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

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September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

Method of Lyapunov functions 197

As it turned out, if the equation (5.34) is convergent then this property remains

valid under small nonlinear perturbations. The following theorem takes place.

Theorem 5.24 Let A, f and F be such that H(A), H(f) and H(F ) are compact

in C(R, [E]), C(R, E) and C(R × E, E) respectively, and the function F satisﬁes

the condition of Lipschitz w.r.t. u ∈ E uniformly w.r.t. t ∈ R with the small

enough constant of Lipschitz (Lip(F ) < ν

(Na)

−1

). Then if the equation (5.34) is

convergent, the perturbed equation

˙u = A(t)u + F (t, u) (5.44)

is convergent too.

Proof. Denote by H(A, f, F ) :=

{(A

, f

, F

) : τ ∈ R}. Let v ∈ B, (B, g, G) ∈

H(A, f, F ), τ ∈ R and ϕ(·, v, B, g, G) be the solution of equation

˙v = B(t)v + g(t) + G(t, v), (5.45)

passing through the point v as t = 0.

According to the conditions of the theorem the equation (5.45) has a unique

solution deﬁned on R and passing through the point v as τ = 0, whatever would

be (B, g, G) ∈ H(A, f, F ) and v ∈ E. To see that is suﬃcient to note that having

the conditions of Theorem 5.24 fulﬁlled we can apply Theorem 1.2

[

132

]

(global

theorem of existence and uniqueness) to the equation (5.45).

Assume Y := H(A, f, F ) and by (Y, R, σ) denote the dynamical system of shifts

on Y . Let X := E × Y . Deﬁne on X a dynamical system by the following rule:

(v; B, g, G) := (ϕ(τ, v, B, g, G); B

, g

, G

). By the conditions of Theorem 5.24

Y is compact and the triplet h(X, R, π), (Y, R, σ), hi, where h := pr

: X → Y

is a non-autonomous dynamical system. Let us show that we can apply Theorem

2.15 and Corollary 2.6 to the constructed dynamical system. Really, the set Y

is compact under the conditions of the theorem. It is clear that the set of the

continuous sections Γ(H(A, f, F ), B ×H(A, f, F )) is isomorphic to the set of all the

continuous mapping x : H(A, f, F ) → E (see paragraph 3.3) and, consequently, is

not empty. Deﬁne on X

×X a scalar non-negative function V by the following rule:

V ((v

; B, g, G), (v

; B, g, G)) := kv

−v

B,1

+∞

|U(t, B)(v

−v

)|dt

for every (B, g, G) ∈ H(A, f, F ) and v

, v

∈ E. From Lemma 5.5 follows that

the constructed function satisﬁes the conditions 1.- 3. of Theorem 2.15. Let

ϕ(·, v

, B, g, G) be the solution of the equation (5.45) passing through the point

(i = 1, 2) as τ = 0. Then the identity

˙ϕ(τ, v

, B, g, G) ≡ B(τ)ϕ(τ, v

, B, g, G) + g(τ) + G(τ, ϕ(τ, v

, B, g, G)).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

198 Global Attractors of Non-autonomous Dissipative Dynamical Systems

holds. From this follows that

ϕ(τ, v

, B, g, G) = U(τ, B)(v

−1

(s, B) (5.46)

[g(s) + G(s, ϕ(s, v

, B, g, G))]ds).

