Elec onic Jou nal o Quali a i e Theo y o Di e en ial Equa ions
P oc. 10 h Coll. Quali a i e Theo y o Di . Equ. (July 1–4, 2015, Szeged, Hunga y)
2016, No. 7, 1–27; doi: 10.14232/ejq de.2016.8.7 h p://www.ma h.u-szeged.hu/ejq de/
A su ey on impulsi e dynamical sys ems
E e aldo Mello Bono o1,Ma heus C. Bo olan2,
Tomás Ca aballoB3and Rodol o Collega i1
1Ins i u o de Ciências Ma emá icas e de Compu ação, Uni e sidade de São Paulo,
Campus de São Ca los, Caixa Pos al 668, São Ca los, SP, B azil
2Depa amen o de Ma emá ica, Uni e sidade Fede al de San a Ca a ina, Campus T indade,
88040-900, Flo ianópolis, B azil
3Depa amen o de Ecuaciones Di e enciales y Análisis Numé ico EDAN, Uni e sidad de Se illa,
Se illa, Spain
Appea ed 11 Agus 2016
Communica ed by Tibo K isz in
Abs ac . In his su ey we p o ide an in oduc ion o he heo y o impulsi e dy-
namical sys ems in bo h he au onomous and nonau onomous cases. In he o me , we
will show wo di e en app oaches which ha e been p oposed o analyze such kind
o dynamical sys ems which can expe ience some ab up changes (impulses) in hei
e olu ion. Bu , unlike he au onomous amewo k, he nonau onomous one is being
de eloped igh now and some p og ess is being ob ained o e he ecen yea s. We
will p o ide some esul s on how he heo y o au onomous impulsi e dynamical sys-
ems can be ex ended o co e such nonau onomous si ua ions, which a e mo e o en
o occu in he eal wo ld.
Keywo ds: impulsi e dynamical sys ems, global a ac o s, nonau onomous dynamical
sys ems, cocycle a ac o s, Na ie –S okes equa ion.
2010 Ma hema ics Subjec Classi ica ion: 35B41, 34A37, 35R12.
1 In oduc ion
The heo y o impulsi e di e en ial equa ions (IDE, o sho ) desc ibes he e olu ion o sys-
ems whe e he con inuous de elopmen o a p ocess is in e up ed by ab up changes o
s a e. These sys ems a e modeled by di e en ial equa ions which desc ibe he pe iod o con-
inuous a ia ion o s a e and condi ions which desc ibe he discon inui ies o i s kind o he
solu ion o o i s de i a i es a he momen s o impulses. Many eal wo ld p oblems can ex-
pe ience ab up ex e nal o ces which can change comple ely hei dynamics. Fo ins ance, an
example o a eal wo ld p oblem ha can be ep esen ed by an impulsi e di e en ial equa ion
is a medicine in ake, whe e he use mus ake egula doses o he medicine, which causes
ab up changes in he amoun o medicine in hei body, o con ol he disease o making i
disappea . Examples ha model eal wo ld p oblems in science and echnology can be ound
BCo esponding au ho . Email: [email p o ec ed]
2E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
in [1,13,19,20]. The eade is also e e ed o [2,3,26] o ob ain mo e de ails abou he heo y
o IDEs, o ins ance, esul s conce ning exis ence and uniqueness o solu ions, dependence o
solu ions on ini ial alues, a ia ion o pa ame e s, oscilla ion and s abili y.
As poin ed ou in [2,26] he e exis di e en kinds o impulses, o ins ance, sys ems wi h
impulses a ixed imes and sys ems wi h impulses a a iable imes. Impulses ha a y in
ime a e mo e a ac i e due o hei complexi y, applicabili y in eal wo ld p oblems, and,
mo eo e , he impulses may occu due o condi ions on he phase space and no in ime.
As an example, we may ci e he billia d- ype sys em which can be modeled by di e en ial
sys ems wi h impulses ac ing on he i s de i a i es o he solu ions. Indeed, he posi ions o
he colliding balls do no change a he momen s o impac (impulse), bu hei eloci ies gain
ini e inc emen s ( he eloci y will change acco ding o he posi ion o he ball).
Solu ions o IDEs wi h impulses a a iable ime may gene a e “impulsi e dynamical sys-
ems” ( amily o piecewise con inuous unc ions ha sa is y he iden i y and semig oup p op-
e ies), o ins ance, when he di e en ial equa ion is au onomous. As in he heo y o IDEs,
he case o impulsi e dynamical sys ems wi h impulses ha a y in ime is mo e di icul o
handle since we do no know p e iously he ime o impulses. Howe e , i p o ides us an
e ec i e ool o desc ibe mo e ypes o discon inuous mo ions.
The heo y o impulsi e dynamical sys ems is a new chap e o he heo y o opological
dynamical sys ems and i was s a ed by Rozko in he pape s [27,28], whe e he in oduced
se e al no ions o impulsi e sys ems wi h impulses a ixed imes. In he ea ly 90’s Kaul (see
[24,25]) cons uc ed he ma hema ical base o his heo y wi h impulses a a iable imes,
and has been ollowed by se e al au ho s in o de o de elop he heo y which is known up
o da e. Fo ins ance, we would like o men ion he pape s by Ciesielski (see [16–18]), whe e
i is analyzed he con inui y o he unc ion φ(see 2.2) ha desc ibes “ he ime o eaching
impulse poin s”, and ecen ly he wo ks by Bono o and his collabo a o s (see [6–10]) whe e
he heo y has been in es iga ed.
Th oughou his wo k, an impulsi e dynamical sys em is a dynamical sys em ha pos-
sesses impulses depending on he s a e (and no on he ime), ha is, he e is a se in he
phase space which is esponsible by he discon inui ies o he solu ions o he sys em. I
is wo h men ioning ha he heo y p esen ed in his wo k p o ides a di e en app oach
om he heo y p esen ed in [21], whe e he au ho ca ies ou a s udy o some ypes o dis-
con inuous di e en ial equa ions. Roughly speaking, Filippo conside s in [21] he equa ion
x0= ( ,x), whe e he igh -hand side unc ion is discon inuous and i is assumed o sa is y
some Ca a héodo y condi ions. Also, he solu ions in his amewo k ha e o be absolu ely
con inuous, which is ano he ele an de ail ha makes Filippo ’s heo y di e en om he
one p esen ed in [2,3,26] and he heo y p esen ed he e, whe e he solu ions can be (and
usually a e) discon inuous.
We aim o p o ide a su ey on he heo y o impulsi e dynamical sys ems in bo h he
au onomous and nonau onomous ields. We s a wi h he au onomous amewo k which
has being s udied o e he las yea s and, o he i s pa o his pape , we will ecall some
esul s es ablished in he pape [5]. In his wo k he au ho s p opose a new app oach o
he impulsi e au onomous heo y, by conside ing p ecompac a ac o s and poin ing ou
se e al imp o emen s ha his p ecompac app oach p o ides, when compa ing wi h he
p e ious heo y in his amewo k. Examples o illus a e he impulsi e au onomous heo y
a e desc ibed in [5], one o hem is ep oduced in his su ey, a he end o he sec ion de o ed
o he au onomous case (see Example 2.22).
To s a o , in Sec ion 2we include some basic de ini ions om he con inuous au-
A su ey on impulsi e dynamical sys ems 3
onomous dynamical sys ems heo y in o de o in oduce he de ini ion o impulsi e dy-
namical sys em. In he sequel we p esen some echnical de ini ions and esul s, known as
“ ube condi ions”, ha is impo an in he de elopmen o his heo y. Then, be o e p esen ing
an impulsi e au onomous example, we in oduce he concep o omega limi se s, which is he
key o cons uc he global a ac o , as well as some esul s on he in a iance and a ac ion
in o de o ob ain an exis ence esul o he global a ac o .
