∞)×Rn× S; [0,∞)) deno e he amily o all con inuous nonnega i e unc ions
V( , x, i) defined on [0,∞)×Rn× S, such ha o each i∈ S, hey a e con inuously
once diffe en iable in and wice in x. Gi en V∈ C1,2, hen we define he I ˆo ope a o
LV : [0,∞)×Rn×Rn× S −→ Rby
LV ( , x, y, i)
=V ( , x, i) + Vx( , x, i) ( , x, y, i) + 1
2 ace[gT( , x, y, i)Vxx( , x, i)g( , x, y, i)]
+
N
∑
j=1
γijV( , x, j).
whe e
V ( , x, i) = ∂V ( , x, i)
∂ , Vx( , x, i) = (∂V ( , x, i)
∂x1
,∂V ( , x, i)
∂x2
, . . . , ∂V ( , x, i)
∂xn),
and
Vxx( , x, i) = (∂2V( , x, i)
∂xl∂xm)n×n
.
To ob ain he main esul s, one mo e gene al condi ion is p esen ed as ollows:
Hypo hesis III (Mono onici y condi ion): The e exis s one Lyapuno unc ion V∈
C1,2, one unc ion U∈ C(Rn; [0,∞)) and some posi i e cons an s c1,c2,λ1and λ2such
ha o any x, y ∈Rn, ≥0, and i∈ S,
c1U(x)≤V( , x, i)≤c2U(x),(2.2)
and
LV ( , x, y, i)≤ −λ1U(x) + λ2U(y),(2.3)
and
lim
|x|→∞ U(x) = ∞.(2.4)
when U(x) = |x|p,Hypo hesis III can be w i en as he ollowing o m:
Hypo hesis IV : The e exis s one Lyapuno unc ion V∈ C1,2, and some posi i e con-
s an s p,c1,c2,λ1and λ2such ha o any x, y ∈Rn, ≥0, and i∈ S,
c1|x|p≤V( , x, i)≤c2|x|p,(2.5)
5

and
LV ( , x, y, i)≤ −λ1|x|p+λ2|y|p,(2.6)
whe e p≥1 and λ2c2< λ1c1.
Rema k 2.1 In [15, 17, 18, 25], Hypo hesis IV has been imposed wi h τ( )≡τo dτ( )
d ∈
(0,1). I should be men ioned ha he es ic i e condi ion ha he de i a i e alue o
ime- a ying delay is less han one is no equi ed in his pape . Thus, he p oposed
me hods in [15, 17, 18, 25] can no be used he e. E en i he asymp o ic beha io o high
nonlinea SDDEs wi h Ma ko ian swi ching has been been conside ed unde he gene al
mono onici y condi ion [31, 32, 33, 34], bu his es ic i e condi ion is also added.
Defini ion 2.2 Le x( ):−τ≤ <σ∞be a con inuous F -adap ed Rn- alued local
p ocess, whe e σ∞is a s opping ime and we se F =F0 o ∈[−τ, 0]. I is called a
local solu ion o Eq. (2.1) wi h ini ial da a φ∈ CF0([−τ, 0]; Rn).I x0=φ={x(θ) :
−τ≤θ≤0}and o all ≥0
x( ∧σk) =φ(0) + ∫ ∧σk
0
(s, x(s), x(s−τ(s)), (s))ds
+∫ ∧σk
0
g(s, x(s), x(s−τ(s)), (s))dB(s)
holds o any k≥1, whe e {σk}k≥1is a nondec easing sequence o fini e s opping imes
such ha σk↑σ∞a.s. Fu he mo e, i lim supk→∞ |x(σk)|=∞is sa isfied whene e
σ∞<∞, i is called a maximal solu ion and σ∞is called he explosion ime. A maximal
local solu ion x( ) : −τ≤ <σ∞, is said o be unique i o any o he maximal local
solu ion ˆx( ) : −τ≤ < ˆσ∞, we ha e σ∞= ˆσ∞a.s. and x( ) = ˆx( ) o all −τ≤ < σ∞
a.s.
Defini ion 2.3 The solu ion o Eq. (2.1) is said o be exponen ially s able in p h(p≥1)
momen wi h decay e o o de γ, i he e exis s a posi i e cons an γsuch ha
lim sup
→∞
log(E|x( )|p)
≤ −γ
holds o any φ∈ CF0([−τ, 0]; Rn). Fu he mo e, he solu ion o Eq. (2.1) is said o be
almos su ely exponen ially s able wi h exponen ial decay e o o de γ, i
lim sup
→∞
log(|x( )|)
≤ −γ a.s.
holds o any φ∈ CF0([−τ, 0]; Rn).