From (5.46) and Lemmas 5.4 and 5.5 follows

kϕ(τ, v

, B, g, G) −ϕ(τ, v

, B, g, G)k

+∞

|U(t, B

)(ϕ(τ, v

, B, g, G) − ϕ(τ, v

, B, g, G))|dt

+∞

|U(t, B

)U(τ, B)(v

−v

−1

(s, B) ×

[G(s, ϕ(s, v

, B, g, G)) −G(s, ϕ(s, v

, B, g, G))]ds|dt ≤

+∞

|U(t + τ, B)(v

−v

)|dt +

+∞

U(t + τ, B)U

−1

(s, B) ×

[G(s, ϕ(s, v

, B, g, G)) −G(s, ϕ(s, v

, B, g, G))]ds|dt ≤

exp(−ντ)|v

−v

+∞

N exp(−ν(t + τ − s))Lip(G) ×

|ϕ(s, v

, B, g, G) − ϕ(s, v

, B, g, G)|ds dt =

exp(−ντ)|v

−v

exp(−ντ)Lip(G) ×

exp(νs|ϕ(s, v

, B, g, G) −ϕ(s, v

, B, g, G)|ds ≤

exp(−ντ)akv

−v

B,1

exp(−ντ)Lip(G) ×

exp(νs)kϕ(s, v

, B, g, G) −ϕ(s, v

, B, g, G)k

B,1

ds.

Let us multiply the both parts of the last inequality by exp(ντ ) and assuming

ϕ(τ) := exp(νs)kϕ(s, v

, B, g, G) − ϕ(s, v

, B, g, G)k

B,1

, we will obtain

ϕ(τ) ≤

−v

B,1

Lip(G)

ϕ(τ)ds. (5.47)

Applying to (5.47) the Gronwall-Bellman inequality we will obtain

ϕ(τ) ≤

−v

B,1

exp(

Lip(G)τ). (5.48)

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Method of Lyapunov functions 199

Consequently,

kϕ(τ, v

, B, g, G) −ϕ(τ, v

, B, g, G)k

B,1

≤ (5.49)

−v

|exp(−(ν −

Lip(G))τ).

From the inequality (5.49) we have

V ((ϕ(τ, v

, B, g, G); B

, g

, G

)), (ϕ(τ, v

, B, g, G); B

, g

, G

)) ≤

M exp(−λτ )V ((v

; B, g, G), (v

; B, g, G))

(here M =

N a

, λ = −

N a

Lip(G) + ν > 0) for all v

, v

∈ E, τ ≥ 0 and (B, g, G) ∈

H(A, f, F ). So, all the conditions of Theorem 2.15 and of Corollary 2.6 are fulﬁlled.

Applying them to the constructed non-autonomous dynamical system we complete

the proof of the theorem. 

Corollary 5.8 Under the conditions of Theorem 5.24 if A, f and F are jointly

τ-periodic (almost periodic, recurrent) then the equation (5.44) is convergent and

every equation (5.45) has a unique τ -periodic (almost periodic, recurrent) globally

uniformly asymptotically stable solution.

Note, that the statements analogous to Lemmas 5.4 and 5.5 in the case when

the operator-function A(t) is stationary are proved in

[

132

]

. Corollary 5.8 in the

case of periodicity of A, f and F (E = R

) are established in

[

270, p.99

]

3. Let H be a real or complex Hilbert space. Consider a diﬀerential equation

˙u = f (t, u), (5.50)

where f ∈ C(R×H, H). Along with the equation (5.50) we will consider its H-class

˙v = g(t, v) (g ∈ H(f )). (5.51)

There takes place

Theorem 5.25 Let f ∈ C(R × H, H) and let the set H(f) be compact in C(R ×

H, H). If there exists a self-adjoint operator-function A ∈ C(R, [H]) satisfying the

following conditions:

1. M

⊂ M

∩ M

;

2. RehA(t)(u − v), f(t, u) − f(t, v)i ≤ −α|u − v| for all t ∈ R and u, v ∈ H (h·, ·i

is a scalar product in H, | · |

= h·, ·i and α > 0 );

3. k

A(t)k ≤ β for all t ∈ R (β ≥ 0);

4. hA(t)u, ui ≥ γ|u|

for all t ∈ R u ∈ H and −2α + β := −λ < 0.

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200 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Then the equation (5.50) is convergent, i.e.

a. whatever would be g ∈ H(f) and v ∈ H, there exists a unique solution ϕ(·, v, g)

of the equation (5.51) passing through the point v as t = 0 and deﬁned on R;

b. for every g ∈ H(f ) the equation (5.51) has a unique compact solution ψ

(t) =

ϕ(t, v

, g), deﬁned on R;

c. there exist positive numbers N and ν (not depending on g ∈ H(f ) and v ∈ H)

such that

|ϕ(t, v, g) −ϕ(t, v

, g)| ≤ N exp(−νt)|v −v

| (5.52)

for all g ∈ H(f), v ∈ H and t ∈ R

Proof. Let us prove the formulated statement by several parts.