In Sec ion 3, we analyze he nonau onomous case, aking in o accoun ha a comple e
desc ip ion o he esul s and hei p oo s can be ound in ou pape [4], while in his su ey
we only in end o p o ide he main ideas o he new heo y highligh ing he di icul ies ha
one can ha e in dealing wi h his much mo e complica ed nonau onomous si ua ion. Needless
o say ha mos p oblems in he eal wo ld a e, by hei own na u e, nonau onomous (o
e en s ochas ic) and, when we wish o ma hema ically analyze hem, we usually app oxima e
hose p oblems by some au onomous models o simpli y he s udy. Howe e , e en being
he au onomous amewo k e y use ul, and p o iding a g ea amoun o esul s, i does no
ake in o accoun he whole ichness o nonau onomous p oblems. In [11,12], one can ind
examples o illus a e how di e en he au onomous and nonau onomous se ings can be.
Men ioning again he medicine in ake example, we could no expec ha he ac ion o he
medicine in he use body depends only on he elapsed ime bu also he ini ial and inal
imes mus play hei ole in he e olu ion o he sys em.
We ollow he same s uc u e han in he au onomous pa , by s a ing wi h a b ie in-
oduc ion on he con inuous nonau onomous dynamical sys ems in o de o de ine he im-
pulsi e nonau onomous dynamical sys ems. We also s a e a esul (see Theo em 3.9) ha is
impo an o ans e p ope ies om he impulsi e skew-p oduc semi low (au onomous) o
he impulsi e nonau onomous dynamical sys em. Nex we p esen he nonau onomous e -
sion o he “ ube condi ions” and some con e gence p ope ies, which a e mo e gene al han
he i s ones because ake in o accoun a second a iable ( he ibe s). Then we de ine he
no ion o impulsi e cocycle a ac o and impulsi e pullback omega limi , and also p esen
some esul s abou in a iance and a ac ion. We would like o men ion ha he de ini ion
o impulsi e pullback omega limi se in oduced in [4] is a li le di e en om he p e ious
one, and his di e ence appea s na u ally when we s a de eloping he impulsi e nonau-
onomous heo y, since in he impulsi e scena io, he con e gence esul s a e ob ained wi h
some “co ec ion imes” (see P oposi ion 3.12). To conclude, we p esen , unde sui able condi-
ions, a esul on he exis ence o impulsi e cocycle a ac o o an impulsi e nonau onomous
dynamical sys em and an example, bo owed om [4, Sec ion 7], whe e a nonau onomous
2D-Na ie –S okes equa ion unde impulses condi ions is conside ed.
Finally, some conclusions, commen s and u u e lines o esea ch a e included in Sec ion 4.
2 Impulsi e dynamical sys ems
To in oduce he heo y o impulsi e dynamical sys em, we i s ecall, e y b ie ly, he heo y
o con inuous au onomous dynamical sys ems (o simply, semig oups).
Le (X,d)be a me ic space and R+be he se o nonnega i e eal numbe s. A semig oup
in Xis a amily o mappings {π( ): ⩾0}, indexed on R+, sa is ying
(i) π(0)x=x o all x∈X;
(ii) π( +s) = π( )π(s) o all ,s⩾0;
(iii) he map R+×X3( ,x)7→ π( )xis con inuous.
4E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
A se A⊂Xis called π-in a ian unde {π( ): ⩾0}i π( )A=A o all ⩾0. Also A
is π-posi i ely (nega i ely) in a ian i π( )A⊆A(π( )A⊇A), o all ⩾0.
Gi en wo subse s A,B⊆X, we say ha Aπ-a ac s Bi
lim
→+∞dH(π( )B,A) = 0,
whe e dH(·,·)deno es he Hausdo semidis ance be ween wo se s, i.e.,
dH(C,D) = sup
x∈C
in
y∈D
d(x,y).
A se A ⊂ Xis called a global a ac o o he semig oup {π( ): ⩾0}i i is compac ,
π-in a ian and π-a ac s all bounded subse s o X.
In his sec ion, we p esen he de ini ions and basic p ope ies o he impulsi e dynamical
sys ems heo y (see [5–7,16,17] o mo e de ails).
Le {π( ): ⩾0}be a semig oup in X. Fo each D⊆Xand J⊆R+we de ine
F(D,J) = [
∈J
π( )−1(D).
A poin x∈Xis called an ini ial poin i F(x, ) = ∅ o all >0.
Now we a e able o de ine he impulsi e dynamical sys ems. An impulsi e dynamical
sys em (IDS, o sho ) (X,π,M,I)consis s o a semig oup {π( ): ⩾0}on a me ic space
(X,d), a nonemp y closed subse M⊆Xsuch ha o e e y x∈M he e exis s ex>0 such
ha
F(x,(0, ex)) ∩M=∅and [
∈(0,ex)
{π( )x} ∩ M=∅, (2.1)
and a con inuous unc ion I:M→Xwhose ac ion will be explained below in he desc ip ion
o he impulsi e ajec o y. Condi ion (2.1) is ou lined in he nex igu e.
Figu e 2.1: The low o he semig oup {π( ): ⩾0}is, in some sense, ans e sal o M.
The se Mis called impulsi e se and he unc ion Iis called impulsi e unc ion. We also
de ine
M+(x) = [
>0
π( )x!∩M
A su ey on impulsi e dynamical sys ems 5
and he unc ion φ:X→(0, +∞]by
φ(x) = (s, i π(s)x∈Mand π( )x/∈M o 0 < <s,
+∞, i M+(x) = ∅.(2.2)
I M+(x)6=∅, he alue φ(x) ep esen s he i s posi i e ime such ha he ajec o y o x
mee s M. In his case, we say ha he poin π(φ(x))xis he impulsi e poin o x.
Rema k 2.1. The de ini ion o he unc ion φabo e makes sense hanks o he ollowing esul .
See [5,24].
P oposi ion 2.2. Le (X,π,M,I)be an IDS and x ∈X. I M+(x)6=∅ hen he e exis s s >0such
ha π(s)x∈M and π( )x/∈M o 0< <s.
Now le us cons uc he impulsi e ajec o y o he IDS.
De ini ion 2.3. The impulsi e ajec o y o x∈Xby he IDS (X,π,M,I)is a map ˜
π(·)x
de ined in an in e al Jx⊆R+, 0 ∈Jx, aking alues in Xwhich is gi en induc i ely by he
ollowing ule: i M+(x) = ∅, hen ˜
π( )x=π( )x o all ∈R+. Howe e , i M+(x)6=∅
hen we deno e x=x+
0and de ine ˜
π(·)xon [0, φ(x+
0)] by
˜
π( )x=(π( )x+
0, i 0 ⩽ <φ(x+
0),
I(π(φ(x+
0))x+
0), i =φ(x+
0).
Now le s0=φ(x+
0),x1=π(s0)x+
0and x+
1=I(π(s0)x+
0). In his case s0<+∞and he p ocess
can go on, bu now s a ing a x+
1. I M+(x+
1) = ∅, hen we de ine ˜
π( )x=π( −s0)x+
1 o
s0⩽ <+∞and in his case φ(x+
1) = +∞. Howe e , i M+(x+
1)6=∅we de ine ˜
π(·)xon
[s0,s0+φ(x+
1)] by
˜
π( )x=(π( −s0)x+
1, i s0⩽ <s0+φ(x+
1),
I(π(φ(x+
1))x+
1), i =s0+φ(x+
1).
Now le s1=φ(x+
1),x2=π(s1)x+
1and x+
2=I(π(s1)x+
1). Assume now ha ˜
π(·)xis de ined
on he in e al [ n−1, n]and ha ˜
π( n)x=x+
n, whe e 0=0 and n=∑n−1
i=0si o n∈N. I
M+(x+
n) = ∅, hen ˜
π( )x=π( − n)x+
n o n⩽ <+∞and φ(x+
n) = +∞. Howe e , i
M+(x+
n)6=∅, hen we de ine ˜
π(·)xon [ n, n+φ(x+
n)] by
˜
π( )x=(π( − n)x+
n, i n⩽ < n+φ(x+
n),
I(π(φ(x+
n))x+
n), i = n+φ(x+
n).
