Nonlinear Dynamical Systems Fourth Class › pessoas › andcarva › SDNL › Aulas › Aula04-englis… · Su cient conditions for the existence of global attractors Upper and lower

Sufficient conditions for the existence of global attractorsUpper and lower semicontinuity of attractors

Gradient Semigroups

Nonlinear Dynamical SystemsFourth Class

Alexandre Nolasco de Carvalho

August 28, 2017

Alexandre N. Carvalho - USP/Sao Carlos Second Semester of 2017


Gradient Semigroups

Sufficient conditions for the existence of global attractors

In what follows we present some results that are commonly used toensure that a semigroup is asymptotically compact and eventuallybounded.



Gradient Semigroups

Theorem (Exercise)

If {T (t) : t ∈ T+} is a point dissipative and eventually compactsemigroup, then {T (t) : t ∈ T+} has a global attractor A.

Proof: From previous results, it is enough to prove that{T (t) : t ∈ T+} is eventually bounded. Given a bounded subset Bof X , it follows from the fact that {T (t) : t ∈ T+} is eventuallycompact that there exists tB ∈ T+ such that T (tB)B is relativelycompact. Hence, it is only necessary to prove that the orbits ofcompact subsets of X are bounded, sinceT (t)T (tB)B ⊂ T (t)T (tB)B.



Gradient Semigroups

Let K be a compact subset of X and B0 be an open boundedsubset of X that absorbs points. Given x ∈ K , from the continuityof T (t) there is a neighborhood Ox of x and tx ∈ T+ such thatT (tx)Ox⊂T (tB0)B0. Since K is compact, there are {Ox1 ,· · · ,Oxp}that cover K .

Let τ = τ(K ) = max{txi : 1 6 i 6 p}, K0 = T (tB0)B0 andK0 = γ+

[0,τ(K0)]K0. Clearly K0 and K0 are compact subsets of X . It

follows that T (t)B0 ⊂ K0 for all t > tB0 and for each compactsubset K of X we have that T (t)K ⊂ K0 for t > τ(K ). Thisproves that the orbit of a compact subset of X is bounded andcompletes the proof of the result.



Gradient Semigroups

TheoremLet X be a Banach space and {T (t) : t ∈ T+} be a semigroup inX . Assume that T (t)=S(t)+K (t) with S(t) and K (t) satisfying

i) For each bounded subset B of X , there is a tB ∈ T+ suchthat K (t)B is relatively compact for each t > tB .

ii) For each bounded subset B of X , there is a tB ∈ T+ suchthat supx∈B ‖S(t)x‖X := sB(t) <∞ for all t > tB and

sB(t)t→∞−→ 0.

Then {T (t) : t ∈ T+} is asymptotically compact. Furtheremore, if{T (t) : t ∈ T+} is point dissipative and eventually bounded, thenit has a global attractor.



Gradient Semigroups

Proof: Given a non-empty, closed and bounded subset B suchthat T (t)B ⊂ B for t > 0 e ε > 0, choose τ ∈ T+ such thatτ > tB and sB(τ) < ε

2 . Since K (τ)B is relatively compact, thereare N = N(τ,B) in N and y1, · · · yN in K (τ)B such thatK (τ)B ⊂ ∪Ni=1B ε

2(yi ). It follows that

ω(B) = ∩t∈T+T (t)B ⊂ T (τ)B ⊂ S(τ)B + K (τ)B

⊂ B ε2(0) + ∪Ni=1B ε

2(yi ) ⊂ ∪Ni=1Bε(yi ).



Gradient Semigroups

Since ε is arbitrary, we have that ω(B) is totally bounded. Hence,ω(B) is closed and totally bounded in the Banach space X and,consequently, compact.

We can easily see that ω(B) is non-empty since, for each sequence{xn} in B and tn ∈ T+ with tn

n→∞−→ ∞ the sequence {T (tn)xn} istotally bounded and so, it has a convergent subsequence.

Now, proceeding as before we conclude that ω(B) attracts B,proving that {T (t) : t ∈ T+} asymptotically compact.

If in addition {T (t) : t ∈ T+} is point dissipative and eventuallybounded then, {T (t) : t ∈ T+} has a global attractor.



Gradient Semigroups

Upper and lower semicontinuity of attractors

Now we study the continuity of attractors relatively toperturbations in the semigroup.