Lemma 2.4 ([37]) Fo γ > 0, he e exis wo posi i e cons an s: λ,λ′wi h λ′< γ, and
a unc ion y: [−τ, ∞)→[0,∞). I he inequali y
y( )≤{λe−γ +λ′∫
0e−γ( −s)supθ∈[−τ,0] y(s+θ)ds, o ≥0,
λe−γ , o ∈[−τ, 0],(2.7)
6
holds, hen we ha e y( )≤˜
Me−µ , o any ∈[−τ, ∞), whe e µis a unique posi i e oo
o he algeb a equa ion: λ′eµτ
γ−µ= 1 and ˜
M= max{λ(γ−µ)
λ′eµτ , λ}>0.
3 Main esul s
Lemma 3.1 Le x( )be a solu ion o Eq. (2.1) wi h he ini ial condi ion φ. Suppose
ha Hypo heses I and III hold. Assume ha he inequali y
λ2c2< λ1c1,
holds, hen we ha e
∆(ε) = ∫∞
0
eε sup
θ∈[−τ,0]
EU(x( +θ))d < ∞,(3.1)
whe e ε∈(0, ε0),ε0is a unique posi i e solu ion o he algeb aic equa ion:
λ2c2eετ
λ1c1−c1c2ε= 1.
P oo : Define he unc ion: H(ε) = λ2c2eετ
λ1c1−c1c2ε−1. I can be p o ed ha H(0) <0,
H((λ1
c2)−) = ∞, and H(ε) is a nondec easing unc ion on (0,λ1
c2). The e o e, he e exis s
a scala ε0∈(0,λ1
c2) sa is ying H(ε0) = 0. Tha is, o any ε∈(0, ε0), we ha e
Λ(ε)≡λ2c2eετ
λ1c1−c1c2ε<1.(3.2)
Using he I ˆo o mula, o any ≥0, i ollows
e
λ1
c2 V( , x( ), ( ))
≤V(0, x(0), (0)) + ∫
0
e
λ1
c2s[λ1
c2
V(s, x(s), (s)) + LV (s, x(s), x(s−τ(s)), (s))]ds
+∫
0
e
λ1
c2sVx(s, x(s), (s))g(s, x(s), x(s−τ(s)), (s))dB(s)
+∫
0∫R
e
λ1
c2s[V(s, x(s), i0+¯
h( (s−), l)−V(s, x(s), (s))]µ(ds, dl),
(3.3)
whe e µ(ds, dl) = ν(ds, dl)−m(dl) is a ma ingale measu e, which is ela ed o he Ma ko
chain bu no he B ownian mo ion.
F om condi ions (2.2) and (2.3), we ob ain
λ1
c2
V(s, x(s), (s)) + LV (s, x(s), x(s−τ(s)), (s)) ≤λ2U(x(s−τ(s)),(3.4)
7
Subs i u ing (3.4) in o (3.3), and hen aking he expec a ion, i yields
e
λ1
c2 EV( , x( ), ( )) ≤EV(0, x(0), (0)) + λ2∫
0
e
λ1
c2sEU(x(s−τ(s))ds.
By using condi ion (2.2), i concludes ha o any ≥0,
EU(x( )) ≤EV(0, x(0), (0))
c1
e−λ1
c2 +λ2
c1∫
0
e−λ1
c2( −s)EU(x(s−τ(s))ds
≤M′e−λ1
c2 +λ2
c1∫
0
e−λ1
c2( −s)EU(x(s−τ(s))ds,
(3.5)
whe e M′=EV(0,x(0), (0))
c1>0.
Fo any ≥τand θ∈[−τ, 0], om (3.5), we ha e
EU(x( +θ)) ≤M′e−λ1
c2( +θ)+λ2
c1∫ +θ
0
e−λ1
c2( +θ−s)EU(x(s−τ(s))ds
≤M′e−λ1
c2( +θ)+λ2
c1∫ +θ
0
e−λ1
c2( +θ−s)sup
u∈[−τ,0]
EU(x(s+u))ds.
Mul iplying by eε (ε∈(0, ε0)) on bo h sides o inequali y abo e in u n, and hen
in eg a ing wi h τ o T(T > τ), i ollows
∫T
τ
eε EU(x( +θ))d
≤M′∫T
τ
eε −λ1
c2( +θ)d +λ2
c1∫T
τ∫ +θ
0
eε −λ1
c2( +θ−s)sup
u∈[−τ,0]
EU(x(s+u))dsd .
