Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Linear almost periodic dynamical systems 417

Theorem 13.5 Assume that A ∈ C(R, [E]) is recurrent (that is, H(A) is a com-

pact minimal set of (C(R, [E]), R, σ)). Then the following conditions are equivalent:

(a) the set

−

, B) = {(v, B) ∈ E × H(A) | sup

t∈R

−

,R)

|ϕ(t, v, B)| < +∞} (13.23)

is closed in E ×H(A), and

(b) there is a positive number M such that

|ϕ(t, v, B)| ≤ M|v| (13.24)

for all t ∈ R

−

, R) and (v, B) ∈ B

−

, B).

Corollary 13.2 Let A ∈ C(R, [E]) be recurrent. Then the following assertions

are equivalent :

(i) all solutions of all equations (13.22) are bounded on R

−

, R), and

(ii) there is an M > 0 such that (13.24) is valid for all v ∈ E, B ∈ H(A), and

t ∈ R

−

, R).

Theorem 13.6 Assume that A ∈ C(R, [E]) is recurrent, the linear non-

autonomous dynamical system generated by equation (13.21) is asymptotic compact,

and all solutions of all equations (13.22) are bounded on R

. Then

(i) there is an M > 0 such that (13.24) is valid for all t ∈ R

, v ∈ E and

B ∈ H(A),

(ii) the set B deﬁned by formula (13.23) is closed in E ×H(A),

(iii) all solutions of all equations (13.22) bounded on R are recurrent,

(iv) for any B ∈ H(A) equation (13.22) has only a ﬁnite number n

of solutions

that are linearly independent and bounded on R, and n

= n

for all B ∈

H(A),

(v) for any v

∈ E and B ∈ H(A) there is a (v, B) ∈ B such that

lim

t→+∞

|ϕ(t, v

, B) − ϕ(t, v, B)| = 0,

that is, any solution of any equation (13.22) is asymptotic recurrent.

We now formulate some suﬃcient conditions for the asymptotic compactness of

the linear non-autonomous dynamical system generated by equation (13.21).

Theorem 13.7 Let A ∈ C(R, [E]), A(t) = A

(t)+A

(t) for all t ∈ R, and assume

that H(A

), i = 1, 2, are compact and the following conditions hold.

(i) The zero solution of the equation

= A

(t)u (13.25)

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418 Global Attractors of Non-autonomous Dissipative Dynamical Systems

is uniformly asymptotically stable, that is, there are positive numbers N and ν

such that

kU(t, A

−1

(τ, A

)k ≤ Ne

−ν(t−τ)

(13.26)

for all t ≥ τ (t, τ ∈ R), where U(t, A

) is the Cauchy’s operator of the equation

(13.25).

(ii) The family of operators {A

(t) | t > 0} is uniformly completely continuous, that

is, for any bounded set A ∈ E the set {A

(t)A | t > 0} is relatively compact.

Then the linear non-autonomous dynamical system generated by equation

(13.21) is asymptotic compact.

Proof. Let B ∈ H(A). Then there are {t

} ⊂ R and B

∈ H(A

), i = 1, 2, such

that B(t) = B

(t) + B

(t) and B

(t) = lim

n→+∞

(t + t

). Note that

ϕ(t, v, B) = U(t, B

)v +

U(t, B

−

1(τ, B

(τ)ϕ(τ, v, B)dτ. (13.27)

By Lemma 13.6,

U(t, B

) = lim

n→+∞

U(t, A

), A

(t) := A

(t + t

and the equality

U(t, A

−

1(τ, A

) = U(t + t

, A

−

1(τ + t

, A

)

and inequality (13.26) imply that

kU(t, B

−

1(τ, B

)k ≤ Ne

−ν(t−τ)

(13.28)

for all t ≥ τ and B

∈ H(A

). By Lemma 13.5, Theorem 13.7 will be proved if we

can prove that the set

{

U(t, B

−

1(τ, B

(τ)ϕ(τ, v, B)dτ | (v, B) ∈ A}

is relatively compact for every t > 0 and every bounded positively invariant set

A ⊆ E × Y. We put

(t)ϕ(t, v, B) | t ∈ R

, (v, B) ∈ A}.

