On the controlled eigenvalue problem for stochastically perturbed multi-channel systems

0.13 MB·English

by  Getachew K. Befekadu

ONTHECONTROLLEDEIGENVALUEPROBLEMFORSTOCHASTICALLY
PERTURBEDMULTI-CHANNELSYSTEMS∗
GETACHEWK.BEFEKADU†
Abstract. In this brief paper, we consider the problem of minimizing the asymptotic exit rate ofdiffusion
6
processesfromanopenconnectedboundedsetpertainingtoamulti-channelsystemwithsmallrandomperturbations.
1
Specifically,weestablishaconnectionbetween:(i)theexistenceofaninvariantsetfortheunperturbedmulti-channel
0 systemw.r.t.certainclassofstate-feedbackcontrollers;and(ii)theasymptoticbehavioroftheprincipaleigenvalues
2 andthesolutionsoftheHamilton-Jacobi-Bellman(HJB)equationscorrespondingtoafamilyofsingularlyperturbed
ellipticoperators. Finally,weprovideasufficientconditionfortheexistenceofaParetoequilibrium(i.e.,asetof
t
c optimalexitratesw.r.t. eachofinputchannels)fortheHJBequations–wherethelattercorrespondtoafamilyof
O nonlinearcontrolledeigenvalueproblems.

4
Keywords. Diffusionprocess,HJBequations,multi-channelsystem,principaleigenvalue, optimalexittime,
] smallrandomperturbations
S
D
AMSsubjectclassifications.37C75,37H10,34D15,34D05,34D10,49L25,34H05
.
h
t
a 1. Introduction. Considerthefollowingcontinuous-timemulti-channelsystem1
m
n
x˙(t)=Ax(t)+ B u (t), x(0)=x , (1.1)
[ i i 0
i=1
3 wherex Rd isthestateofthesXystem,ui i Rri isthecontrolinputtotheith-
6v channel,∈andXA⊆∈Rd×dandBi ∈Rd×ri,fori=1∈,2U,..⊂.,n,areconstantmatrices.
5 LetD beanopenconnectedboundedsetwithsmoothboundary.Forthemulti-channel
2 ⊂X
systemin(1.1),weconsiderthefollowingclassofstate-feedbackcontrollers
1
0 n
. Γ γ1,γ2,...,γn Rri Λ(D)= , (1.2)
1 ⊆ ∈ i=1 6 ∅
0 whereΛ(D) D ∂Dnis(cid:0)the maximalin(cid:1)variYantset(un(cid:12)(cid:12)dertheactioonofSt , exp A+
15 ni=1Biγi t⊂, t≥∪0)suchthat (cid:12) (cid:8)(cid:0)
:
v P (cid:1) (cid:9) StΩ=Ω Λ(D), t 0, (1.3)
i ⊂ ∀ ≥
X
foranysetΩ.
r
a In what follows, we provide a connection between the existence of an invariantset for the
system St in D ∂D and the asymptotic behavior of the principal eigenvalue for singu-
∪
larlyperturbedellipticoperatorwhichisassociatedwiththefollowingstochasticdifferential
equation(SDE)
n
dXǫ,γ(t)= A+ B γ Xǫ,γ(t)dt+√ǫσ Xǫ,γ(t) dW(t), Xǫ,γ(0)=x , (1.4)
i i 0
i=1
where (cid:16) X (cid:17) (cid:0) (cid:1)
∗Received by the editors January 15, 2015. This research was carried-out while the author was working at
theUniversityofNotreDame. Theauthoracknowledges supportfromtheDepartmentofElectricalEngineering,
UniversityofNotreDame.
†NRC/AFRL&DepartmentofIndustrialSystemEngineering,UniversityofFlorida-REEF,1350N.Poquito
Rd,Shalimar,FL32579,USA([email protected]).
1Thisworkis,insomesense,acontinuationofourpreviouspaper[3].
1
2 GETACHEWK.BEFEKADU
- Xǫ,γ() isanRd-valueddiffusionprocess,ǫ isasmallparameterlyinginaninter-
·
val (0,ǫ∗) (which represents the level of random perturbation in the system), and
γ ,γ ,...,γ isann-tupleofstate-feedbacksfromtheclassΓ,
1 2 n
- σ(cid:0): Rd Rd×d(cid:1)isLipschitzcontinuouswiththeleasteigenvalueofσ()σT()uni-
→ · ·
formlyboundedawayfromzero,i.e.,
σ(x)σT(x) κI , x Rd,
d×d
≥ ∀ ∈
forsomeκ>0,and
- W()(withW(0)=0)isand-dimensionalstandardWienerprocess.