(3.6)
No e ha o any θ∈[−τ, 0] and ≥τ, he o mula o in eg a ion by pa s implies
∫T
τ∫ +θ
0
eε −λ1
c2( +θ−s)sup
u∈[−τ,0]
EU(x(s+u))dsd
≤eετ ∫T
τ
e−(λ1
c2−ε)( +θ)∫ +θ
0
e
λ1
c2ssup
u∈[−τ,0]
EU(x(s+u))dsd
≤e
λ1
c2τ
λ1
c2−ε∫τ
0
e
λ1
c2ssup
u∈[−τ,0]
EU(x(s+u))ds
+eετ
λ1
c2−ε∫T
0
eεs sup
u∈[−τ,0]
EU(x(s+u))ds.
(3.7)
8
Subs i u ing (3.7) o (3.6) implies
∫T
τ
eε EU(x( +θ))d
≤M′e
λ1
c2τ∫T
τ
eε −λ1
c2 d +λ2c2e
λ1
c2τ
λ1c1−c1c2ε∫τ
0
e
λ1
c2ssup
u∈[−τ,0]
EU(x(s+u))ds
+λ2c2eετ
λ1c1−c1c2ε∫T
0
eεs sup
u∈[−τ,0]
EU(x(s+u))ds.
(3.8)
F om (3.8), we ha e
∫T
0
eε EU(x( +θ))d
=∫τ
0
eε EU(x( +θ))d +∫T
τ
eε EU(x( +θ))d
≤¯
M+ Λ(ε)∫T
0
eεs sup
u∈[−τ,0]
EU(x(s+u))ds,
(3.9)
whe e ¯
M=∫τ
0eε EU(x( +θ))d +c2M′eετ
λ1−c2ε+λ2c2e
λ1
c2τ
λ1c1−c1c2ε∫τ
0e
λ1
c2ssupu∈[−τ,0] EU(x(s+u))ds.
Combing (3.2) and (3.9), i gi es
∫T
0
eε sup
θ∈[−τ,0]
EU(x( +θ))d ≤¯
M
1−Λ(ε)<∞.
Le T→ ∞, he desi ed esul (3.1) is ob ained. 2
Rema k 3.2 F om (3.1), i ollows ha
∆ = ∫∞
0
sup
θ∈[−τ,0]
EU(x( +θ))d < ∞,(p≥1).(3.10)
Theo em 3.3 Suppose ha he condi ions o Lemma 3.1 hold, o any ini ial da a
φ∈ CF0([−τ, 0]; Rn), he e is a unique solu ion x( ) o Eq. (2.1) on ∈[−τ, ∞)wi h
p obabili y one.
P oo : By Hypo hesis I, o any ini ial da a φ∈ CF0([−τ, 0]; Rn), by using Theo em
7.12 (see, pp. 278 [18]), i is shown ha he e exis a unique maximal local s ong solu ion
x( ) on [−τ, σe], whe e σeis he explosion ime. To show ha his solu ion is global, we
only need o p o e σe=∞,a.s. No e ha φ∈ CF0([−τ, 0]; Rn), consequen ly, he e mus
exis a posi i e numbe k0such ha ||φ||C≤k0. Fo each in ege k > k0, define he
s opping ime
τk= in { ∈[0, σe) : |x( )| ≥ k}.
9
F om (3.27), i yields
P({α2i−1<∞, βh=∞} ∩ { sup
0≤ ≤T
|U(x(α2i−1+ )) −U(y(α2i−1))|< ε}
≥P({α2i−1<∞, βh=∞} ∩ { sup
0≤ ≤T
|x(α2i−1+ )−x(α2i−1)|< δ})
> ε.
(3.29)
Se
ˆ
Ωi={sup
0≤ ≤T
|U(x(α2i−1+ )) −U(y(α2i−1))|< ε},
and no e ha
α2i(ω)−α2i−1(ω)≥T, i ω∈ {α2i−1<∞, βh=∞} ∩ ˆ
Ωi.
Using (3.25) and (3.29), we ha e
∞ ≥ ε
∞
∑
i=1
E{I{α2i<∞,βh=∞}[α2i−α2i−1]}
≥ε
∞
∑
i=1
E{I{α2i<∞,βh=∞}∩ˆ
Ωi[α2i−α2i−1]}
≥εT
∞
∑
i=1
P({α2i<∞, βh=∞} ∩ ˆ
Ωi)
> εT
∞
∑
i=1
ε=∞,
which is a con adic ion. Hence, (3.18) holds (i.e. lim →∞ U(x( )) = 0).