1. Since M

⊂ M

∩ M

, then from

[

300

]

–

[

302

]

follows the existence of the

continuous mapping q : H(f) → H(A) × H(

A) satisfying the condition

q(f) = (A,

A) and q(g

) = (q(g)

q(g)

)

for all g ∈ H(f) and τ ∈ R.

2. Let us show that under the conditions of the theorem every function g ∈

H(f) satisﬁes the conditions analogous to 1.–4. Really, let g ∈ H(f ) and q(g) =

) ∈ H(A) ×H(

A). Then there exists {τ

} ⊂ R such that g = lim

n→+∞

and

) = ( lim

n→+∞

, lim

n→+∞

). From the condition 2. of the theorem follows

that

RehA(t + τ

)(u − v), f(t + τ

, u)f(t + τ

, v)i ≤ −α|u − v|

(5.53)

for all t ∈ R, u v ∈ H (n=1,2,. . . ). Passing to limit in (5.53) as n → +∞ we will

obtain

RehB

(t)(u − v), g(t, u) − g(t, v)i ≤ −α|u −v|

In the same way we can show that the operator-function B

∈ H(A) (g ∈ H(f ))

satisﬁes the conditions 3. and 4. with the same constants β and γ that has the

operator-function A ∈ C(R, [H]).

3. A slight modiﬁcation of the reasoning in the work

[

267

]

(see also

[

329

]

) allows

to prove that if the conditions of the theorem are fulﬁlled then, whatever would

be u ∈ H, the equation (5.50) has a unique solution ϕ(·, u, f ) deﬁned on R and

satisfying the condition ϕ(0, u, f) = u. Taking into account that along with the

function f every other function g ∈ H(f) satisﬁes the conditions of the theorem,

we make the conclusion that for every v ∈ H and g ∈ H(f ) the equation (5.51) has

a unique solution ϕ(·, v, g) deﬁned on R and satisfying the condition ϕ(0, v, g) = v.

4. Assume Y := H(f) and by (Y, R, σ) denote a dynamical system of shifts on

Y . Let X := H × H(f). Deﬁne on X a dynamical system by the following rule:

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

Method of Lyapunov functions 201

π((v, g), τ) := (ϕ(τ, v, g), g

). Note that under the conditions of the theorem the

set Y is compact. The triplet h(X, R, π), (Y, R, σ), hi, where h := pr

: X → Y ,

is a non-autonomous dynamical system. Let us show that we can apply to the

constructed dynamical system Theorem 2.15 and Corollary 2.6. Really, the set Y

is compact under the conditions of the theorem. The set of continuous sections

Γ(Y, X) is not empty. Deﬁne on X

×X a non-negative scalar function V by the

following rule:

V ((v

, g), (v

, g)) := hB

(0)(v

−v

), v

−v

for every g ∈ H(f) and v

, v

∈ H (B

= q(g)).

It is clear that the function V satisﬁes the conditions 2. and 3. of Theorem 2.15.

Really, according to the condition 4.

γ|v

−v

≤ hA(t)(v

−v

), v

−v

i ≤ δ|v

−v

(5.54)

for all v

, v

∈ H and t ∈ R, where δ := sup{kA(t)k | t ∈ R} (δ < +∞, as M

⊂ M

and H(f ) is compact). For g ∈ H(f ) there exists {τ

} ⊂ R such that A

→ B

From (5.54) and the last correspondence follows that

γ|v

−v

≤ hB

(t)(v

−v

), v

−v

i ≤ δ|v

−v

for all g ∈ H(f), t ∈ R and v

, v

∈ H. In particular, we have

γ|v

−v

≤ V ((v

, g), (v

, g)) ≤ δ|v

−v

(5.55)

for all g ∈ H(f) and v

, v

∈ H.