Then

U(t, B

−

1(τ, B

(τ)ϕ(τ, v, B)dτ (13.29)

∈ t ·

conv{U(t, B

−

1(τ, B

)w | 0 ≤ τ ≤ t, B

∈∈ H(A

), w ∈ K

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Linear almost periodic dynamical systems 419

Since H(A

), H(A), and K

are compact sets, formula (13.29), condition (ii)

of Theorem 13.7, and Lemma 13.6 imply that

{U(t, B

−

1(τ, B

)w | 0 ≤ τ ≤

t, B

∈ H(A

), w ∈ K

} is compact, which completes the proof of the theorem. 

Theorem 13.8 Let H(A) be compact and assume that there is a ﬁnite-

dimensional projection P ∈ [E] such that

(i) the family of projections {P (t) | t ∈ R}, where P (t) := U (t, A)P U

−1

(t, A), is

relatively compact in [E], and

(ii) there are positive numbers N and ν such that

kU(t, A)QU

−1

(τ, A)k ≤ N e

−ν(t−τ)

for all t ≥ τ, where Q := I − P.

Then the linear non-autonomous dynamical system generated by equation

(13.21) is asymptotically compact.

Proof. Since the family of projections {P (t) | t ∈ R} is relatively compact,

we can assume that the sequence {P (t

)} converges. Let P (B) := lim

n→+∞

P (t

We claim that the family H :=

{P (t) | t ∈ R} is uniformly completely continu-

ous, where the bar denotes closure in [E]. Indeed, let A be a bounded subset of

E, {x

} ⊆ {QA | Q ∈ H}, and ε

↓ 0. Then there are t

∈ R and v

∈ A such

that ρ(x

, P (t

) ≤ ε

. Since the sequence {P (t

)} is relatively compact, we can

assume that it converges. Let L := lim

n→+∞

P (t

). Then L is completely continuous,

which implies that the sequence {x

} := {Lv

} is relatively compact. Note that

ρ(x

, x

) ≤ ρ(x

, P (t

) + ρ(P (t

, Lv

) ≤ ε

+ kP (t

) − Lk|v

which implies that ρ(x

, x

) → 0 as n → +∞. Hence, {x

} is relatively compact.

Assume that B ∈ H(A) and {t

} ⊂ R are such that

B = lim

n→+∞

, P (B) := lim

n→+∞

P (A

where P (A

) := U (t

, A)P U

−1

, A). The assertions proved above imply that

the family {P (B) | B ∈ H(A)} is uniformly completely continuous. Note that

Q(B) = lim

n→+∞

Q(A

), where Q(B) := I − P (B) and Q(A

) := I − P (A

Moreover, condition (ii) of Theorem 13.8 implies that

kU(t, A

)Q(A)U

−1

(τ, A

)k ≤ Ne

−ν(t−τ)

(13.30)

for all t ≥ τ. Passing to the limit in (13.30) as n → +∞ and taking Lemma 13.6

into account, we obtain that

kU(t, B)Q(B)U

−1

(τ, B)k ≤ Ne

−ν(t−τ)

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420 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all t ≥ τ and B ∈ H(A). We complete the proof of the theorem by observing

that U(t, B)Q(B) + U (t, B)P (B) = U(t, B) and applying Lemma 13.5. 

Linear functional-diﬀerential equations. Let r > 0, C([a, b], R

) be the

Banach space of all continuous functions ϕ : [a, b] → R

with the norm sup . For

[a, b] := [−r, 0] we put C := C([−r, 0], R

). Let c ∈ R, a ≥ 0, and u ∈ C([c − r, c +

a], R

). We deﬁne u

∈ C for any t ∈ [c, c + a] by the relation u

(θ) := u(t + θ), −r ≥

θ ≥ 0. Let A = A(C, R

) be the Banach space of all linear operators that act from

C → R

equipped with the operator norm, let C(R, A) be the space of all operator-

valued functions A : R → A with the compact-open topology, and let (C(R, A), R, σ)

be the dynamical system of shifts on C(R, A). Let H(A) :=

| τ ∈ R}, where

is the shift of the operator-valued function A by τ and the bar denotes closure

in C(R, A).