S ep 4: Now, i is necessa y o show ha Ke (U)=∅. F om (3.18), i is seen ha
he e exis s an Ω0⊂Ω wi h P(Ω0) = 1 such ha
lim
→∞ U(x( )) = 0 and sup
0≤ <∞
|x( )|<∞, o any ω∈Ω0.(3.30)
Choose any ω∈Ω0, hen {x( )} ≥0is bounded in Rn. Then, he e mus be an inc easing
sequence { k}k≥1such ha k→ ∞ and {x( k)}k≥1con e ges o some ¯x∈Rn. Thus,
U(¯x) = lim
k→∞ U(x( k)) = 0,
which implies ha ¯x∈Ke (U). Tha is, Ke (U)=∅.
S ep 5: I is necessa y o show ha o any ω∈Ω0,
lim
→∞ d(x( ), Ke (U)) = 0.(3.31)
16

I his is alse, hen he e exis s some ¯ω∈Ω0such ha
lim sup
→∞
d(x( , ¯ω), Ke (U)) >0.
Thus, he e exis s a subsequence {x( k,¯ω)}k≥0o {x( , ¯ω)} ≥0sa is ying
lim sup
k→∞
d(x( k,¯ω), Ke (U)) >¯ε,
o some ¯ε > 0. Since {x( k,¯ω)}k≥0is bounded, we can find a subsequence con e ging o
some ˜x∈Rn. Clea ly, ˜x /∈Ke (U) and U(˜x)>0. Howe e , om (3.30),
U(˜x) = lim
k→∞ U(x( k,¯ω)) = 0.
This is a con adic ion. The e o e, (3.31) mus be sa isfied. In addi ion, i U(x) = 0 ⇔
x= 0, hen Ke (U) = 0. Consequen ly, om (3.31), we deduce ha
lim
→∞ x( ) = 0. a.s.
The p oo is he e o e comple e. 2
Co olla y 3.8 Le x( ;φ)be a solu ion o Eq. (2.1) wi h he ini ial condi ion φ. Suppose
ha Hypo heses I and IV a e sa is ied, o any ini ial da a φ∈ CF0([−τ, 0]; Rn), he
p h(p≥1)-momen Lyapuno exponen o he solu ion o he Eq. (2.1) obeys
lim
→∞ sup 1
log(E|x( ;φ)|p)≤ −¯µ,
whe e ¯µ∈(0,λ1
c2)is a oo o he algeb a equa ion : λ2c2eµτ
λ1c1−c1c2µ= 1. Tha is, he solu ion
o he Eq. (2.1) is exponen ially s able in p h(p≥1) mean.
Co olla y 3.9 Le x( )be a solu ion o Eq. (2.1) wi h he ini ial condi ion φ. Suppose
ha Hypo heses I and IV hold, o any ini ial da a φ∈ CF0([−τ, 0]; Rn), he sample
Lyapuno exponen o he solu ion o he Eq. (2.1) obeys
lim
→∞ sup 1
log(|x( ;φ)|)≤ −ε
p,a.s.
whe e p≥1, and ε∈(0, ε0), whe e ε0is gi en in Lemma 3.1. Tha is, he solu ion o he
Eq. (2.1) is almos su ely exponen ially s able.
4 Two Examples
In o de o illus a e he ad an ages o he main esul s, wo examples a e p o ided.
17
Example 4.1: Le B( )be a scala B ownian mo ion on (Ω,F,{F } ≥0,P). Conside
one dimensional s ochas ic diffe en ial equa ions wi h ime- a ying delay and Ma ko ian
swi ching:
dx( ) = ( , x( ), x( −τ( )), ( ))d +g( , x( ), x( −τ( )), ( ))dB( ), ≥0,(4.1)
wi h he ini ial alue {x(θ) : −τ≤θ≤0}=φ∈ CF0([−τ, 0]; Rn) and (0) = i0∈ S and
(0) = 1 ∈ S ={1,2}, whe e x( ) and x( −τ( )) a e he s a e scala and he delayed s a e
scala , espec i ely. τ( ) is a bounded measu able unc ion wi h 0 ≤τ( )≤τ( ≥0, τ >
0), and ( ) is a igh -con inuous Ma ko chain aking alues in Swi h he gene a o
Γ = (γij)2×2=[−2 2
1−1].