DefinitionLet X and Λ be metric spaces and {Aλ}λ∈Λ ⊂ 2X .

1. We say that {Aλ}λ∈Λ is upper semicontinuous at λ0 if

distH(Aλ,Aλ0) = supxλ∈Aλ

dist(xλ,Aλ0)λ→λ0−→ 0.

2. We say that {Aλ}λ∈Λ is lower semicontinuous at λ0 if

distH(Aλ0 ,Aλ) = supx∈Aλ0

dist(x ,Aλ)λ→λ0−→ 0.



Gradient Semigroups

To prove upper and lower semicontinuity we will use the followingresult

LemmaLet X and Λ be metric spaces and {Aλ}λ∈Λ ⊂ 2X .

1. If any sequence {xλn} with xλn ∈ Aλn , λnn→∞−→ λ0, has a

convergent subsequence with limit in Aλ0 , then {Aλ}λ∈Λ isupper semicontinuous at λ0.

2. If Aλ0 is compact and for each x ∈ Aλ0 and sequence

λnn→∞−→ λ0, there is a subsequence {λnk} of {λn} and

sequence {xλnk } with xλnk ∈ Aλnk that converges to x , then{Aλ}λ∈Λ lower semicontinuous at λ0.



Gradient Semigroups

Proof: 1) If any sequence {xλn} with xλn ∈ Aλn , λnn→∞−→ λ0, has a

convergent subsequence with limit in Aλ0 , and {Aλ}λ∈Λ is notupper semicontinuous at λ0, then there are ε > 0 and sequence{λn} with λn

n→∞−→ λ0 such that supx∈Aλndist(x ,Aλ0)>2ε, n ∈ N.

Hence, for some xλn ∈ Aλn , we have that dist(xλn ,Aλ0) > ε,n ∈ N. This is a contradiction with the fact that {xλn} has aconvergent subsequence to an element of Aλ0 .



Gradient Semigroups

2) If Aλ0 is compact and for each x ∈ Aλ0 and λnn→∞−→ λ0 there is

a subsequence λnkk→∞−→ λ0, and sequence {xλnk } with xλnk ∈ Aλnk

that converges to x and {Aλ} is not lower semicontinuous λ0,there are ε > 0 and sequence λn

n→∞−→ λ0 such thatsupx∈Aλ0

dist(x ,Aλn) > 2ε, n ∈ N.



Gradient Semigroups

Thus, since Aλ0 is compact, there is a xλn ∈ Aλ0 such thatdist(xλn ,Aλn) > 2ε, n ∈ N and we may assume that {xλn}converges to some x ∈ Aλ0 and that dist(x ,Aλn) > ε, n ∈ N.

From our assumptions, there is a subsequence λnkk→∞−→ λ0 and a

sequence yλnk ∈ Aλnk such that yλnkk→∞−→ x and

ε 6 dist(x ,Aλnk ) 6 dist(x , yλnk )k→∞−→ 0 gives a contradiction.



Gradient Semigroups

DefinitionWe say that the family of semigroups {Tη(t) : t ∈ T+}η∈[0,1], is

continuous at η = 0 if Tη(t, x)η→0−→ T0(t, x), uniformly for (t, x) in

compact subsets of T+ × X .



Gradient Semigroups

Theorem (Upper semicontinuity)

Let {Tη(t) : t ∈ T+}η∈[0,1] be a family of semigroups which iscontinuous at η = 0. If {Tη(t) : t ∈ T+} has a global attractor Aηfor each η ∈ [0, 1] and ∪η∈[0,1]Aη is compact, the family{Aη : η ∈ [0, 1]} is upper semicontinuous at η = 0.



Gradient Semigroups

Proof: Consider the subsequences ηk → 0, uηk ∈ Aηk . Since⋃η∈[0,1]Aη is compact in X , there is a u0 ∈ X such that

uηkk→∞−→ u0.

To complete the proof of upper semicontinuity it remains to showthat u0 ∈ A0. To that end, it is sufficient to prove that there is aglobal bounded solution of {T0(t) : t ∈ T} through u0.



Gradient Semigroups

From the invariance of the attractors Aηk , for each k ∈ N, there isa global bounded solution ψ(ηk ) : T→ X through uηk . For t > 0, itfollows from the continuity of [0, 1] 3 η → Tη(t)x ∈ X uniformlyin compact subsets of T+ × X that

ψ(ηk )(t) = Tηk (t)uηk → T0(t)u0,

uniformly for t in bounded subsets of T+.