·
Foranyfixedǫ (0,ǫ∗),letτǫ,γ bethefirstexittimeofthediffusionprocessXǫ,γ(t)from
∈
thesetD,i.e.,
τǫ,γ =inf t>0 Xǫ,γ(t) / D . (1.5)
∈
n (cid:12) o
Further,letPx0,γ andEx0,γ ξ ,asusual(cid:12)(cid:12),denotetheprobabilityofaneven andthe
ǫ,0 A ǫ,0 A
expectation of a random variable ξ, respectively, for the diffusion process Xǫ,γ(t) starting
(cid:8) (cid:9) (cid:8) (cid:9)
fromx D.
0
∈
Here, it is worthmentioningthat, in system reliabilityanalysisand otherstudies, oneoften
requirestoconfineacontrolleddiffusionprocessXǫ,γ(t)toagivenopenconnectedbounded
setDaslongaspossible.Astandardformulationforsuchaproblemistomaximizethemean
exittimeEx0,γ τǫ,γ fromthesetD. Wealsoobservethatamoresuitableobjectivewould
ǫ,0
beto minimizetheasymptoticratewith whichthe diffusionprocessXǫ,γ(t)exitsfromthe
setD. Further,(cid:8)thissu(cid:9)ggestsminimizingtheprincipaleigenvalueλγ
ǫ,0
1
λγ = limsup logPx0,γ τǫ,γ >t , (1.6)
ǫ,0 − t→∞ t ǫ,0
n o
withrespecttocertainclassofadmissiblecontrols(includingtheaboveclassofstate-feedbacks
in(1.2)).2 NotethatifthedomainDcontainsanequilibriumpointforthedeterministicmulti-
channelsystemin(1.1),whensuchasystemiscomposedwiththen-tupleofstate-feedbacks
fromΓ,thentheprincipaleigenvalueλγ ,whichisassociatedwiththesingularlyperturbed
ǫ,0
ellipticoperator
n ǫ
γ x = ▽ , A+ B γ x + tr σ(x)σT(x)▽2 , (1.7)
−Lǫ,0 · · i=1 i i 2 ·
(cid:0)(cid:1)(cid:0) (cid:1) D (cid:0)(cid:1) (cid:0) X (cid:1) E n (cid:0)(cid:1)o
withzeroboundaryconditionson∂D,satisfies(seealsoCorollary2.3)
logλγ =ǫ−1rγ +o(ǫ−1) as ǫ 0,
− ǫ,0 →
whererγ isgivenbythefollowing
1
rγ =limsup inf Iγ(ϕ(t)) ϕ(t) D ∂D, t [0,T] (1.8)
T→∞ ϕ(t)∈ΨT(cid:26) T (cid:12) ∈ ∪ ∈ (cid:27)
(cid:12)
(cid:12)
2Recently,theauthorsin[1]haveprovidedinterestingresultsoncontrolledequilibriumselectioninstochasti-
callyperturbeddynamics,butinaslightlydifferentcontext.
Onthecontrolledeigenvalueproblem 3
andtheactionfunctionalIγ(ϕ),with(γ ,γ ,...,γ ) Γ,isgivenby
T 1 2 n ∈
1 T dϕ(t) n T −1
Iγ(ϕ(t))= A+ B γ ϕ(t) σ(ϕ(t))σT(ϕ(t))
T 2 dt − i=1 i i
Z0 (cid:20) (cid:0) Xdϕ(t) (cid:1) (cid:21) (cid:16)n (cid:17)
A+ B γ ϕ(t) dt, (1.9)
i i
× dt − i=1
(cid:20) (cid:21)
(cid:0) X (cid:1)
wherethesetΨ consistingofallabsolutelycontinuousfunctionsϕ C ([0,T],Rd), with
T
∈
ϕ(0) = x , a compact set in D (e.g., see [12], [13], [8, Chapter 14] or [7] for additional
0
discussions).