At last, let us show that the condition 4. of Lemma 2.13 is fulﬁlled too. Really,

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) =

(0)(ϕ(τ, v

, g) − ϕ(τ, v

, g)), ϕ(τ, v

, g) −ϕ(τ, v

, g)i =

(τ)(ϕ(τ, v

, g) − ϕ(τ, v

, g)), ϕ(τ, v

, g) −ϕ(τ, v

, g)i.

By direct calculation we can verify that

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) =

(τ)(ϕ(τ, v

, g) −ϕ(τ, v

, g)), ϕ(τ, v

, g) −ϕ(τ, v

, g)i+

2RehB

(τ)(ϕ(τ, v

, g) −ϕ(τ, v

, g)), g(τ, ϕ(τ, v

, g)) −g(τ, ϕ(τ, v

, g))i.

From the last equality and (5.53) and also from the condition 3. of Theorem 5.25

follows the inequality

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) ≤ (5.56)

β|ϕ(τ, v

, g) − ϕ(τ, v

, g)|

−−2α|ϕ(τ, v

, g) −ϕ(τ, v

, g)|

−λ|ϕ(τ, v

, g) −ϕ(τ, v

, g)|

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202 Global Attractors of Non-autonomous Dissipative Dynamical Systems

On the other hand, from the inequality (5.55) we obtain

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) ≤ δ|ϕ(τ, v

, g) −ϕ(τ, v

, g)|

. (5.57)

From the inequality (5.56) and (5.57) we have

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) ≤ −λδV ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)).

Integrating the last inequality, we get the following inequality

V ((ϕ(τ, v

, g), g

), (ϕ(τ, v

, g), g

)) ≤ exp(−λδτ)V ((v

, g), (v

, g))

for all g ∈ H(f), v

, v

∈ H and τ ∈ R

. So, all the conditions of Theorem 2.15 are

fulﬁlled and applying it and Corollary 2.6 to the constructed dynamical system we

complete the proof of Theorem 5.25. 

Corollary 5.9 Under the conditions of Theorem 5.25 if the right hand side f

of the equation (5.50) is τ-periodic (almost periodic, recurrent), then the equation

(5.50) is convergent and every equation (5.51) has a unique τ -periodic (almost pe-

riodic, recurrent) solution complying with the estimation (5.52).

Remark 5.8 1. In the case when f is τ-periodic, Corollary 5.9 reinforces the

main result of the work

[

192

]

(in

[

192

]

under the same conditions on basis of the

principle of Lere-Showder there was proved only the existence of a unique τ -periodic

solution in the case when H = E

2. If the right hand side f is almost periodic and the operator-function A(t)

does not depend on time (

A(t) ≡ 0), then the statement of Corollary 5.9 coincides

with the main result of the work

[

267

]

. Also note, that the results of the work

[

267

]

are achieved on basis of the series of delicate estimations. Our method is of purely

topological character.

3. Note, at last, that in the work

[

187

]

there are studied so-called α-contracting

non-autonomous dynamical systems (i.e. such systems that satisfy the condition:

ρ(x

t, x

t) ≤ αρ(x

, x

)

for every t ∈ R

, x

∈ X and h(x

) = h(x

), where 0 ≤ α ≤ 1) in the relation

with the problem of stability and asymptotic stability in the sense of Poisson of the

solutions of diﬀerential equations with monotone right hand side.

In the case of ﬁnite-dimensional Hilbert space (H = E

), when the function f

is recurrent and A(t) (

A(t) ≡ 0) is stationary, Theorem 5.25 admits the following

ampliﬁcation.

Theorem 5.26 Let f ∈ C(R × H, H) satisfy the condition of Lipschitz and be

recurrent w.r.t. t ∈ R uniformly w.r.t. u on compacts from H. If there exists a

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

Method of Lyapunov functions 203

self-adjoint operator A ∈ [H] and a positively deﬁned function c : R

→ R

such

that

RehA(u − v), f(t, u) −f(t, v)i ≤ −c(|u −v|) (∀ u, v ∈ H) (5.58)

and c(r)r

−1

→ +∞ as r → +∞ (for example, c(r) = r

1+α

, α > 0), then Equation

(5.50) is convergent.