Gradient Semigroups

Now, we construct a global solution through u0 in the followingway. If η0

k := ηk , k ∈ N, given j ∈ N∗ there is a subsequence {ηjk}of {ηj−1

k } and u−j such that ψ(ηjk )(−j) k→∞−→ u−j (recall that

{ψ(ηk )(−j)}j∈N is in⋃

η∈[0,1]

Aη).

From the convergence of x 7→ Tη(1)x to x 7→ T0(1)x uniformly incompact subsets of X it follows that, for 0 6 i 6 j ,

ψ(ηjk )(−j) = Tηk (i)ψ(ηjk )(−j + i)→ u−j = T0(i)u−j+i .



Gradient Semigroups

Defining

ψ(0)(t) :=

{T0(t)u0, for t > 0T0(t + j)u−j , for − j 6 t < −j + 1, j ∈ N∗

we have that ψ(0) : T→ X is a global solution of {T0(t) : t ∈ T}through u0 and

ψ(ηkk )(t)k→∞−→ ψ(0)(t), ∀t ∈ T.

Since ψ(0) : T→ X is bounded, its image must be contained in A0

and in particular u0 ∈ A0. Now the result follows from the previouslemma.



Gradient Semigroups

RemarkThe assumption that ∪η∈[0,1]Aη be compact in the previoustheorem can be weakened assuming instead some collectivecompactness; that is, given a sequence {ηn} in [0, 1] withηn

n→∞−→ 0 and a sequence {xn} with xn ∈ Aηn , then {xn} has aconvergent subsequence. Note that, this implies the existence ofη0 ∈ (0, 1] such that ∪η∈[0,η0]Aη is bounded. A proof of theprevious theorem with this weaker hypothesis is completelyanalogous to the proof given.



Gradient Semigroups

Given a semigroup {T (t) : t ∈ T+} and A an invariant set, wehave already seen that, through each a ∈ A, there is a globalsolution. The simplest possible global solutions are the constantones; that is, the stationary solutions.

Recall that y∗ ∈ X is an equilibrium point for {T (t) : t ∈ T+} ifthe set {y∗} is the orbit of a global constant solution; that is,T (t)y∗ = y∗, for all t > 0. We will denote by E the set of allequilibrium points of {T (t) : t ∈ T+}.



Gradient Semigroups

The orbits of all global solutions that tends to y∗ as t tends to−∞ is an invariant set that is called the unstable set W u(y∗) ofy∗; that is,

W u(y∗)={y ∈ X : there is a global solution φy : T→ X

such that, φy (0) = y and φy (t)t→−∞−→ y∗}.



Gradient Semigroups

Given a neighborhood V of y∗, the set of points y of V such thatthere is a global solution φy : T→ X with, φy (0) = y ,

φy (t)t→−∞−→ y∗ and φy (t) ∈ V for all t ∈ T− is called the local

unstable set de y∗ and is denoted by W uloc(y∗). When V = Bδ(y

∗)we denote the local unstable set by W u

δ (y∗).



Gradient Semigroups

The orbit γφ of a global non-constant solution φ : T→ X thattens to an equilibrium y∗ as t tends to ±∞ is called a homoclinicorbit in y∗.

We observe that, whenever there is a homoclinic orbit in y∗,W u

loc(y∗) does not coincide with the intersection of W u(y∗) with aneighborhood of y∗.



Gradient Semigroups

Theorem (Lower Semicontinuity)

Let {Tη(t) : t ∈ T+}η∈[0,1] be a family of semigroups with globalattractors {Aη}η∈[0,1] which is continuous at η = 0. Assume that

a) If Eη the set of stationary solutions of {Tη(t) : t ∈ T+}, thereis a p ∈ N such that Eη = {y∗,η1 , · · · , y∗,ηp }, for all η ∈ [0, 1],

and sup16i6p d(y∗,ηi , y∗,0i )η→0−→ 0.

b) There is a δ > 0 such that {W uδ (y∗,ηj )} is lower semicontinous

at η = 0.

c) A0 = ∪pj=1Wu(y∗,0j )

Then, {Aη : η ∈ [0, 1]} is lower semicontinuous at η = 0.