On the other hand, if the maximum invariant set Λ(D) for the deterministic multi-channel
systemin(1.1),w.r.t. thestate-feedbacksfromΓ,isnonempty. Then,thefollowingasymp-
totic condition also holds true (see Lemma 2.5 in the Appendix section (cf. [10, Theo-
rem2.1]))
1
lim limsup logPx0,γ τǫ,γ >t < , x D. (1.10)
−ǫ→0 t→∞ t ǫ,0 ∞ 0 ∈
(cid:8) (cid:9)
Moreover,theprincipaleigenvalueλγ turnsouttobetheboundaryvaluebetweenthoseR<
ǫ,0
rγ forwhichEx0,γ exp(ǫ−1Rτǫ,γ) < andthoseR>rγ forwhichEx0,γ exp(ǫ−1
ǫ,0 ∞ ǫ,0
Rτǫ,γ) = (see also [8, pp. 373–382]for additionaldiscussions). Note that estimating
∞ (cid:8) (cid:9) (cid:8)
the asymptotic exit rate with which the controlled-diffusionprocess Xǫ,γ(t) exits from the
(cid:9)
domain D is also related to a singularly perturbed eigenvalue problem. For example, the
asymptoticbehaviorforthe principaleigenvaluecorrespondingto the followingeigenvalue
problem
Lγǫ,∗0ψγ∗∗(x)+λγǫ,∗0ψγ∗∗(x)=0, ∀x∈D (1.11)
ψγ∗∗(x)=0, ∀x∈∂D (cid:27)
whereψγ∗∗ ∈Wl2o,cp(D)∩C(D∪∂D),forp>2,withψγ∗∗ >0onD,hasbeenwellstudiedin
thepast(e.g.,see[5]inthecontextofanasymptoticbehaviorfortheprincipaleigenfunction;
andsee[6]or[4]inthecontextofanasymptoticbehaviorfortheequilibriumdensity).
Before concludingthis section, let us introducethe followingdefinition for minimalaction
state-feedbacks,w.r.t.theactionfunctionalIγ(ϕ),whichisusefulforthedevelopmentofour
T
mainresult.
DEFINITION 1.1. The n-tuple (γ∗,γ∗,...,γ∗) Γ is said to be minimal action state-
1 2 n ∈
feedbacksif
(γ∗,γ∗,...,γ∗) argminIγ(ϕ). (1.12)
1 2 n ∈ T
Inthefollowingsection,wepresentourmainresults–whereweestablishaconnectionbe-
tweentheasymptoticexitratewithwhichthecontrolleddiffusionprocess(w.r.t. eachofthe
inputchannels)exitsfromthesetDandtheasymptoticbehaviorofprincipaleigenvaluesfora
familyofsingularlyperturbedellipticoperatorswithzeroboundaryconditionon∂D. Later,
such a formulation allows us to provide a sufficient condition for the existence of a Pareto
equilibrium(i.e.,asetofoptimalexitratesw.r.t.eachoftheinputchannels)fortheHJBequa-
tions–where thelatter correspondto a familyof nonlinearcontrolledeigenvalueproblems
(e.g.,see[11]or[9,Chapter8]foradditionaldiscussionsoneigenvalueproblems).
4 GETACHEWK.BEFEKADU
2. Main Results. In this section, we consider a family of SDEs (w.r.t. each of input
channels)
dXǫ,i(t)= A+ B γ Xǫ,i(t)dt+B u (t)dt+√ǫσ Xǫ,i(t) dW(t),
j j i i
j6=i
(cid:16) X (cid:17) Xǫ,i(0)=x , (cid:0)i=1,2(cid:1),...,n, (2.1)
0
whereu ()isa -valuedmeasurablecontrolprocesstotheith-channel(i.e.,anadmissible
i i
controlfro·mtheUset i Rri)suchthatforallt>s,W(t) W(s)isindependentofui(ν)
U ⊂ −
forν s(nonanticipativitycondition)and
≤
t
E u (τ)2dτ < , t s,
i
| | ∞ ∀ ≥
Zs
fori=1,2,...,n.