Now le sn=φ(x+
n),xn+1=π(sn)x+
nand x+
n+1=I(π(sn)x+
n). This p ocess ends a e a
ini e numbe o s eps i M+(x+
n) = ∅ o some n∈N, o i may p oceed inde ini ely, i
M+(x+
n)6=∅ o all n∈Nand in his case ˜
π(·)xis de ined in he in e al [0, T(x)), whe e
T(x) = ∑+∞
i=0si.

6E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
Figu e 2.2: Sys em (X,π)wi h Figu e 2.3: Impulsi e ajec o y o x
con inuous ajec o ies. in he sys em (X,π,M,I).
Rema k 2.4.
• We will always assume ha all impulsi e ajec o ies exis o all ime ⩾0, i.e., T(x) =
+∞ o all x∈X, since we a e in e es ed in he asymp o ic beha io o impulsi e
dynamical sys ems.
• A simple consequence o he de ini ion o impulsi e ajec o ies is ha i we assume
ha I(M)∩M=∅, hen no poin x∈Mis in any impulsi e ˜
π- ajec o y, excep i he
ajec o y s a s a x.
The de ini ions o ˜
π-in a iance and ˜
π-a ac ion a e analogous o he no ions o π-in a i-
ance and π-a ac ion, espec i ely, simply eplacing πby ˜
π.
2.1 Tube condi ions on impulsi e dynamical sys ems
In o de o ob ain some esul s in he impulsi e heo y o dynamical sys ems ( o example,
in a iance and a ac ion esul s), we mus ensu e ha he con inuous semi low possesses
a nice beha io nea he impulsi e se Mand, o his pu pose we in oduce he so-called
“ ube condi ions”. They a e impo an o deduce a esul ensu ing he nega i e in a iance o
impulsi e ω-limi s. Fo mo e de ails and p oo s see also [5,16,18].
De ini ion 2.5. Le {π( ): ⩾0}be a semig oup on X. A closed se Scon aining x∈Xis
called a sec ion h ough xi he e exis s λ>0 and a closed subse Lo Xsuch ha :
(a) F(L,λ) = S;
(b) F(L,[0, 2λ]) con ains a neighbo hood o x;
(c) F(L,ν)∩F(L,ζ) = ∅, i 0 ⩽ν<ζ⩽2λ.
We say ha he se F(L,[0, 2λ]) is a λ- ube (o simply a ube)and he se Lis a ba .
A su ey on impulsi e dynamical sys ems 7
2λ
λ
π(x,λ)
x
LS
q q
Figu e 2.4: Tube F(L,[0, 2λ]).
De ini ion 2.6. Le (X,π,M,I)be an IDS. We say ha a poin x∈Msa is ies he s ong ube
condi ion (STC), i he e exis s a sec ion S h ough xsuch ha S=F(L,[0, 2λ]) ∩M. Also,
we say ha a poin x∈Msa is ies he special s ong ube condi ion (SSTC)i i sa is ies
STC and he λ- ube F(L,[0, 2λ]) is such ha F(L,[0, λ]) ∩I(M) = ∅.
We inish his pa p esen ing wo p oposi ion. The i s one yields o a be e unde s and-
ing abou he beha io o impulsi e ajec o ies nea he impulsi e se Mand will be use ul o
ob ain some esul s la e . I s a es ha he impulsi e low ˜
π( )canno each he “ igh side” o
he impulsi e se M o la ge alues o . The second p oposi ion summa izes some impo an
con e gence esul s ha also will be use ul o ob ain u he esul s. Fo de ails and p oo s
he eade may see [5].
P oposi ion 2.7 ([5]).Le (X,π,M,I)be an IDS such ha I(M)∩M=∅and le y ∈M sa is y
SSTC wi h λ- ube F(L,[0, 2λ]). Then ˜
π( )X∩F(L,[0, λ]) = ∅ o all >λ.
P oposi ion 2.8. Le (X,π,M,I)be an IDS.
(i) Suppose ha I(M)∩M=∅and each poin o M sa is ies STC. Le x ∈X M and le
{xn}n∈Nbe a sequence in X such ha xnn→+∞
−→ x. Then, gi en ⩾0, he e exis s a sequence
{ηn}n∈N⊆[0, +∞)such ha ηnn→+∞
−→ 0and ˜
π( +ηn)xnn→+∞
−→ ˜
π( )x.
(ii) Suppose ha each poin in M sa is ies STC. Le x ∈X M and le {xn}n∈Nbe a sequence
in X M such ha xnn→+∞
−→ x. Then i αnn→+∞
−→ 0and αn⩾0, o all n ∈N, we ha e
˜
π(αn)xnn→+∞
−→ x.
(iii) Le z ∈M sa is y STC wi h λ- ube F(L,[0, 2λ]). Assume ha he e exis s a sequence {zn}n∈N
such ha zn∈F(L,(λ, 2λ]) and znn→+∞
−→ z. Then he e exis a subsequence {znk}k∈No
{zn}n∈Nand a sequence {ek}k∈Nsuch ha ek>0and ek→0as k →+∞, yk=π(ek)znk∈
M, φ(znk) = ekand yk
k→+∞
−→ z.
2.2 A ac o s
We s a wi h a i s app oach abou a ac o s o he IDS. In [6], he au ho s p opose he
ollowing de ini ion o global a ac o o an impulsi e dynamical sys em.
8E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
De ini ion 2.9. A compac subse Ao Xis a global a ac o o an IDS (X,π,M,I)i he
ollowing condi ions a e ul illed:
(i) A ∩ M=∅;
(ii) Ais ˜
π-in a ian ;
(iii) A˜
π-a ac s all bounded subse s o X.
Rema k 2.10.
1. This de ini ion is consis en wi h he no ion o a global a ac o o semig oups, ha is,
when M=∅, bo h de ini ions coincide; and in ac , his no ion o a global a ac o is
use ul o desc ibe he asymp o ic dynamics o ˜
πin many cases.
2. Since Ais a compac se and Mis a closed se , condi ion (i) implies ha he e exis s
a posi i e dis ance be ween Aand M. Then he asymp o ic beha io o he impulsi e
dynamical sys ems is quali a i ely no di e en om he asymp o ic beha io o he
o iginal dynamical sys em, hus, his no ion does no conside some IDS. Le us see an
example bo owed om [5] o illus a e hese ac s.
Example 2.11. Conside he ollowing con inuous di e en ial equa ion
˙
x=(1, i x<0,
1−x, i x≥0, (2.3)
wi h he ini ial condi ion x(0) = x0∈Rand conside he ac ion o he impulsi e unc ion
I(0) = −1. The solu ions o (2.3) wi hou he ac ion o Ia e gi en by
π( )x0=






+x0,x0<0, ∈[0, −x0),
−e− −x0+1, x0<0, ∈[−x0,+∞),
(x0−1)e− +1, x0⩾0, ∈[0, +∞).
This p oblem has only one bounded in a ian se ; namely he asymp o ically s able equilib-
ium solu ion {1}, and i is also he global a ac o o (2.3). Now, he solu ions o (2.3) wi h
he ac ion o I, a e gi en by
˜
π( )x0=






+x0,x0<0, ∈[0, −x0),
+x0−n,x0<0, ∈[−x0+n−1, −x0+n),n∈N,
(x0−1)e− +1, x0≥0, ∈[0, +∞).
(2.4)
We can see ha he dynamics is qui e di e en , since he e appea ed he “impulsi e pe i-
odic o bi ” [−1, 0). No e ha in his case he e is no subse o Rsa is ying all he condi ions
o De ini ion 2.9. Bu we can dis inguish some in e es ing se s:
• The se A1= [−1, 0)∪ {1}is ˜
π-in a ian and ˜
π-a ac ing bounded se s, A1∩M=∅,
bu A1is no compac .
• The se A2= [−1, 0]∪ {1}˜
π-a ac s bounded se s, A2is compac , bu A2∩M6=∅and
A2is nei he ˜
π-posi i ely no ˜
π-nega i ely in a ian .