Gradient Semigroups

Proof: From d) we have that A0 = ∪pj=1Wu(y∗,0j ). It follows

that, if u0 ∈ A0, there exists a global solution φ(0) : T→ Xthrough u0 (φ(0)(0) = u0), 1 6 ` 6 p and τ ∈ T+ such thatφ(0)(−τ) ∈W u

δ (y∗,0` ).



Gradient Semigroups

From c), there is a u−τη ∈W uδ (y∗,ηj ) such that u−τη

η→0−→ φ0(−τ)

and a global solution ψ(η) : T→ X of {Tη(t) : t∈T+} with

ψ(η)(0)=u−τη such that ψ(η)(0)η→0−→φ(0)(−τ).

From a) and the continuity of {Tη(t) : t∈T+} at η=0 we have that

Tη(τ)ψ(η)(0)→ u0.

The result now follows from Lemma 1.



Gradient Semigroups

Gradient Semigroups

Next we consider the gradient semigroups. This class of semigroupsappear naturally in several applications and its characteristics allowus to describe quite precisely the structure of their attractors.

DefinitionLet {T (t) : t ∈ T+} be a semigroup and E be its set of equilibria.We say that {T (t) : t ∈ T+} is gradient if it has a Lyapunovfunctional; that is, if there is a continuous function V : X → Rwith the following properties:

(i) T+ 3 t 7→ V (T (t)x) is decreasing for each x ∈ X ;

(ii) If x is such that V (T (t)x) = V (x) for all t ∈ T+, then x ∈ E .



Gradient Semigroups

For gradient semigroups the following characterization result holds:

LemmaIf {T (t) : t ∈ T+} is a gradient semigroup, then ω(x) is a subsetof E for each x ∈ X . If there is a global solution φ : T→ Xthrough x , then αφ(x) is a subset of E .

If {T (t) : t ∈ T+} is gradient, has a global attractor A and E onlyhas isolated points, then E is finite and for each x ∈ X , ω(x) is aunitary set. In this case, if x ∈ A and φ : T→ A is a globalsolution through x , then αφ(x) and ω(x) are unitary sets.



Gradient Semigroups

Proof: If ω(x) = ∅ the result is trivial. If ω(x) 6= ∅ we have that

V (T (t)x)t→∞−→ c for some c ∈ R, since it is decreasing and has a

convergent subsequence. Hence V (ω(x)) = {c}.

Since T (t)ω(x) ⊂ ω(x), t ∈ T+, we have that each pointy ∈ ω(x) is such that V (T (t)y) = V (y) = c , t ∈ T+, and fromproperty (ii) in Definition 3 we have that y ∈ E .



Gradient Semigroups

Suppose that there is a global solution φ : T→ X through x . Ifαφ(x) = ∅ the result is trivial. On the other hand, if y ∈ αφ(x),

there is a c ∈ R such that V (φ(−t))t→∞−→ c , since it is increasing

and has a convergent subsequence. Hence V (αφ(x)) = {c}.

Since T (t)αφ(x) ⊂ αφ(x), t ∈ T+, we have that for eachy ∈ αφ(x), V (T (t)y) = V (y) = c , t ∈ T+, and y ∈ E .



Gradient Semigroups

Now assume that {T (t) : t ∈ T+} has a global attractor A. SinceA is compact and E ⊂ A, it follows that if all points of E areisolated, then E is finite.

It remains to show that, if the set of stationary solutions is finite,then ω(x) and αφ(x) are unitary sets. If T = R this followsimmediately from the fact that ω(x) and αφ(x) are connected.



Gradient Semigroups

If T = Z and ω(x) = {y∗1 , · · · , y∗` } ⊂ E with ` > 2, then there is adisjoint convering {Ni : 1 6 i 6 `} of N with the property thateach Ni is infinite and lim

n∈Nin→∞

T (n)x = y∗i , 1 6 i 6 `.

Choose a sequence {kn : n ∈ N} such that k2n−1 ∈ N1 andk2n = k2n−1 + 1 ∈ Nj , for some 2 6 j 6 `. Then,y∗1 = T (1)y∗1 = limk→∞ T (k2n)x = y∗j , which is a contradiction.

This shows that, if the set of stationary solutions is finite, thenω(x) is a unitary set. The proof that αφ(x) is a unitary set whenT = Z is entirely analogous. This concludes the proof.


Documents

Nonlinear Dynamical Systems Fourth Class › pessoas › andcarva › SDNL › Aulas › Aula04-englis… · Su cient conditions for the existence of global attractors Upper and lower