Next,let
λu = λu1, λu2, ..., λun , (2.2)
ǫ ǫ,1 ǫ,2 ǫ,n
with (cid:0) (cid:1)
1
λui = limsup logPx0,ui τǫ >t , i=1,2,...,n,
ǫ,i − t→∞ t ǫ,i i
(cid:8) (cid:9)
wherethe probabilityPx0,ui is conditionedon the initial pointx D and the admissible
ǫ,i 0 ∈
controlu . Moreover,τǫ isthefirstexittimeforthediffusionprocessXǫ,i(t)fromthe
i ∈Ui i
setD,i.e.,τǫ =inf t>0 Xǫ,i(t) / D fori=1,2,...,n.
i ∈
Further,letusintrod(cid:8)ucethe(cid:12)followingset(cid:9)
(cid:12)
Σǫ = λu Rn γ ,γ ,...,γ Γ . (2.3)
ǫ ∈ + 1 2 n ∈
n (cid:12)(cid:12)(cid:0) (cid:1) o
REMARK 2.1. NoticethatthesetΣǫ (w.r.t.(cid:12)theclassofstate-feedbacksΓ)isaclosedsubset
ofRn.
Definethepartialordering onΣǫ byλu′ λu′′,i.e.,
≺ ǫ ≺ ǫ
λuǫ,′11,λuǫ,′22,...,λuǫ,′nn ≺ λuǫ,′11′,λuǫ,′22′,...,λuǫ,′nn′ , (2.4)
u′ u′′ (cid:0) (cid:1) (cid:0) (cid:1)
ifλ i λ i foralli = 1,2,...,n,withstrictinequalityforatleastonei 1,2,...,n .
Furtǫh,ier≤,weǫ,siaythatλu∗ ΣǫisaParetoequilibrium(w.r.t.theclassofstate∈-fe{edbacksΓ)}if
thereisnoλu Σǫfoǫrw∈hichλu λu∗.
ǫ ∈ ǫ ≺ ǫ
Then, we have the following proposition that provides a sufficient condition for the exis-
tence ofa Pareto equilibriumλu∗ Σǫ (i.e., a setof optimalexitratesw.r.t. each ofinput
ǫ ∈
channels).
PROPOSITION 2.2. Supposethatthe statementin Definition1.1holdstrue atleast forone
n-tuple of state-feedbacks (γ∗,γ∗,...,γ∗) Γ. Then, there exists a Pareto equilibrium
λu∗ Σǫ (i.e.,asetofoptima1lex2itrateswn.r.t∈. eachofinputchannels)suchthat
ǫ ∈
λuǫ,∗11,λuǫ,∗22,...,λuǫ,∗nn ≺ λǫu,11,λuǫ,22,...,λuǫ,nn on Σǫ, (2.5)
(cid:0) (cid:1) (cid:0) (cid:1)
Onthecontrolledeigenvalueproblem 5
u∗
where the principaleigenvaluesλ i are the unique solutionsfor the HJB equationscorre-
ǫ,i
spondingtothefollowingfamilyofnonlinearcontrolledeigenvalueproblems
max ui ψ∗ (x,u )+λu∗i ψ∗ (x)=0 , x D
ui∈Ui Lǫ,i ui i ǫ,i ui ∀ ∈ (2.6)
n ψ∗ (x)=0, x ∂D o
ui ∀ ∈
withψ∗ W2,p(D) C(D ∂D),forp>2,withψ∗ >0onD,and
ui ∈ loc ∩ ∪ ui
ǫ
ui x = ▽ , A+ B γ∗ x+B u + tr σ(x)σT(x)▽2 , (2.7)
−Lǫ,i · · j6=i j j i i 2 ·
(cid:0)(cid:1)(cid:0) (cid:1) D (cid:0)(cid:1) (cid:0) X (cid:1) E n (cid:0)(cid:1)o
fori=1,2,...,n.
Proof. Supposethereexistsann-tupleofstate-feedbacks(γ∗,γ∗,...,γ∗) Γthatsatisfies
thestatementinDefinition1.1. Then,ourfirstclaimforψ∗ 1 W22,p D n C∈D ∂D ,with
ui ∈ loc ∩ ∪
p > 2, followsfromEquation(2.6)(cf. [11, Theorems1.1,1.2and1.4]). Thatis, ifu∗ for
i 1,2,...,N ,isameasurableselectorofargmax uiψ∗ x (cid:0),w(cid:1)ithx(cid:0) D. Th(cid:1)eni,by
∈{ } Lǫ,i ui ∈
theuniquenessclaimfortheeigenvalueproblem(cf. Equation(1.11)),wehave
(cid:8) (cid:0) (cid:1)(cid:9)
λu∗i = limsup1logPx0,u∗i τǫ >t , (2.8)
ǫ,i − t→∞ t ǫ,i i
n o
where the probability Px0,u∗i is conditioned on x , u∗ and γ∗,...,γ∗ ,γ∗ ,...,γ∗ .