Consider the linear functional-diﬀerential equation with delay

= A(t)u

(13.31)

along with the family of equations

= B(t)v

, (13.32)

where B ∈ H(A). Let ϕ(t, v, B) be the solution of equation (13.32) satisfying the

condition ϕ(0, v, B) = v and deﬁned for all t ≥ 0. Let Y := H(A) and denote

the dynamical system of shifts on H(A) by (Y, R, σ). Let X := C × Y and let

π := (ϕ, σ) be the dynamical system on X deﬁned by the equality π(τ, (v, B)) :=

(ϕ(τ, v, B), B

). The non-autonomous system h(X, R

, π), (Y, R, ), hi(h := pr

X → Y ) is linear.

Lemma 13.7 Let H(A) be compact in C(R, A). Then the linear non-autonomous

dynamical system h(X, R

, π), (Y, R, σ), hi generated by equation (13.31) is com-

pletely continuous, that is, for any bounded set A ⊆ X there is an l = l(A) > 0

such that π

A is relatively compact.

Proof. This follows from general properties of solutions of linear functional- diﬀer-

ential equations with delay (see, for example,

[

175

]

, Lemmas 2.2.3 and 3.6.1) since

Y = H(A) is compact. 

Applying the results obtained in section 13.1 to the linear non-autonomous

dynamical system generated by equation (13.31), we obtain the following assertions.

Theorem 13.9 Let A ∈ C(R, A) be recurrent. Then the following conditions are

equivalent :

(i) all solutions of all equations (13.32) are bounded on R

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Linear almost periodic dynamical systems 421

(ii) there is a positive number M such that |ϕ(t, v, B)| ≤ M|v| for all t ≥ 0, v ∈ C,

and B ∈ H(A).

Theorem 13.10 Let A ∈ C(R, A) be recurrent, and assume that all solutions of

all equations (13.32) are bounded on R

. Then

(i) the set of all the solutions of all equations (13.32) that are bounded on R is

closed in C(R, C) × H(A),

(ii) all the solutions of all equations (13.32) that are bounded on R are recurrent,

(iii) for any B ∈ H(A) equation (13.32) has only a ﬁnite number n

of solutions

that are linearly independent and bounded on R, and n

= n

for all B ∈

H(A),

(iv) all solutions of all equations (13.32) are asymptotic recurrent.

Now consider the neutral functional-diﬀerential equation

= A(t)u

, (13.33)

where A ∈ C(R, A) and the operator D ∈ A is atomic at zero

[

175, p.67

]

. Like

(13.31), equation (13.33) generates a linear non-autonomous dynamical system

h(X, R

, π), (Y, R, σ), hi, where X := C × Y, Y := H(A), and π := (ϕ, σ).

Lemma 13.8 Let H(A) be compact, and assume that the operator D is sta-

ble, that is, the zero solution of the homogeneous diﬀerence equation Dy

= 0 is

uniformly asymptotically stable

[

175

]

. Then the linear non-autonomous dynamical

system (X, R

, π), (Y, R, σ), hi generated by equation (13.33) is asymptotically com-

pact.

Proof. This follows from Theorem 12.3.2 and Lemma 12.4.1 in

[

175

]

, since Y =

H(A) is compact. 

Therefore, Theorems 13.9 and 13.10 hold for equations (13.33).

Linear partial diﬀerential equations. Consider the diﬀerential equation

(13.21) with unbounded coeﬃcients. Let A ∈ C(R, Λ), where Λ is a complete metric

space of linear closed operators that act on E (for example, Λ := {A

+B | B ∈ [E]},

where A

is a closed operator that acts on E). Consider the H-class (13.22) of

equation (13.21), where B ∈ H(A). We assume that the following conditions are

fulﬁlled for equation (13.21) and its H-class:

(i) for any v ∈ E and B ∈ H(A) equation (13.22) has precisely one solution

ϕ(0, v, B) that is deﬁned on R

and satisﬁes the condition ϕ(0, v, B) = v;

(ii) the map ϕ : (t, v, B) → ϕ(t, v, B) is continuous in the topology of R

× E ×

C(R, A);

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422 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(iii) for every t ∈ R

the map U(t, ·) : H(A) → [E] is continuous, where U(t, B)

is the Cauchy’s operator of equation (13.22), that is, U (t, B)v := ϕ(t, v, B) for

all t ∈ R

and v ∈ E.