A su ey on impulsi e dynamical sys ems 9
• The se A3= [−1, 1]˜
π-a ac s bounded se s, A3is compac , i is ˜
π-posi i ely in a ian ,
bu i is no ˜
π-nega i ely in a ian and A3∩M6=∅.
Inspi ed by he ideas om his las example, in [5] he au ho s p o ide ano he de ini ion
o global a ac o , in o de o co e a la ge class o impulsi e dynamical sys ems. Thei
de ini ion is he ollowing.
De ini ion 2.12. A subse A ⊂ Xwill be called a global a ac o o he IDS (X,π,M,I)i i
sa is ies he ollowing condi ions:
(i) Ais p ecompac and A=A M;
(ii) Ais ˜
π-in a ian ;
(iii) A˜
π-a ac s bounded subse s o X.
Rema k 2.13.
• The main di e ence be ween De ini ion 2.12 and De ini ion 2.9 is he compac ness. In
De ini ion 2.12, he global a ac o does no need o be compac and now he a ac o
can “ ouch” he impulsi e se M, while compac se s which do no in e sec Mha e o
be a a posi i e dis ance om M.
• I is easy o see ha , wi h De ini ion 2.12, i Aexis s, i is unique.
• We ecall now ha a unc ion ψ:R→Xis a global solu ion o ˜
πi
˜
π( )ψ(s) = ψ( +s), o all ⩾0 and s∈R.
Mo eo e , i ψ(0) = xwe say ha ψis a global solu ion h ough x. Then, wi h De ini-
ion 2.12, i he IDS (X,π,M,I)possesses a global a ac o Aand I(M)∩M=∅we
ha e
A={x∈X: he e exis s a bounded global solu ion o ˜
π h ough x}.
Coming back o Example 2.11, we can see ha se A1is he global a ac o o he IDS,
acco ding o De ini ion 2.12. This example shows how di e en he con inuous and he im-
pulsi e dynamics can be, as well as ha a e y la ge amoun o impulsi e dynamical sys ems,
which do no i he heo y in [6], can now be conside ed.
In wha ollows we will p esen some de ini ions and esul s o ensu e he exis ence o a
global a ac o o an IDS (X,π,M,I)as de ined in De ini ion 2.12. We will include a ske ch
o some p oo s and o all he de ails he eade may see [5].
We s a gi ing he de ini ion o impulsi e ω-limi .
De ini ion 2.14. We ep esen he impulsi e posi i e o bi o x∈Xs a ing a s⩾0 by he
se
˜
γ+
s(x) = {˜
π( )x: ⩾s}.
Also we se ˜
γ+(x) = γ+
0(x).
Gi en a subse B⊆Xwe de ine ˜
γ+
s(B) = Sx∈B˜
γ+
s(x)and we de ine he impulsi e ω-limi
o Bas he se
˜
ω(B) =
⩾0
˜
γ+
(B)
16 E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
P oposi ion 3.12. Le [(ϕ,θ)(X,Σ),M,I]be an INDS.
(i) Suppose ha I(M)∩M=∅and each poin o M sa is ies ϕ-STC. Le also x ∈X M and
{xn}n∈Nbe a sequence in X such ha xnn→+∞
−→ x. Then, gi en ⩾0,σ∈Σand a sequence
{σn}n∈N⊂Σwi h σnn→+∞
−→ σ, he e exis s a sequence {ηn}n∈N⊆[0, +∞)such ha ηnn→+∞
−→
0and ˜
ϕ( +ηn,σn)xnn→+∞
−→ ˜
ϕ( ,σ)x.
(ii) Suppose ha each poin in M sa is ies ϕ-STC, x ∈X M, σ∈Σ,{xn}n∈Nbe a sequence in
X M such ha xnn→+∞
−→ x and σnn→+∞
−→ σ. Then i αnn→+∞
−→ 0and αn⩾0, o all n ∈N, we
ha e ˜
ϕ(αn,σn)xnn→+∞
−→ x.
(iii) Assume ha each x ∈M sa is ies ϕ-SSTC and I(M)∩M=∅. Le ˆ
B be a nonau onomous se ,
{ n}n∈N⊂R+,σ∈Σ,{ηn}n∈N⊂R+and {xn}n∈Nbe sequences such ha ηnn→+∞
−→ 0,
xn∈B(θ− nσ) o each n ∈N. I {˜
ϕ( n+ηn,θ− nσ)xn}n∈Nis con e gen wi h limi
y∈M and {en}n∈N⊂R+is a sequence wi h enn→+∞
−→ 0, hen he e is a subsequence
{˜
ϕ( nk+ηnk,θ− nkσ)xnk}k∈Nsuch ha φ(˜
ϕ( nk+ηnk,θ− nkσ)xnk,θηnkσ)k→+∞
−→ 0and ei he
˜
ϕ(enk,θηkσ)˜
ϕ( nk+ηnk,θ− nkσ)xnk
k→+∞
−→ y
o
˜
ϕ(enk,θηkσ)˜
ϕ( nk+ηnk,θ− nkσ)xnk
k→+∞
−→ I(y).
In pa icula ,
˜
ϕ(αk,θηkσ)˜
ϕ( nk+ηnk,θ− nkσ)xnk
k→+∞
−→ I(y),
whe e αk=φ(˜
ϕ( nk+ηnk,θ− nkσ)xnk,θηnkσ).
3.2 Impulsi e cocycle a ac o s
He e we will p esen he no ion o a ac o o an INDS (impulsi e cocycle a ac o ), de ine
and es ablish some p ope ies o he impulsi e omega limi se s in o de o ob ain an exis ence
esul o impulsi e cocycle a ac o s. We will see ha his no ion o a ac o is no a na u al
gene aliza ion o he global a ac o gi en in [5] (see De ini ion 2.12), since he esul s on
he in a iance in he impulsi e case canno be ob ained as a na u al gene aliza ion o he
con inuous case. A mo e comple e analysis can be ound in [4] and some o he p oo s will be
ep oduced he e o illus a e he echniques.
Le us in oduce he no ion o a ac o o an INDS (wi h espec o a uni e se).
De ini ion 3.13. Gi en a uni e se D, a compac nonau onomous se ˆ
Ais called a D-impulsi e
cocycle a ac o o he INDS [(ϕ,θ)(X,Σ),M,I]i :
(i) ˆ
A M={A(σ) M}σ∈Σis ˜
ϕ-in a ian ;
(ii) ˆ
Ais (˜
ϕ,D)-pullback a ac ing;
(iii) ˆ
Ais minimal, ha is, i ˆ
Cis a closed nonau onomous se sa is ying (ii), hen A(σ)⊆C(σ)
o each σ∈Σ.
Rema k 3.14. No e ha , in he i ial case (i.e., Σ={σ}), he de ini ion o he cocycle a ac-
o educes o a compac se Asuch ha A Mis in a ian and a ac s bounded se s o X

A su ey on impulsi e dynamical sys ems 17
which is no he de ini ion o a global a ac o o he au onomous case, as gi en in De ini-
ion 2.12. We again emphasize ha he nonau onomous amewo k is mo e challenging han
he au onomous one, and so, i is easonable ha we ind mo e es ic i e condi ions in he
de ini ion o impulsi e cocycle a ac o s.
Now we s a e he de ini ion o he impulsi e omega limi se along wi h i s cha ac e iza-
ion.
De ini ion 3.15. Gi en a nonau onomous se ˆ
B.
={B(σ)}σ∈Σand σ∈Σwe de ine he impul-
si e pullback omega-limi o ˆ
Ba he ibe σas he se
˜
ω(ˆ
B,σ) =
s≥0[
≥s[
e∈[0,s−1)
˜
ϕ( +e,θ− σ)B(θ− σ)
and he impulsi e pullback omega-limi o ˆ
Bas he nonau onomous se
˜
ω(ˆ
B).
={˜
ω(ˆ
B,σ)}σ∈Σ.