In (4.1), we assume ha , g : [0,∞)×R×R× S → Rwi h
( , x, y, i) = {−0.15x−2x3+ 0.4y, i i= 1,
−2x−0.5xy4+ 0.82y, i i= 2,
and
g( , x, y, i) = {2x2,i i= 1,
xy2,i i= 2.
Define a Lyapuno unc ion
V( , x, i) = {x2,i i= 1,
0.5x2,i i= 2,
hen, i is compu ed o he I ˆo ope a o o Eq. (4.1) ha
LV ( , x, y, 1) = 2x[−0.15x−2x3+ 0.4y]+4x4+
2
∑
j=1
γ1jV( , x, j)
=−1.3x2+ 0.8xy
≤ − 0.9x2+ 0.4y2,
and
LV ( , x, y, 2) = x[−2x−0.5xy4+ 0.82y]+0.5x2y4+
2
∑
j=1
γ2jV( , x, j)
=−1.5x2+ 0.82xy
≤ − 1.09x2+ 0.41y2.
Hence, we ha e
LV ( , x, y, i)≤ −0.9x2+ 0.41y2,
wi h λ1= 0.9, λ2= 0.41, c1= 0.5 and c2= 1. Then, λ2c2< λ1c1holds, which implies ha
he exis ence and uniqueness, he exponen ial s abili y in mean squa e, he almos su e
18
0 5 10 15
−0.5
0
0.5
1
1.5
2
Time (a)
E|x( )|2
x( )
Figu e 1: Asymp o ic beha io in mean squa e o he global solu ion o Eq. (4.1)
0 5 10 15
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (a)
x( )
x( )
Figu e 2: Asymp o ic beha io in almos su e sense o he global solu ion o Eq. (4.1)
exponen ial s abili y and he almos su e asymp o ical s abili y o he global solu ion o
Eq. (4.1) a e gua an eed. When he ini ial condi ion x( ) = −1 ( ∈[−2.3,0]), (0) = 1,
and τ( ) = 1.1|sin( )|+1.2 a e fixed, Fig. 1 and Fig. 2 illus a e he asymp o ic beha io
in mean squa e and in almos su e sense o he global solu ion o Eq. (1), espec i ely.
Example 4.2: One coupled sys em consis s o a mass-sp ing-dampe (MSD) model [39].
An ac ua o is aken o a ans e sys em. The ma hema ical exp ession o he sys em is
DDEs, which a e w i en as
M¨y( ) + C˙y( ) + Ky( ) = 0 (4.2)
on ≥0, whe e M,C,Ka e he mass, s iffness and damping o a mass-sp ing-dampe
model, and y( ), ˙y( ), ¨y( ) deno e he posi ion, eloci y and accele a ion o MSD a ime .
I his physical model is affec ed by he ex e nal o ce, hen Eq. (4.2) is u he desc ibed
as
M¨y( ) + C˙y( ) + Ky( ) + F( ) = 0 (4.3)
on ≥0, whe e F( ) deno es he ex e nal o ce, M= 10, C= 25, and K= 15. Assume
ha his ex e nal o ce is subjec o he en i onmen al noise and ab up changes in he
19
pa ame e s, which is cha ac e ized by
F( ) = F1( ˙y( ),˙y( −τ( )), ( )) + F2( ˙y( ), y( −τ( )),˙y( −τ( )), ( )) ˙
B( )
whe e ˙
B( )is a scala whi e noise ( i.e. ˙
B( )is a scala B ownian mo ion), τ( ) is he
ime- a ying delay, ( ) is a Ma ko ian swi ching aking alues in S={1,2}wi h i s
gene a o Γ = [−2 2
3−3],
F1( ˙y( ),˙y( −τ( )), ( )) = {5.4 ˙y( ) ˙y2( −τ( )),i i= 1,
15 ˙y3( ) ˙y2( −τ( )),i i= 2,
and
F2( ˙y( ), y( −τ( )),˙y( −τ( )), ( ))
={6 ˙y( ) ˙y( −τ( )) + 3y( −τ( )) + 3 ˙y( −τ( )),i i= 1,
10 ˙y2( ) ˙y( −τ( )) + 2y( −τ( )) + 2 ˙y( −τ( )),i i= 2.