Under the above assumptions equation (13.21) generates a linear non-

autonomous dynamical system to which the results obtained in section 13.1 can

be applied. Therefore, Theorems 13.5 and 13.6 hold in this case.

In conclusion we consider a partial diﬀerential equation that satisﬁes conditions

(i)-(iii).

Example 13.1 A closed linear operator A : D(A) → E with dense domain D(A)

is said

[

188

]

to be sectorial if one can ﬁnd a φ ∈ (0,

), an M ≥ 1, and a real number

a such that the sector

a,φ

:= {λ | |arg (λ − a)| ≤ π, λ 6= a}

lies in the resolvent set ρ(A) of A and k(λI −A)

−1

k ≤ M|λ −a|

−1

for all λ ∈ S

a,φ

An important class of sectorial operators is formed by elliptic operators

[

188

]

[

202

]

Consider the diﬀerential equation

= (A

+ A

(t))u, (13.34)

where A

is a sectorial operator that does not depend on t ∈ R, and A

∈ C(R, [E]).

The results of

[

238

]

[

188

]

imply that equation (13.34) satisﬁes conditions (i)-(iii).

Therefore, analogies of Theorems 13.5 and 13.6 hold for equation (13.34).

Remark 13.4 Statements similar to Theorems 13.5 and 13.6 hold for diﬀerence

equations and can be deduced from the results of section 13.1 by applying these

results to linear non-autonomous dynamical systems with discrete time generated by

the corresponding diﬀerence equations.

13.3 Finite-dimensional systems

Throughout this section we assume that the Banach space E is ﬁnite-dimensional

and its norm | · | is induced by the scalar product h·, ·i, that is, | · |

:= h·, ·i. We

propose several conditions for equations (13.21) in a ﬁnite-dimensional space that

are equivalent to the uniform bistability of the zero solution of equation (13.21), and

prove that the uniform Lyapunov stability of the zero solution of equation (13.21)

implies that there is a frame of solutions of equation (13.21) bounded on R whose

Gram determinant is separated from zero.

Let x

, ..., x

be a set of vectors in E. Let us recall

[

]

that the Gram de-

terminant Γ(x

, ..., x

) of the vectors x

, ..., x

is deﬁned to be the determinant

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Linear almost periodic dynamical systems 423

|hx

, x

i,j=1

. The Gram determinant of the vectors x

, ..., x

is no-negative, and

equals zero only if the vectors x

, ..., x

are linearly dependent.

Theorem 13.11 The following assertions are equivalent:

(i) there is an M > 0 such that (13.23) is valid for all t ∈ R and (x, y) ∈ B;

(ii) B is closed;

(iii) B is a subbundle of F , that is, B is closed and there is a k such that dimB

= k

for all y ∈ Y ;

(iv) all the motions in F that are non-trivial and bounded on R are separated from

zero, that is, inf{ |ϕ(t, x, y)| : t ∈ R} > 0 for any (x, y) ∈ B such that x 6= 0;

(v) one can ﬁnd a y

∈ Y and a basis ξ

, ..., ξ

∈ B

such that

inf

t∈R

Γ(ξ

, ..., ξ

; t) := α > 0, (13.35)

where ξ

= (x

, y

), i = 1, ..., k and Γ(ξ

, ..., ξ

; t) is the Gram determinant of

the vectors π(t, ξ

) ∈ E × Y, i = 1, ..., k.

Proof. Assertions (i) and (ii) are equivalent by Theorem 13.1. By Theorems 13.1

and 13.2, (ii) implies (iii) (the reverse implication is obvious). The equivalence of

conditions (iii) and (iv) follows from

[

]

, Theorem 8.22.

Assume that (iv) is fulﬁlled, let y

∈ Y, and let ξ

, ..., ξ

∈ B

be a basis in B.

We claim that (13.35) holds. Assume the contrary. Then one can ﬁnd a {t

} ⊂ R

such that |t

| → +∞ and Γ(ξ

, ..., ξ

; t

) → 0 as n → +∞. Since all non-zero

motions in F are separated from zero, Lemma 13.1 implies that ξ

, ..., ξ

and y

are jointly recurrent. Without loss of generality, we can assume that the sequences

{π(t

, ξ

)}, i = 1, ..., k and {σ(t

, y)} are convergent.