Lemma 3.16. I ollows ha
˜
ω(ˆ
B,σ) = nx∈X: he e exis sequences { n}n∈N,{en}n∈N⊆R+and {xn}n∈N⊆B(θ− nσ)
wi h nn→+∞
−→ +∞,enn→+∞
−→ 0such ha ˜
ϕ( n+en,θ− nσ)xnn→+∞
−→ xo
and ˜
ω(ˆ
B,σ)is closed.
Rema k 3.17. No e ha i M=∅, hen ˜
ω(ˆ
B,σ) = ω(ˆ
B,σ), o each nonau onomous se ˆ
B.
De ini ion 3.18. An INDS [(ϕ,θ)(X,Σ),M,I]is said o be pullback D-asymp o ically compac ,
i o each σ∈Σ,ˆ
D∈Dand sequences { n}n∈N⊆R+,{xn}n∈N⊂Xsuch ha nn→+∞
−→ +∞
and xn∈D(θ− nσ), implies ha he sequence {˜
ϕ( n,θ− nσ)xn}n∈Npossesses a con e gen
subsequence.
De ini ion 3.19. A nonau onomous se ˆ
Bis said o be pullback D-abso bing o he INDS
[(ϕ,θ)(X,Σ),M,I], i o each σ∈Σand ˆ
D∈D, he e exis s 0= 0(σ,ˆ
D)⩾0 such ha
˜
ϕ( ,θ− σ)D(θ− σ)⊆B(σ) o all ⩾ 0.
Following he same scheme as he au onomous case, we will p esen esul s on he impul-
si e pullback omega limi and inish his sec ion gi ing a esul on he exis ence o a impulsi e
cocycle a ac o . The esul s can be ound in [4, Sec ion 4 and Sec ion 5] and we will include
some o he p oo s in o de o illus a e he echniques.
P oposi ion 3.20 ([4, P oposi ion 4.8]).I he INDS [(ϕ,θ)(X,Σ),M,I]is pullback D-asymp o ically
compac , each poin o M sa is ies ϕ-SSTC, I(M)∩M=∅,ˆ
B∈Dand σ∈Σ, hen he nonau-
onomous se ˜
ω(ˆ
B)is nonemp y, compac and pullback a ac s ˆ
B, ha is, o each σ∈Σ
lim
→+∞dH(˜
ϕ( ,θ− σ)B(θ− σ),˜
ω(ˆ
B,σ)) = 0.
18 E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
P oo . Le σ∈Σand ake sequences nn→+∞
−→ +∞( n≥0)and xn∈B(θ− nσ),n∈N. By he
asymp o ic compac ness, he sequence {˜
ϕ( n,θ− nσ)xn}n∈Nhas a con e gen subsequence o
a poin x∈X. I is easy o e i y ha x∈˜
ω(ˆ
B,σ), which p o es ha ˜
ω(ˆ
B,σ)is nonemp y.
Now, since ˜
ω(ˆ
B,σ)is closed, o show i s compac ness i is su icien o p o e ha i
{zn}n∈N⊆˜
ω(ˆ
B,σ)is a sequence, hen we can ob ain a con e gen subsequence. So, le
{zn}n∈N⊆˜
ω(ˆ
B,σ), hen o each n∈N, one can ob ain sequences { n
k}k∈N⊂R+,{en
k}k∈N⊂
R+and {xn
k}k∈N⊂B(θ− n
kσ)such ha n
k
k→+∞
−→ +∞,en
k
k→+∞
−→ 0 and
˜
ϕ( n
k+en
k,θ− n
kσ)xn
k
k→+∞
−→ zn.
Thus, he e is a na u al kn≥nsuch ha
d(˜
ϕ( n
kn+en
kn,θ− n
knσ)xn
kn,zn)≤1
n.
No e ha
˜
ϕ( n
kn+en
kn,θ− n
knσ)xn
kn=˜
ϕ(en
kn,σ)˜
ϕ( n
kn,θ− n
knσ)xn
kn,
o n,k∈N.
Since [(ϕ,θ)(X,Σ),M,I]is pullback D-asymp o ically compac , we may assume wi hou loss
o gene ali y ha he e is w∈Xsuch ha
˜
ϕ( n
kn,θ− n
knσ)xn
kn
n→+∞
−→ w.
I w/∈M, hen using i em (ii) o P oposi ion 3.12, we ge
˜
ϕ(en
kn,σ)˜
ϕ( n
kn,θ− n
knσ)xn
kn
n→+∞
−→ w,
which shows ha zkn
n→+∞
−→ w.
I w∈M, we may assume by i em (iii) o P oposi ion 3.12 ha ei he
˜
ϕ(en
kn,σ)˜
ϕ( n
kn,θ− n
knσ)xn
kn
n→+∞
−→ w
o
˜
ϕ(en
kn,σ)˜
ϕ( n
kn,θ− n
knσ)xn
kn
n→+∞
−→ I(w),
which shows ha {zn}n∈Nadmi s a con e gen subsequence.
Now, assume ha he las s a emen does no hold, ha is, he e exis σ∈Σ,e0>0 and
sequences nn→+∞
−→ +∞and zn∈B(θ− nσ)such ha
d(˜
ϕ( n,θ− nσ)zn,˜
ω(ˆ
B,σ)) ⩾e0,n∈N.
Bu ˜
ϕ( n,θ− nσ)znn→+∞
−→ x o some x∈Xalong some subsequence. Clea ly x∈˜
ω(ˆ
B,σ)
and
0=d(x,˜
ω(ˆ
B,σ)) ⩾e0,
which gi es us a con adic ion and p o es he esul .
P oposi ion 3.21 ([4, P oposi ion 4.9]).Le [(ϕ,θ)(X,Σ),M,I]be an INDS such ha I(M)∩M=∅
and each poin o M sa is ies ϕ-STC. Then o any nonemp y nonau onomous se ˆ
B, i s impulsi e
ω-limi ˜
ω(ˆ
B) M.
={˜
ω(ˆ
B,σ) M}σ∈Σis posi i ely ˜
ϕ-in a ian .
A su ey on impulsi e dynamical sys ems 19
P oo . Fix σ∈Σand ⩾0. Le x∈˜
ω(ˆ
B,σ) M. Then he e exis { n}n∈N,{en}n∈N⊂R+and
{xn}n∈N⊆B(θ− nσ)wi h nn→+∞
−→ +∞,enn→+∞
−→ 0 such ha ˜
ϕ( n+en,θ− nσ)xnn→+∞
−→ x. Since
x/∈Mand Mis closed, we may assume ha ˜
ϕ( n+en,θ− nσ)xn/∈M o all n∈N. The e o e,
by i em (i) o P oposi ion 3.12, he e exis s a sequence {ηn}n∈N⊂Rsuch ha ηnn→+∞
−→ 0 and
˜
ϕ( n+ +ηn+en,θ−( + n)θ σ)xn=˜
ϕ( +ηn,θenσ)˜
ϕ( n+en,θ− nσ)xnn→+∞
−→ ˜
ϕ( ,σ)x.
Hence, ˜
ϕ( ,σ)x∈˜
ω(ˆ
B,θ σ). I =0, he e is no hing o do. I >0 obse e ha
˜
ϕ( ,σ)x/∈M, since any impulsi e ajec o y s a ing a a poin o X Mne e eaches Min
ini e ime (no e ha I(M)∩M=∅). This shows he posi i e ˜
ϕ-in a iance o ˜
ω(ˆ
B) M.
Be o e es ablishing he nega i e in a iance o impulsi e pullback ω-limi se s, we need
an auxilia y esul .
Lemma 3.22 ([4, Lemma 4.10]).Le [(ϕ,θ)(X,Σ),M,I]be an INDS wi h I(M)∩M=∅. Assume
ha e e y poin om M sa is ies ϕ-SSTC and le ˆ
B be a nonau onomous se . I y ∈˜
ω(ˆ
B,σ)∩M hen
I(y)∈˜
ω(ˆ
B,σ) M.
P oposi ion 3.23 ([4, P oposi ion 4.11]).Le [(ϕ,θ)(X,Σ),M,I]be an INDS wi h I(M)∩M=∅.