Le x1( ) = y( ) and x2( ) = ˙y( ), Eq. (4.3) can be w i en as highly nonlinea SDDEs
wi h Ma ko ian swi ching:
dx( ) = ( , x( ), x( −τ( )), ( ))d +g( , x( ), x( −τ( )), ( ))dB( )(4.4)
whe e x( ) = col[x1( ), x2( )],
( , x( ), x( −τ( )),1) = [x2( )
−1.5x1( )−2.5x2( )−0.54x2( )x2
2( −τ( )) ],
( , x( ), x( −τ( )),2) = [x2( )
−1.5x1( )−2.5x2( )−1.5x3
2( )x2
2( −τ( )) ],
g( , x( ), x( −τ( )),1) = [0
0.6x1( −τ( )) + 0.3x2( −τ( )) + 0.3x2( )x2( −τ( )) ],
and
g( , x( ), x( −τ( )),2) = [0
x1( −τ( )) + 0.2x2( −τ( )) + 0.2x2( )x2( −τ( )) ].
Fo Eq. (4.4), conside a Lyapuno unc ion
V( , x, i) = {|x|2,i i= 1,
0.8|x|2,i i= 2,
wi h |x|2=x2
1+x2
2.
Then, o Eq. (4.4), he I ˆo ope a o is compu ed as
LV ( , x( ), x( −τ( )), i)
= 2qixT( ) ( , x( ), x( −τ( )), ( )) + qi ace[gT( , x( ), x( −τ( )), i)
×g( , x( ), x( −τ( )), i)] +
2
∑
j=1
γijV( , x( ), j),
20
012345
−0.5
0
0.5
1
1.5
2
Time (a)
E|x( )|2
x1( )
x2( )
Figu e 3: Asymp o ic beha io in mean squa e o he global solu ion o Eq. (4.4)
0 1 2 3 4 5 6 7 8
−1
−0.5
0
0.5
1
1.5
2
Time (a)
x( )
x1( )
x2( )
Figu e 4: Asymp o ic beha io in almos su e sense o he global solu ion o Eq. (4.4)
whe e q1= 1, q2= 0.8.
Consequen ly, when i= 1, we ha e
LV ( , x( ), x( −τ( ),1) ≤ − 2[x2
1( ) + x2
2( )] −1.08x2
2( )x2
2( −τ( ))
+ [0.6x2( )x2( −τ( )) + 0.3x1( −τ( )) + 0.3x2( −τ( ))]2
−0.4[x2
1( ) + x2
2( )]
≤ − 2.4|x( )|2+ 0.27|x( −τ( ))|2,
and when i= 2,
LV ( , x( ), x( −τ( ),2) ≤ − 1.6[x2
1( ) + x2
2( )] −2.4x4
2( )x2
2( −τ( ))
+ 0.8[x2
2( )x2( −τ( )) + 0.2x1( −τ( )) + 0.2x2( −τ( ))]2
+ 0.6[x2
1( ) + x2
2( )]
≤ − |x( )|2+ 0.096|x( −τ( ))|2.
Thus, o any i∈ S.
LV ( , x( ), x( −τ( )), i)≤ −|x( )|2+ 0.27|x( −τ( ))|2.
21

wi h λ1= 1, λ2= 0.27, c1= 0.8 and c2= 1. Thus, λ2c2< λ1c1is sa isfied. Consequen ly,
he exis ence and uniqueness, he exponen ial s abili y in mean squa e, he almos su e ex-
ponen ial s abili y and he almos su e asymp o ical s abili y o he global solu ion o Eq.
(4.4) a e gua an eed. When aking he ini ial condi ion x( ) = col[−sin( ),0.5 cos( )] ( ∈
[−2.3,0]), (0) = 1, and τ( ) = 1.1|cos( )|+ 1.2, Fig. 3 and Fig. 4 show he asymp o ic
beha io in mean squa e and in almos su e sense o he global solu ion o Eq. (4.4),
espec i ely.
5 Conclusion
The me hod o Lyapuno unc ion has been widely used in he s udy o he s abili y
o SDDEs wi h Ma ko ian swi ching. Howe e , so a , mos o he exis ing esul s in
his a ea usually equi e ha he delay is a cons an o he ime- a ying delay wi h i s
de i a i e alue being less han one, which limi s hei applica ions o some ex en . To
emo e his es ic i e condi ion, fi s ly, wo in eg al lemmas ha e been p oposed. Then,
by using he in eg al inequali y, some s ochas ic analysis echnique and he nonnega i e
semi-ma ingale con e gence heo em, he exis ence-uniqueness heo em and he s abili y
analysis o he global solu ion o highly nonlinea hyb id SDDEs ha e been discussed.
Finally, wo examples ha e been p o ided o illus a e he effec i eness o he heo e ical
esul s ob ained.
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