Proof. In the same way that in Theorem 5.25 we establish that whatever would

be v ∈ H and g ∈ H(f), the equation (5.51) has a unique solution ϕ(·, v, g) deﬁned

on R and satisfying the condition ϕ(0, v, g) = v. Let us deﬁne a function V :

H(f) ×H → R

by the following rule

V(g, v) := hAv, vi. (5.59)

From the equality (5.59) follows that

V(g

, ϕ(τ, v, g)) = 2RehAϕ(τ, v, g), ϕ(τ, v, g)i. (5.60)

Since c(r)r

−1

→ +∞ as r → +∞, there exists r

> 0 such that

c(r)r

−1

> |f

|kAk (5.61)

for all r > r

, where |f

| := sup{|f(t, 0)| : t ∈ R}. Let t > 0, v ∈ H and g ∈ H(f)

be such that |ϕ(τ, v, g)| ≥ r

for all τ ∈ [0, t]. From (5.58), (5.60) and (5.61) follows

that V(g

, ϕ(τ, v, g)) < V(g, v), and according to Theorem 5.7 the equation (5.50)

is dissipative.

Let us denote by V : H(f) × H × H → R

a function deﬁned by the equation

V (g, u, v)) := hA(u − v), (u −v)i. (5.62)

By direct calculation we obtain

V (g

, ϕ(τ, u, g), ϕ(τ, v, g)) = 2RehA(ϕ(τ, u, g) − ϕ(τ, v, g)),

g(τ, ϕ(τ, u, g)) − g(τ, ϕ(τ, v, g))i. (5.63)

from (5.58) and (5.63) follows that V (g

, ϕ(t, u, g), ϕ(t, v, g)) < V (g, u, v) for all

t > 0, g ∈ H(f) and u, v ∈ H (u 6= v). According to Theorem 2.17 the equation

(5.50) is convergent. 

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204 Global Attractors of Non-autonomous Dissipative Dynamical Systems

5.5 Dissipativity and convergence of some equations of 2nd and

3rd order

1. Let us consider the system of two diﬀerential equation of the following kind

(

˙x = y − F (x) + Q(x, y, t)

˙y = −g(x),

(5.64)

where F and g are continuous functions deﬁned on R and Q(x, y, t) is continuous

w.r.t. the set of variables, is bounded and uniformly continuous w.r.t. t ∈ R

uniformly w.r.t. (x, y) on compacts from R

. As we know, we can reduce the

equation

¨x + h(x) ˙x + g(x) = p(t) (5.65)

to the system (5.64) if we introduce a new variable y = ˙x + F (x) − g(t) where

F (x) =

h(τ)dτ , g(t) =

p(s)ds. Along with the system (5.64) consider its

H-class

(

˙x = y − F (x) + G(x, y, t)

˙y = −g(x),

(5.66)

(G ∈ H(Q) =

: τ ∈ R}).

Theorem 5.27 Let the following conditions be satisﬁed:

(1) xg(x) > 0 for |x| ≥ 1;

(2)

+∞

g(x)dx = −

−∞

g(x)dx = +∞;

(3) there exists constant N > 0 such that |Q(x, y, t)| ≤ N for |x| ≤ 1.

(4) [F (x)−Q(x, y, t)]sgn x ≥ 0 for |x| ≥ 1; besides for every G ∈ H(Q) there exists

> 0 such that F (x) − G(x, y, θ

) 6= 0 for |x| ≥ 1 and y ∈ R.

(5) when |x| ≥ 1 the inequality |F (x) − Q(x, y, t)| ≥ α(y) ≥ 0 is valid, where

α(y) on every interval is a measurable in the sense of Lebesgue function and

+∞

−∞

α(y)dy > 0.

Then the system (5.64) is dissipative.