Assume ha e e y poin om M sa is ies ϕ-SSTC and le ˆ
B be a nonau onomous se . I ˜
ω(ˆ
B)is
compac and pullback a ac s ˆ
B, hen ˜
ω(ˆ
B) M is nega i ely ˜
ϕ-in a ian .
P oo . Le ⩾0, σ∈Σand x∈˜
ω(ˆ
B,θ σ) M. Then he e exis sequences { n}n∈N,{en}n∈N⊂
R+and {xn}n∈N⊆B(θ− n+ σ)wi h nn→+∞
−→ +∞and enn→+∞
−→ 0 such ha
˜
ϕ( n+en,θ− n+ σ)xnn→+∞
−→ x.
Now, since ˜
ω(ˆ
B)is compac and pullback a ac s ˆ
Band we ha e i em (ii) o P oposi-
ion 3.12, we can assume ha {˜
ϕ( n− +en,θ− n+ σ)xn}n∈Npossesses a con e gen subse-
quence (which we deno e by he same no a ion and we al eady assumed ha n> , since
nn→+∞
−→ +∞and is ixed). Thus yn.
=˜
ϕ( n− +en,θ− n+ σ)xnn→+∞
−→ y∈˜
ω(ˆ
B,σ).
Case 1: y∈X M.
By i em (i) o P oposi ion 3.12, he e exis s a nonnega i e sequence ηnn→+∞
−→ 0 such ha
˜
ϕ( +ηn,θenσ)ynn→+∞
−→ ˜
ϕ( ,σ)y.
Bu ˜
ϕ( +ηn,θenσ)yn=˜
ϕ( n+en+ηn,θ− n+ σ)xn, and using i em (ii) o P oposi ion 3.12 we
know ha ˜
ϕ( +ηn,θenσ)ynn→+∞
−→ x. The e o e, x=˜
ϕ( ,σ)y∈˜
ϕ( ,σ)( ˜
ω(ˆ
B,σ) M).
Case 2: y∈M.
In his case, using i em (iii) o P oposi ion 3.12 and Lemma 3.22, we ob ain a subsequence
{ynk}k∈Nsuch ha γk=φ(ynk,θenkσ)k→+∞
−→ 0 and
z+
k
.
=˜
ϕ(γk,θenkσ)ynk
k→+∞
−→ I(y).
=z∈˜
ω(ˆ
B,σ) M.
Now, by i em (i) o P oposi ion 3.12, he e exis s a non-nega i e sequence αk
k→+∞
−→ 0 such
ha
˜
ϕ( +αk,θγk+enkσ)z+
k
k→+∞
−→ ˜
ϕ( ,σ)z.
20 E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
Bu ˜
ϕ( +αk,θγk+ekσ)z+
k=˜
ϕ( nk+enk+γk+αk,θ− nk+ σ)xnkand again, using i em (ii) o
P oposi ion 3.12, we ha e ˜
ϕ( +αk,θγk+enkσ)z+
nk
k→+∞
−→ x. The e o e,
x=˜
ϕ( ,σ)z∈˜
ϕ( ,σ)( ˜
ω(ˆ
B,σ) M).
Now we p esen a esul which gua an ees he exis ence o an impulsi e cocycle a ac o .
Theo em 3.24 ([4, Theo em 5.1]).Le [(ϕ,θ)(X,Σ),M,I]be an INDS pullback D-asymp o ically
compac such ha I(M)∩M=∅and e e y poin om M sa is ies ϕ-SSTC. Assume ha he e exis s
a pullback (˜
ϕ,D)-abso bing nonau onomous se ˆ
K∈D. Then, he nonau onomous se ˆ
A de ined by
A(σ) = ˜
ω(ˆ
K,σ)
is a D-impulsi e cocycle a ac o o he INDS [(ϕ,θ)(X,Σ),M,I].
P oo . By P oposi ion 3.20 we ha e ˆ
Ais nonemp y, compac and pullback D-a ac s ˆ
K. The
in a iance o ˆ
A M ollows om P oposi ion 3.21 and P oposi ion 3.23. Since ˆ
Kis pullback
(˜
ϕ,D)-abso bing hen we deduce ha ˆ
Kis (˜
ϕ,D)-pullback a ac ing. Suppose he e exis s a
nonau onomous closed se ˆ
C ha pullback D-a ac s e e y nonau onomous se ˆ
B∈D. Since
˜
ω(ˆ
B) Mis ˜
ϕ-in a ian , we ha e
dH(˜
ω(ˆ
B,σ) M,C(σ)) = dH(˜
ϕ( ,θ− σ)˜
ω(ˆ
B,θ− σ) M,C(σ)) →+∞
−→ 0,
ha is, ˜
ω(ˆ
B,σ) M⊆C(σ), o e e y ˆ
B∈Dand σ∈Σ.
Now, le x∈˜
ω(ˆ
B,σ)∩M. Then he e exis sequences { n}n∈N,{en}n∈N⊂R+and xn∈
B(θ− nσ)wi h nn→+∞
−→ +∞,enn→+∞
−→ 0 such ha ˜
ϕ( n+en,θ− nσ)xnn→+∞
−→ x. Le zn.
=
˜
ϕ( n,θ− nσ)xn,n∈N. We may assume ha znn→+∞
−→ z∈˜
ω(ˆ
B,σ). By i ems (ii) and (iii) o
P oposi ion 3.12, we ha e (possibly, aking subsequences) ei he
(1) ˜
ϕ(en,σ)znn→+∞
−→ zo
(2) ˜
ϕ(en,σ)znn→+∞
−→ I(z),
and since ˜
ϕ(en,σ)zn=˜
ϕ( n+en,θ− σ)xnn→+∞
−→ xand I(M)∩M=∅, i em (2) canno happen
and we mus ha e z=x∈Mand ˜
ϕ( n,θ− nσ)xnn→+∞
−→ x. Since ˆ
Cpullback D-a ac s
nonau onomous se s, we ha e x∈C(σ). Then ˜
ω(ˆ
B,σ)⊆C(σ) o e e y ˆ
B∈Dand σ∈Σ,
which implies in pa icula ha ˜
ω(ˆ
K,σ)⊂C(σ)and he e o e A(σ)⊆C(σ)and ends he
p oo .
Rema k 3.25. Wi h De ini ion 3.13, i ˆ
Aexis s, i is uniquely de e mined.
To inish his sec ion, we s a e an impo an cha ac e iza ion o he impulsi e cocycle a -
ac o .
De ini ion 3.26. We say ha a unc ion ψ:R→Xis a global solu ion o ˜
ϕa σi
˜
ϕ( −s,θsσ)ψ(s) = ψ( ) o all ⩾s,s∈R.
Mo eo e , i ψ(0) = xwe say ha ψis a global solu ion h ough x. We say ha a global
solu ion is bounded i ψ(R)is a bounded subse o X.
A su ey on impulsi e dynamical sys ems 21
P oposi ion 3.27 ([4, P oposi ion 5.5]).A ligh o De ini ion 3.13, i he INDS [(ϕ,θ)(X,Σ),M,I]
has an impulsi e cocycle a ac o ˆ
A∈Dwi h uni e se Dconsis ing o all nonau onomous se s ˆ
B
such ha Sσ∈ΣB(σ)is bounded in X and I(M)∩M=∅, hen
A(σ) M={x∈X:ψis a bounded global solu ion o ˜
ϕa σ h ough x}.
P oo . I ψ(·)is a bounded global solu ion o ˜
ϕa σ h ough x hen ψ(R)∩M=∅, since i
ψ( 0)∈M o some 0∈R hen ˜
ϕ( 0−s,θsσ)ψ(s) = ψ( 0)∈M o each ssuch ha 0−s>0
which canno happen ( he impulsi e cocycle om xcanno each Min posi i e ime o any
x∈X, because I(M)∩M=∅). Hence, ψ(R)∩M=∅. By i s in a iance we can see ha
x∈A(σ)and he e o e, x∈A(σ) M.