Let

:= lim

n→+∞

π(t

, ξ

), i = 1, ..., k, q := lim

n→+∞

σ(t

, y

Then

Γ(η

, ..., η

) = lim

n→+∞

Γ(π(t

, ξ

), ..., π(t

, ξ

)) = 0.

Hence, η

, ..., η

are linearly dependent. Repeating the above argument, we deduce

from the last fact that ξ

, ..., ξ

are linearly dependent, which contradicts their

choice. Hence, (iv) implies (v).

We claim that (v) implies (iv). First, we prove that for any q ∈ Y one can ﬁnd

a basis η

, ..., η

such that Γ(η

, ..., η

; t) ≥ α > 0. Indeed, since Y is minimal, there

is a sequence {t

} ⊂ R such that σ(t

, y

) → q. Since ξ

, ..., k ∈ B

, we can assume

that the sequences {π(t

, ξ

)}, i = 1, ..., k are convergent. Let η

:= lim

n→+∞

π(t

, ξ

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

424 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Then η

, ..., η

∈ B

and

Γ(η

, ..., η

; t) = lim

n→+∞

Γ(ξ

, ..., ξ

; t + t

) (13.36)

for all t ∈ R. Therefore, η

, ..., η

are linearly independent. Hence, n

:= dimB

≥

:= dimB

. Since Y is minimal, the reverse inequality also holds. Therefore,

= n

for all q ∈ Y. This implies that η

, ..., η

is a basis in B

. Thus, we

have shown that for any q ∈ Y there is a basis η

, ..., η

∈ B

that satisﬁes condition

(13.36). For any q ∈ Y and (x, q) ∈ B

we have inf{ |ϕ(t, x, q)| : t ∈ R} > 0. Indeed,

if we assume the contrary, then there are (x

, q) ∈ B

, x

6= 0, and |t

| → +∞ such

that

|ϕ(t

, x

, y)| → 0 (13.37)

as n → +∞. Let η

, ..., η

be a basis in B

, and let η

= (x

, q). Then (13.37) implies

that

Γ(η

, ..., η

) → 0. (13.38)

Since η

, ..., η

and , η

, ..., η

are two bases in B

, there is a non-degenerate linear

transformation that transforms the ﬁrst basis into the second, and vice versa. For-

mula (13.38) implies that a similar relation holds for the basis η

, ..., η

, which

contradicts inequality (13.36). This contradiction completes the proof of the impli-

cation (v)=⇒ (iv). The theorem is proved. 

Applying Theorem 13.11 to the linear non-autonomous dynamical system gen-

erated by equation (13.21), we obtain the following assertion.

Theorem 13.12 Let A ∈ C(R, [E]) be recurrent. Then the following conditions

are equivalent:

(a) the set B := {(u, B) ∈ E×H(A) | sup

t∈R

|ϕ(t, u, B)| < +∞} is closed in E×H(A);

(b) there is a positive number M such that

|ϕ(t, u, B)| ≤ M|u| (13.39)

for all t ∈ R and (u, B) ∈ B;

have the same dimension, that is, all the equations

(13.22) have the same number of solutions that are linearly independent and

bounded on R;

(d) all non-trivial solutions of all equations in the H-class (13.22) bounded on R

are separated from zero, that is,

inf

t∈R

|ϕ(t, u, B)| > 0 (13.40)

for all (u, B) ∈ B, u 6= 0;

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Linear almost periodic dynamical systems 425

(e) there is a basis ϕ

, ..., ϕ

(k = dimB

and B

:= {(u, A) | (u, A) ∈ B}) that

consists of solutions of equation (13.21) bounded on R and satisﬁes the condition

Γ(ϕ

, ..., ϕ

; t) > 0 for all t ∈ R, where Γ(ϕ

, ..., k; t) := |hϕ

(t), ϕ

(t)i|

i,j=1

the Gram determinant of ϕ

, ..., ϕ

Corollary 13.3 Let A ∈ C(R, [E]) be recurrent. Then the following conditions

are equivalent:

(i) all solutions of all equations from the H-class (13.22) are bounded on R;

(ii) there is an M > 0 such that |ϕ(t, u, B)| ≤ M|u| for all t ∈ R and u ∈ E;

(iii) all non-zero solutions of all equations from the H-class (13.22) are bounded

on R and separated from zero, that is, (13.40) is valid for all (u, B) ∈ B such

that u 6= 0;

(iv) there are positive numbers C and such that kU(t, A)k ≤ C for all t ∈ R and

inf{|det U(t, A)| : t ∈ R} = α, where U (t, A) is the Cauchy operator of

equation (13.21).