Proof. In the same way that in the work

[

160

]

we show that there exists η > 0

such that under certain τ every solution of the equation (5.66) gets in the rectangle

{(x, y) : |x| ≤ 1, |y| ≤ η} (τ depends on the system (5.66) and the according

solution). Now to complete the proof of the theorem it is suﬃcient to refer to

Theorem 5.8. 

Let us consider a more general equation

¨x + h(x) ˙x + g(x) = R(x, ˙x, t).

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Method of Lyapunov functions 205

Assume y = ˙x + F (x) where F (x) =

h(τ)dτ . Then we obtain the following

system

(

˙x = y − F (x)

˙y = −g(x) + R(x, y, t)

(5.67)

Along with the system (5.67) consider its H-class

(

˙x = y − F (x)

˙y = −g(x) + G(x, y, t)

(5.68)

where G ∈ H(R).

Theorem 5.28 Let the functions h, g and R be continuous and R be bounded

and uniformly continuous w.r.t. t ∈ R uniformly w.r.t. (x, y) on compacts from R

Suppose that in addition there exist constants L > N > 0 and K > 0 such that

(1) |R(x, y, t)| ≤ N for all x, y, t ∈ R;

(2) g(x)sgn x ≥ L for |x| ≥ 1;

(3) f (x) ≥ K for |x| ≥ 1.

Then the system (5.67) is dissipative.

Proof. As well as in the work

[

270

]

let us consider the following function V(x, y) =

− yF (x) +

F (x) + 2

g(τ)dτ. The derivative of this function taken in virtue

of the system (5.68) equals to

V(x, y, t) = −[y − F (x)]

f(x) − g(x)F (x) + [2y − F (x)]G(x, y, t).

According to

[

270

]

there exists a > 0 such that for x

+ y

≥ a

(x, y, t) < 0 and

V(x, y) → +∞ for x

+ y

→ +∞. According to Theorem 5.7 the system (5.67) is

dissipative. 

Consider the diﬀerential equation of the 3rd order

...

+ a¨x + b ˙x + f(x) = p(x, ˙x, ¨x, t)

where a, b ∈ R. Assume y = ax + ˙x, z = bx + a ˙x + ¨x. Then this equation is

equivalent to the system











˙x = y − ax

˙y = z − bx

˙z = −f(x) + p(x, y, z, t)

(5.69)

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206 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Along with the system (5.69) consider its H-class











˙x = y − ax

˙y = z − bx

˙z = −f (x) + g(x, y, z, t)

(5.70)

where g ∈ H(p).

Theorem 5.29 Let the function f and p(x, y, z, t) be continuous and let the

following conditions be satisﬁed:

1. a, b > 0;

2. 0 <

f(x)

< ab for |x| ≥ 1;

3. lim

|x|→+∞

|f(x)| = +∞;

4. lim

|x|→+∞

|f(x) − abx| = +∞;

5. p is bounded and uniformly continuous w.r.t. t ∈ R uniformly w.r.t. (x, y, t) on

compacts from R

;

6. there exists A > 0 such that |p(x, y, z, t)| ≤ A for all x, y, z and t ∈ R.

Then the system (5.69) is dissipative.

Proof. To prove the formulated statement, following

[

270

]

we will consider the

function

V(x, y, z) =

x − ay + z)

(z − bx)

+ a

[f(τ ) − abτ]dτ.

Under the conditions of Theorem 5.29 V(x, y, z) → +∞ for x

+ y

+ z

→ +∞

and the derivative of this function taken by virtue of the system (5.70) being to

(x, y, z, t) = −a(a

x − ay + z)

+[abx − f (x)][2(a

x − ay + z) −bx][(a

−b)x −ay + 2z]g(x, y, z, t).

In the same way that in

[

270

]

we can show that there exists a > 0 such that out

of the ball of radius a the function V decreases along the trajectories of (5.70) and,

according to Theorem 5.7, the system (5.69) is dissipative. 

2. Let us give two criterions of convergence of the equation (5.65) that is equiv-

alent to the system

(

˙x = y

˙y = −x − h(x)y + p(t),

or, respectively, to the system

(

˙x = −F (x) + v

˙v = −x + p(t)

(5.72)