Fo he e e se inclusion, i x∈A(σ) M hen x∈˜
ϕ(1, θ−1σ)A(θ−1σ)and he e exis s
x−1∈A(θ−1σ)such ha ˜
ϕ(1, θ−1σ)x−1=x. Again, since x−1∈A(θ−1σ) he e exis s x−2∈
A(θ−2σ)such ha ˜
ϕ(1, θ−2σ)x−2=x−1. Induc i ely, we can cons uc a sequence {x−n}n∈N
such ha ˜
ϕ(1, θ−n−1σ)x−n−1=x−n o all n⩾0, wi h x0=x. Then we can de ine
ψ( ) = (˜
ϕ( +n,θ−nσ)x−n, i ∈[−n,−n+1],n∈N,
˜
ϕ( ,σ)x0, i ⩾0.
Since ˆ
A∈D, i is clea ha his global solu ion is bounded and comple es he p oo .
3.3 Nonau onomous 2D-Na ie –S okes equa ions wi h impulses
He e we p esen an example o illus a e he impulsi e nonau onomous heo y. A mo e de-
ailed desc ip ion can be ound in [4, Sec ion 6].
The Na ie –S okes equa ions model luid low and a e ob ained used he conse a ion o
linea momen um, which is
u −ν∆u+ (u· ∇)u+∇p=g( ), (3.2)
oge he wi h an incomp essibili y condi ion
∇ · u=0,
whe e u( ,x)deno es he ec o eloci y, ν>0 is he kinema ic iscosi y, gis a body o ce
and pis a scala p essu e.
Now, we will conside his model wi h impulses in he s a e space, ha can be imagined
as o ced changes on he ec o eloci y o he luid, in o de o p e en p oblems ha may
occu when he luid eaches ce ain speeds. One can modi y his model a li le and add a
componen ( ,x) o he posi ion o he luid, and we could imagine impulses as a way o
a oid ba ie s and obs acles along he luid ajec o y.
We use he e he app oached adop ed in [15], and we ea his p oblem in Ω= [0, 2π]2(a
pe iodic domain) and we equi e ze o o al momen um, ha is, i RΩu0=0 and RΩg( ) = 0
o all ⩾0, hen RΩu( ) = 0 o all ⩾0.
W i ing ˙
Z2=Z2 {0, 0}, le ˙
Hsbe he subspace o he Sobole space Hswhich consis s o
all di e gence- ee, ze o a e age, pe iodic eal unc ions
˙
Hs.
=(u=∑
k∈˙
Z2
ˆ
ukeik·x:ˆ
uk=ˆ
uk,∑
k∈˙
Z2
|k|2s|ˆ
uk|2<∞,k·ˆ
uk=0),

22 E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
wi h he no m
kuk2
s=∑
k∈˙
Z2
|k|2s|ˆ
uk|2.
We ha e ha H.
=˙
H0is he na u al phase space o his p oblem, and we w i e k · k o he
no m in H( he usual L2-no m). Remembe also ha he space o di e gence- ee unc ions is
pe pendicula (in L2(Ω)) o he space o g adien s, since in eg a ing by pa s, we ha e
ZΩu· ∇p=−ZΩ(∇ · u)p=0,
and so we use he Le ay p ojec o P, which is he o hogonal p ojec ion o L2(Ω)in o he
space os di e gence- ee ields. Applying his p ojec o o (3.2) we ob ain
du
d +νAu +B(u,u) = ( ), (3.3)
whe e A=−P∆is he S okes ope a o , B(u,u) = P[(u· ∇)u]and ( ) = Pg( ). In he pe iodic
case, we ha e ha Au =−∆Pu and so Au =−∆u, o u∈˙
Hs.
We de ine he ac ional powe As/2 o Aby D(As/2) = ˙
Hsand
As/2 ∑
k∈˙
Z2
ˆ
ukeik·x!=∑
k∈˙
Z2
|k|sˆ
ukeik·x.
We no e ha he no ms k · k1and he no m kA1/2 · k a e equi alen , and also ha ˙
H1is
compac ly embedded in H. Also, we deno e he dual space o ˙
H1by H−1.
A simple in eg a ion by pa s leads o he ollowing an isymme ic iden i y
(B(u, ),w) = −(B(u,w), ),
which implies in pa icula ha
(B(u, ), ) = 0. (3.4)
Also, wi h a li le mo e e o and using he incomp essibili y condi ion, one can p o e
ha in he wo-dimensional pe iodic case, we ha e
(B(u,u),Au) = 0.
Then, we can summa ize he esul s o [15, Sec ion 11.1] in he nex p oposi ion.
P oposi ion 3.28. Assume ha k ( )k⩽α o all ⩾0, hen we ha e:
(i) equa ion (3.3)de ines a nonau onomous dynamical sys em (ϕ,θ)(H,R), whe e θ s= +s o all
⩾0and s ∈Rand
ϕ( ,s)u0=u( +s,s,u0)
is he unique solu ion in H o (3.3), wi h u(s,s,u0) = u0∈H;
(ii) ϕ(·,s)u0∈L∞(0, T;H)∩L2(0, T;D(A1/2)) and ϕ (·,s)u0∈L2(0, T;D(A−1/2)) o e e y
T>0;
(iii) o u0∈H and s ∈R
kϕ( ,s)u0k2⩽e−νλ1 ku0k2+α2
ν2λ2
1
, o all ⩾0,
whe e λ1is he i s eigen alue o A.
A su ey on impulsi e dynamical sys ems 23
Now we assume ha Mis an impulsi e se in H o (ϕ,θ)(H,R), and assume ha e e y
poin o Msa is ies ϕ-SSTC. Also, le I:M→Hbe an impulsi e unc ion such ha
(H1) I(M)∩M=∅;
(H2) kI( )k2⩽µ, o all ∈M.
(H3) Assume ha he e exis s ξ>0 such ha φ( ,s)⩾2ξ, o all ∈I(M)and s∈R.
Le ˜
ϕ( ,s)u0be he associa ed impulsi e solu ion o







du
d +νAu +B(u,u) = ( ),
u(0) = u0∈H,
I:M→H.
(3.5)
We assume ha k ( )k ≤ α o all ≥0.
Now we summa ize some esul s (see [4, Sec ion 6]) which a e use ul o ob ain an exis ence
esul o impulsi e cocycle a ac o o his example.
P oposi ion 3.29.
(i) ([4, Lemma 6.2])Fo each >0and s ∈R, he map ϕ( ,s):H→H is compac .
(ii) ([4, Lemma 6.3])We ha e k˜
ϕ( ,s)u0k2⩽µ+α2
ν2λ2
1
, o all u0∈I(M), ⩾0and s ∈R.
(iii) ([4, P oposi ion 6.4])I B ⊂H is a bounded subse hen he e exis s 0= 0(B)⩾0such ha
k˜
ϕ( ,s)u0k2⩽µ+α2
ν2λ2
1
, i ⩾ 0, o all u0∈B and s ∈R.
(i ) ([4, Lemma 6.5])I G is a p ecompac subse o H and τ∈[0, ξ), hen ˜
ϕ(τ,s)G is p ecompac
in H o each s ∈R.
Using he esul s in P oposi ion 3.29, we can cons uc a compac nonau onomous se
ˆ
K={K(s)}s∈Rwhich ˜
ϕ-pullback abso bs all bounded subse s o H. We will ep oduce i s
p oo he e.
Theo em 3.30 ([4, Theo em 6.6]).The e exis s a compac nonau onomous se ˆ
K={K(s)}s∈Rwhich
˜
ϕ-pullback abso bs all nonau onomous se s ˆ
D wi h Ss∈RD(s)bounded in H, and such ha Ss∈RK(s)
is bounded in H.
P oo . Le B0=u∈H:kuk2⩽µ+α2
ν2λ2
1. Fi s ly, we ix τ∈(ξ, 2ξ). We claim ha G(s) =
˜
ϕ(τ,θ−τs)B0is p ecompac o each s∈R. Indeed, we can w i e B0=C1∪C2∪C3whe e
C1={u∈B0:φ(u,θ−τs)⩾2ξ},C2={u∈B0:ξ<φ(u,θ−τs)⩽2ξ}and
C3={u∈B0:φ(u,θ−τs)⩽ξ}.