Proof. This follows from Theorem 13.12, since Γ(ϕ

, ..., ϕ

; t) = |det U(t, A)|

where ϕ

, ..., ϕ

are the column-vectors of the matrix U(t, A). 

Remark 13.5 The equivalence of conditions (i) and (ii) was established in

[

]

[

204

]

. The implication (iv)=⇒ (i) sharpens a result in

[

204

]

Theorem 13.13 Assume that all solutions of all equations from the H-class

(13.22) are bounded on R

. Then there is an M > 0 such that |ϕ(t, u, B)| ≤ M|u|

for all t ∈ R and (u, B) ∈ B, that is, B is closed.

Proof. This follows from Theorems 13.1 and 13.4. 

In conclusion we consider examples that illustrate the above results.

Example 13.2 Let a ∈ C(R, R) be the Bohr almost periodic function deﬁned by

the equality

a(t) :=

∞

k=0

(2k + 1)

3/2

sin

2k + 1

, (13.41)

and let

h(t) :=

a(s)ds =

∞

k=0

(2k + 1)

1/2

sin

2(2k + 1)

Note that a(t + t

) → −a(t) uniformly on R, where t

:= (2n + 1)!!. Therefore,

−a ∈ H(a) :=

| τ ∈ R}. Using the inequality |sin t| ≥

|t| with |t| ≤ 1, we

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

426 Global Attractors of Non-autonomous Dissipative Dynamical Systems

obtain that

h(t) =

∞

k=0

(2k + 1)

1/2

sin

2(2k + 1)

≥

k≥

(

|t|

−1)

(2k + 1)

5/2

≥

|s|≥

(

|t|

−1)

(2s + 1)

5/2

3/2

24|t|

3/2

√

|t|

1/2

→ +∞

as |t| → +∞. This implies that the module of all non-zero solutions of the equation

= a(t)x (13.42)

tend to +∞ as |t| → +∞, whereas those of the equation

= b(t)y, (13.43)

with b := −a ∈ H(a) tend to zero.

The above example is a slight modiﬁcation of the well-known example of Favard

(see

[

137, p.435

]

[

150

]

). Our case diﬀers from Favard’s example in that the

solutions of equation (13.43) are not only bounded on R, but they tend to zero as

|t| → +∞. Thus, a non-zero solution of equation (13.43) is asymptotically stable,

but the zero solution of equation (13.42) is not, even though a ∈ H(b).

Example 13.3 Assume that a ∈ C(R, R) is deﬁned by the formula a(t) := −1 +

sin t

, t ∈ R. For equation (13.42) the sets

:= {(x, b) | x ∈ R, b ∈ H(a)} = R × H(a),

B := {(0, b) | b ∈ H(a)} ∪ {(x, θ) | x ∈ R},

where θ ∈ C(R, R) is the function identically equal to zero, are closed. Thus, the

recurrence of A in Theorem 13.12 (Theorem 13.13) is a suﬃcient condition, but this

condition is not necessary.

13.4 Relationship between diﬀerent types of stability

In 1962 W. Hahn

[

168

]

posed the problem of whether asymptotic stability implies

uniform stability for linear equation

= A(t)x (x ∈ R

) (13.44)

with almost periodic coeﬃcients. C. C. Conley and R. K. Miller

[

123

]

gave a negative

answer to this by constructing a scalar equation x

= a(t)x with the property that

every solution ϕ(t, x, a) → 0 as t → +∞, but the null solution is not uniformly

stable (see also

[

]

). From the results of R. J. Sacker and G. R. Sell

[

275

]

and

I. U. Bronshteyn

[

33, p.141

]

follows, that for linear system (13.44) with recurrent