ǫ,i 0 i 1 i−1 i+1 n
Moreover,foranyotheradmissiblecontrolsv ,wehave
i i
∈U (cid:0) (cid:1)
u∗iψ∗ x,v +λu∗iψ∗ x 0, t 0. (2.9)
Lǫ,i ui i ǫ,i ui ≤ ∀ ≥
LetQ Rd beasmoothbound(cid:0)edop(cid:1)endomainc(cid:0)on(cid:1)tainingD ∂D. Letψˆ andλˆvi bethe
princip⊂aleigenfunction-eigenvaluepairsfortheeigenvaluepro∪blemof vivion∂Q.ǫ,Fiurther,
Lǫ,i
let
τˆǫ =inf t>0 Xǫ,i(t) / Q . (2.10)
i ∈
n (cid:12) o
Then,undervi,fori 1,2,...,N andfors(cid:12)omexˆ D,wehave
∈{ } ∈
ψˆ xˆ Ex0,vi exp λˆvit ψˆ x(t) 1 τˆǫ >t ,
vi ≥ ǫ,i ( ǫ,i vi i )
n o
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
inf ψˆ x(t) exp λˆvi t Px0,u∗i τǫ >t . (2.11)
≥x∈D vi ǫ,i ǫ,i i
(cid:12) (cid:12) n o
Leadingto (cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
λˆvi limsup1logPx0,u∗i τǫ >t . (2.12)
ǫ,i ≥− t→∞ t ǫ,i i
n o
LettingQshrinktoDandusingProposition4.10of[11],thenwehaveλˆvi λu∗i.Thus,we
ǫ,i → ǫ,i
have
λu∗i limsup1logPx0,u∗i τǫ >t . (2.13)
ǫ,i ≤− t→∞ t ǫ,i i
n o
CombiningwithEquation(2.8),thisestablishestheoptimalityofu∗ andthefactthatλu∗i is
i ǫ,i
theoptimalexitrate.
6 GETACHEWK.BEFEKADU
Conversely,letv∗ beanadmissibleoptimalcontrol,thenwehave
i
Luǫ,∗iiψv∗i∗ x,vi∗ +λˆǫv,i∗iψv∗i∗ x =0 (2.14)
(cid:0) (cid:1) (cid:0) (cid:1)
and
Luǫ,∗iiψu∗∗i x,vi∗ +λuǫ,∗iiψu∗∗i x ≤0, ∀t>0, (2.15)
withλˆvi∗ =λu∗i,fori=1,2,..(cid:0).,n. (cid:1) (cid:0) (cid:1)
ǫ,i ǫ,i
Foruermthe1r,.4n(ao)t]ic).e tThhatenψ,u∗w∗i eisseaestchaalatrvm∗uisltiapllseooaf mψv∗ai∗xiamnidz,inagt msoemaseuxrˆab∈leDsele(ccft.or[i1n1,ETqhuea--
i
tion(2.6).
Ontheotherhand,forafixedsmallǫ,definethefollowingcontinuousfunctional(i.e.,autility
functionoveraclosedsetΣǫ Rn thatcanbeconvexfied)
∈ +
Σǫ λu Uǫ λu , ω, λu R, (2.16)
∋ ǫ 7→ ǫ ǫ ∈
(cid:0) (cid:1) (cid:10) (cid:11)
whereω >0fori=1,2,...,n.
i
NotethattheutilityfunctionUǫsatisfiesthepropertyUǫ λu′ <Uǫ λu′′ ,wheneverλu′
λu′′ onΣǫ w.r.t. theclassofstate-feedbacksΓ. Then,fromǫ theArrowǫ-Barankin-Blackǫwe≺ll
ǫ (cid:0) (cid:1) (cid:0) (cid:1)
theorem(e.g.,see[2]),onecanseethatthesetin
λu Σǫ ω >0, i=1,2,...,n, min ω, λu = ω, λu∗ (2.17)
ǫ ∈ ∃ i λu∈Σǫ ǫ ǫ
(cid:26) (cid:12) ǫ (cid:27)
(cid:12) (cid:10) (cid:11) (cid:10) (cid:11)
isdenseinthesetof(cid:12)allParetoequilibria.Thisfurtherimpliesthat,foranychoiceofω >0,
i
i = 1,2,...,n, the minimizer ω, λu = n ω λui over Σǫ satisfies the Pareto equi-
ǫ i=1 i ǫ,i
librium condition w.r.t. some minimal action state-feedbacks (γ∗,γ∗,...,γ∗) Γ. This
(cid:10) (cid:11) P 1 2 n ∈
completestheproofofProposition2.2.