Then we ha e
G(s) = ϕ(τ,θ−τs)C1∪˜
ϕτ−ξ,θ−τ+ξsϕ(ξ,θ−τs)C2∪ϕτ−ξ,θ−τ+ξs˜
ϕ(ξ,θ−τs)C3,
since φ( ,s)⩾2ξ o all ∈I(M)and s∈R, and τ−ξ∈(0, ξ).
By P oposi ion 3.29, since C1and ˜
ϕ(ξ,θ−τs)C3a e bounded (see i em (ii)), i ollows ha
se s ϕ(τ,θ−τs)C1and ϕτ−ξ,θ−τ+ξs˜
ϕ(ξ,θ−τs)C3a e p ecompac in H(see i em (i)). Also,
24 E. M. Bono o, M. C. Bo olan, T. Ca aballo and R. Collega i
since ϕ(ξ,θ−τs)C2is p ecompac in H, i ollows ha ˜
ϕτ−ξ,θ−τ+ξsϕ(ξ,θ−τs)C2is also
p ecompac in H(i em (i )).
The e o e, K(s).
=G(s)is compac in H, o each s∈R. Clea ly, we ha e ha
sup
∈K(s)
k k2⩽β+α2
ν2λ2
1
,
whe e β=max{µ,L0}and L0=supu∈B0kuk2.
Now i emains o p o e ha ˆ
K˜
ϕ-pullback abso bs nonau onomous bounded se s ˆ
Dwi h
Ss∈RD(s)bounded in H. To his end, le ˆ
Da nonau onomous se in Hwi h B.
=Ss∈RD(s)
bounded in Hand ix s∈R.
We know, by i em (iii) o P oposi ion 3.29, ha he e exis s 0= 0(B)>0 such ha
˜
ϕ( ,θ− −τs)B⊂B0, o all ⩾ 0.
Thus
˜
ϕ( +τ,θ− −τs)B=˜
ϕ(τ,θ−τs)˜
ϕ( ,θ− −τs)B⊂˜
ϕ(τ,θ−τs)B0⊂K(s),
which shows ha i ⩾ 0+τ
˜
ϕ( ,θ− s)D(θ− s)⊂˜
ϕ( ,θ− s)B⊂K(s),
and p o es ha ˆ
Kis a ˜
ϕ-pullback abso bs ˆ
D.
As a consequence o his las heo em we ob ain ha he INDS [(ϕ,θ)(H,R),M,I]de ined
by (3.5) has an impulsi e cocycle a ac o (see [4, Co olla y 6.7]).
4 Conclusion, commen s and u u e di ec ions
In his su ey pape we desc ibed he heo ies o impulsi e dynamical sys ems in bo h au-
onomous and nonau onomous amewo ks. In he i s pa o his su ey we p esen ed wo
di e en app oaches o s udy he asymp o ic dynamical beha io o au onomous sys ems,
p oposed by Bono o and Demune (see [6,7]) and Bono o e al. (see [5]), espec i ely. In
[6,7], he de ini ion o global a ac o s o impulsi e au onomous dynamical sys ems was i s in-
oduced, whe e he a ac o is in a ian , consis s o a compac se which does no in e sec
he impulsi e se Mand a ac s bounded se s. This de ini ion is consis en wi h he no ion
o global a ac o s o semig oups ( hey coincide when M=∅) and desc ibes he asymp o ic
beha io o many impulsi e dynamical sys ems. Howe e , i is no sui able o a la ge class
o impulsi e dynamical sys ems. Fo example, when he global a ac o is compac and is
disjoin wi h he closed se M, he compac ness o he global a ac o implies a sepa a ion
be ween hem and hence he asymp o ic beha io o he impulsi e dynamical sys em is no
quali a i ely di e en om he asymp o ic beha io o he o iginal sys em wi hou impulse
(see, e.g., Example 2.11). La e in [5] he no ion o p ecompac global a ac o s was in oduced,
whe e he global a ac o can “ ouch” he impulsi e se M,i.e., he bounda y o he global
a ac o can ha e poin s which belong o M. The simplici y o au onomous amewo k allows
us o s udy a ious ypes o impulsi e dynamical sys ems, along wi h many in e es ing new
applica ions. In his su ey we illus a ed one o he h ee in e es ing applica ions p esen ed
in [5].
A su ey on impulsi e dynamical sys ems 25
In he second pa o his su ey we desc ibed he ecen ly de eloped heo ies o nonau-
onomous impulsi e dynamical sys ems, wi h mul iple lines o p ospec i e esea ch. In pa ic-
ula , we ecalled he main esul s o ou ecen wo k [4], whe e we p oposed he i s app oach
in he nonau onomous heo y o s udy impulsi e dynamical sys ems. This is done by de in-
ing he no ion o impulsi e nonau onomous dynamical sys ems, in which he ajec o ies ha e
o be de ined in a ca e ul manne o ob ain hei ela ionship wi h he associa ed impulsi e
skew-p oduc semi low (see Theo em 3.9). The main goal is o cons uc a p ope no ion o
impulsi e cocycle a ac o s and de elop hei exis ence. To his end, we in oduced a di e -
en no ion o omega limi se (see De ini ion 3.15), o o e come he di icul ies encoun e ed in
p o ing he usual p ope ies such as in a iance and pullback a ac ion in he nonau onomous
heo y.
I is wo h men ioning again ha he heo y o impulsi e dynamical sys ems is s ill in he
ea ly s age o in es iga ion and has many in e es ing opics o be disco e ed, especially in
he nonau onomous amewo k. We ha e made an ini ial s ep owa d es ablishing he mo-
de n heo y o impulsi e dynamical sys ems, by de eloping a de ini ion o impulsi e nonau-
onomous dynamical sys ems and p esen ing an exis ence esul o impulsi e cocycle a ac o .
Ye he e a e many o he in e es ing and impo an p oblems along his di ec ion o be in es-
iga ed, e en in he au onomous amewo k. Fo example, on he one hand, he e a e no
s udies o da e on he semi-con inui y and geome ical s uc u es o a ac o s o impulsi e
dynamical sys ems, and on he o he hand, he e a e no many examples om applica ions
analyzed in a de ailed way. The main easons a e he di icul ies in o de o check some o he
hypo heses ensu ing he gene a ion o an impulsi e sys em, as well as he condi ions equi ed
o he exis ence o a ac o s. The e o e, his is a ield o be explo ed in a mo e de ailed way
in he u u e and we plan o wo k on his di ec ion. Ano he majo esea ch di ec ion would
be de eloping a se o analog heo ies o impulsi e dynamical sys ems whe e he nonau-
onomous cha ac e in ol es unce ain y, i.e., noise. This leads o a amewo k o andom
impulsi e dynamical sys ems, a b and new a ea o esea ch.
Acknowledgemen s
E. M. B. is pa ially suppo ed by FAPESP g an 2014/25970-5 and CNPq g an 307317/2013-7.
M. C. B. and T. C. a e pa ially suppo ed by FEDER and Minis e io de Economía y Compe i-
i idad (Spain) unde g an MTM2015-63723-P, and Conseje ía de Inno ación, Ciencia y Em-
p esa (Jun a de Andalucía) unde P oyec o de Excelencia P12-FQM-1492. R. C. is suppo ed by
FAPESP g an s 2013/23933-2 and 2014/20691-0. The au ho s also would like o hank he e -
e ee o he help ul commen s and sugges ions which allowed us o imp o e he p esen a ion
o his su ey.
Re e ences
[1] N. U. Ahmed, Exis ence o op imal con ols o a gene al class o impulsi e sys ems on
Banach spaces, SIAM J. Con ol Op im. 42(2003), No. 2, 669-685. MR1982287;u l
[2] D. D. Baino , P. S. Simeono ,Sys ems wi h impulsi e e ec . S abili y, heo y and applica ions,
Wiley, New Yo k, 1989. MR1010418