We conclude this section with the following corollary that gives a lower-bound for rγ∗ in
(1.8) (w.r.t. some minimal action state-feedbacks (γ∗,γ∗,...,γ∗) Γ) (cf. [8, Corol-
1 2 n ∈
lary11.2,pp.377])
COROLLARY 2.3. Supposethatthe statementin Proposition2.2holds. Then, rγ∗ in (1.8),
w.r.t. theminimalactionstate-feedbacks(γ∗,γ∗,...,γ∗) Γ,satisfies3
1 2 n ∈
rγ∗ max ru∗i, (2.18)
≥i∈{1,2,...,n}
3Noticethat
rγ∗ =limsup inf 1 Iγ∗(ϕ(t)) ϕ(t)∈D∪∂D, t∈[0,T] ,
T→∞ ϕ(t)∈ΨT(cid:26) T (cid:12) (cid:27)
(cid:12)
(cid:12)
where
ITγ∗(ϕ(t))= 12Z0T(cid:20)dϕd(tt) −(cid:0)A+Xni=1Biγi∗(cid:1)ϕ(t)(cid:21)T(cid:16)σ(ϕ(t))σT(ϕ(t))(cid:17)−1
×(cid:20)dϕd(tt) −(cid:0)A+Xni=1Biγi∗(cid:1)ϕ(t)(cid:21)dt.
Onthecontrolledeigenvalueproblem 7
whereru∗i isgivenby
ru∗i = ǫlogλu∗i as ǫ 0. (2.19)
− ǫ,i →
REMARK 2.4. Here,weremarkthat,foreachi 1,2,...,n ,wehave
∈{ }
λγ λu∗i 0, ǫ (0,ǫ∗), (γ ,γ ,...,γ ) Γ,
ǫ,0− ǫ,i ≥ ∀ ∈ 1 2 n ∈
whichsuggeststhattheabovecorollaryisusefulforselectingthemostappropriateminimal
actionstate-feedbacksfrom Γ thatconfinethe diffusionprocessXǫ(t) to the prescribed set
Dforalongerduration(cf. Equations(1.6)and(2.2)).
Appendix. The followinglemma (whoseproofis an adaptationof [10, Theorem2.1])
providesaconditionunderwhichthemaximalinvariantsetΛ(D)isnonempty.
LEMMA 2.5.
(a) Ifforsomex D,
0
∈
1
limsuplimsup logPx0,γ τǫ,γ >t < , (A.1)
− ǫ→0 t→∞ t ǫ,0 ∞
n o
thenthemaximalinvariantsetΛ(D)isnonempty.
(b) Ifforsomex D,
0
∈
limsuplogEx0,γ τǫ,γ = , (A.2)
ǫ→0 ǫ,0 ∞
n o
thenthemaximalinvariantsetΛ(D)isnonempty.
Proof. SupposethatthemaximumclosedinvariantsetΛ(D)isempty(i.e.,theinvariantset
forStinD ∂Disempty).Then,thereexistsanopenboundeddomainD˜ D ∂Dsuch
thatthecorr∪espondingsetΛ(D˜)isalsoempty. ⊃ ∪
Note that it is easy to check that if D D , then Λ(D ) Λ(D ). Take the following
2 1 2 1
⊂ ⊂
sequence D ofopendomainssuchthat
m
(cid:8) (cid:9)
D D D and D =D ∂D. (A.3)
1 2 3 m
⊃ ⊃ ⊃··· m≥1 ∪
\
IfΛ(D )= forallm 1,then
m
6 ∅ ≥
Λ= Λ(D ). (A.4)
m
m≥1
\
Moreover,sinceΛ(D )isclosed,wehave
m
Λ(D ) Λ(D ) Λ(D ) . (A.5)
1 2 3
⊃ ⊃ ⊃···
NotethatΛisaninvariantclosedsetfortheunperturbedsystem
n
x˙γ(t)= A+ B γ xγ(t), (γ ,γ ,...,γ ) Γ, xγ =x
i=1 i i 1 2 n ∈ 0 0
(cid:16) X (cid:17)
8 GETACHEWK.BEFEKADU
andΛ D ∂D. Thus, =Λ Λ(D). Thiscontradictsourearlierassumption. Then,for
⊃ ∪ ∅6 ⊂
somem 1,wehave
0
≥
Λ(D )= . (A.6)
m0 ∅
LetD˜ =D ,foranyǫ (0,ǫ∗)andx D˜ ∂D˜,introducethefollowing
m0 ∈ 0 ∈ ∪
n
τǫ,γ =inf t>0 exp A+ B γ t x / D˜ ∂D˜ . (A.7)
D˜ i=1 i i 0 ∈ ∪
n (cid:12) n(cid:16) X (cid:17) o o
Then,wecanshowthatτǫ,γ < (cid:12) foranyx D˜ ∂D˜.
D˜ ∞ 0 ∈ ∪
Noticethat,ifτǫ,γ = ,thenD˜ ∂D˜ exp A+ n B γ t x forallt 0. Then,
D˜ ∞ ∪ ∋ i=1 i i 0 ≥
wehavethefollowing n(cid:16) (cid:17) o
P
n
exp A+ B γ t x , t 0 Λ(D˜)= , (A.8)
i i 0
i=1 ≥ ⊂ ∅
n n(cid:16) X (cid:17) o o
whichshowthatτǫ,γ isfinite.
D˜
Notethat,fromupper-semicontinuityofτǫ,γ,wehave
D˜
T = sup τǫ,γ < . (A.9)
D˜ ∞
x0∈D˜∪∂D˜
Moreover,foranyδ >0,let
n
lim sup Ex0,γ dist Xǫ,γ(t),exp A+ B γ t x >δ =0, t 0. (A.10)
ǫ→0x0∈D ǫ,0 i=1 i i 0 ≥
n (cid:16) n(cid:16) X (cid:17) o (cid:17) o
FromEquations(A.7)–(A.10),wehave
sup Px0,γ τǫ,γ >T 0 as ǫ 0. (A.11)
ǫ,0 → →
x0∈D
n o
Then,usingtheMarkovproperty,wehave
Px0,γ τǫ,γ >ℓT =Ex0,γχ Ex0,γχ Ex0,γχ ,
ǫ,0 ǫ,0 τǫ,γ>T ǫ,0 τǫ,γ>T ··· ǫ,0 τǫ,γ>T
n o ℓ
sup Px0,γ τǫ,γ >T , (A.12)
≤ ǫ,0
x0∈D
(cid:16) n o(cid:17)
whereχ istheindicatorfortheevent .
A
A
SincePx0,γ τǫ,γ >t decreasesint,thenwehave
ǫ,0
n o
1 1
limsup logPx0,γ τǫ,γ >t log sup Px0,γ τǫ,γ >T . (A.13)
t→∞ t ǫ,0 ≤ T x0∈D ǫ,0
n o (cid:16) n o(cid:17)
TakingintoaccountEquation(A.11),then,foranyx D,wehavethefollowing
0
∈
1
limsup logPx0,γ τǫ,γ >t as ǫ 0. (A.14)
t→∞ t ǫ,0 →−∞ →
n o
Hence,ourassumptionthatΛ(D)= isinconsistent.
∅
Onthecontrolledeigenvalueproblem 9
Toproofpart(ii),noticethat
∞
Ex0,γ τǫ,γ T Px0,γ τǫ,γ >(ℓ 1)T . (A.15)
ǫ,0 ≤ ℓ=1 ǫ,0 −
n o X n o
AssumptionΛ(D)= gives,inviewofEquations(A.11),(A.12)and(A.13)forsufficiently
∅
smallǫ>0andforanyx D,
0
∈
∞ ℓ−1
Ex0,γ τǫ,γ T sup Px0,γ τǫ,γ >T
ǫ,0 ≤ ℓ=1 x0∈D ǫ,0
n o X h n oi
< , (A.16)
∞
whichcontradictswithEquation(A.2). ThiscompletestheproofofLemma2.5.
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