Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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ADVANCED STABILITY THEORY 235

Next, we provide two converse theorems for stability of time-varying

systems.

Theorem 4.8. Assume that the nonlinear time-varying dynamical

system (4.57) is u niformly asymptotically s table and f : [0, ∞) ×D → R

continuously diﬀerentiable. Let δ > 0 be such that B

(0) ⊂ D is contained

in the domain of attraction of (4.57) and assume

∂f

∂x

is bounded on B

(0)

uniformly in t. Then there exist a continuously diﬀerentiable function

V : [0, ∞) × B

(0) → R and class K functions α(·), β(·), and γ(·) such

that

α(kxk) ≤ V (t, x) ≤ β(kxk), (t, x) ∈ [0, ∞) × B

(0), (4.98)

V (t, x) ≤ −γ(kxk), (t, x) ∈ [0, ∞) ×B

(0). (4.99)

Proof. The proof is a direct consequence of T heorem 4.3 with n

= n,

= 1, x

(t − t

) = x(t), x

(t − t

) = t, f

, x

) = f(t, x), f

, x

) = 1,

and V (x

, x

) = V (t, x).

Theorem 4.9. Assume that the nonlinear time-varying dynamical

system (4.57) is (uniformly) exponentially stable and f : [0, ∞) × D → R

is continuously diﬀerentiable. Let δ > 0 be such that B

(0) ⊂ D is

contained in the domain of attraction of (4.57) and assume

∂f

∂x

is bounded

on B

(0) uniformly in t. Th en , for every p > 1, there exist a continuously

diﬀerentiable function V : [0, ∞) × B

(0) → R and positive constants α, β,

and γ such that

αkxk

≤ V (t, x) ≤ βkxk

, (t, x) ∈ [0, ∞) ×B

(0), (4.100)

V (t, x) ≤ −γkxk

, (t, x) ∈ [0, ∞) × B

(0). (4.101)

Proof. The proof is a direct consequence of T heorem 4.4 with n

= n,

= 1, x

(t − t

) = x(t), x

(t − t

) = t, f

, x

) = f(t, x), f

, x

) = 1,

and V (x

, x

) = V (t, x).

Finally, we state Lyapunov’s and Chetaev’s instability theorems for

time-varying systems. The proofs of these results are virtually identical

to the proofs of their time-invariant counterparts given in Section 3.6, and

hence, are left as an exercise for the reader.

Theorem 4.10. Consider the nonlinear dynamical system (4.57).

Assume that there exist a continuously diﬀerentiable fu nction V : [0, ∞) ×

D → R and class K functions β(·) and γ(·) such that

V (t, 0) = 0, t ∈ [0, ∞), (4.102)

V (t, x) ≤ β(kxk), (t, x) ∈ [0, ∞) × D, (4.103)

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236 CHAPTER 4

V (t, x) ≥ γ(kxk), (t, x) ∈ [0, ∞) × D. (4.104)

Furth ermore, assu me that for every suﬃciently small δ > 0 there exist a

time t

≥ 0 an d x

∈ D such that kx

k < δ and V (t

, x

) > 0. Then the

zero solution x(t) ≡ 0 to (4.57) is u nstable.

Theorem 4.11. Consider the nonlinear dynamical system (4.57).

Assume that there exist a continuously diﬀerentiable fu nction V : [0, ∞) ×

D → R, a class K function β(·), a function W : [0, ∞) × D → R, a time

≥ 0, and scalars ε, λ > 0 such that

V (t

, 0) = 0, (4.105)

V (t, x) ≤ β(kxk), (t, x) ∈ [0, ∞) × D, (4.106)

W (t, x) ≥ 0, (t, x) ∈ [0, ∞) ×B

(0), (4.107)

V (t, x) ≥ λV (t, x) + W (t, x). (4.108)

Furth ermore, assume that for every suﬃciently small δ > 0 there exists

∈ D such that kx

k < δ and V (t

, x

) > 0. Then the zero solution

x(t) ≡ 0 to (4.57) is unstable.

Theorem 4.12. Consider the nonlinear dynamical system (4.57).

Assume that there exist a continuously diﬀerentiable fu nction V : [0, ∞) ×

D → R, a class K function γ(·), a scalar ε > 0, and an open set Q ⊆ B

(0)

such that

V (t, x) > 0, (t, x) ∈ [0, ∞) × Q (4.109)

sup

t≥0

sup

x∈Q

V (t, x) < ∞, (4.110)

0 ∈ ∂Q, (4.111)

V (t, x) = 0, (t, x) ∈ [0, ∞) × (∂Q ∩ B

(0)), (4.112)

V (t, x) ≥ γ(kxk), (t, x) ∈ [0, ∞) ×Q. (4.113)

Then the zero solution x(t) ≡ 0 to (4.57) is unstable.

4.4 Lagrange Stability, Bounde d ne ss, and Ultimate Boundedne ss

In the previous sections we introduced the concepts of stability and

partial stability for nonlinear dynamical systems. In certain engineering

applications, however, it is more natural to ascertain whether for every

system initial condition in a ball of radiu s δ the solution of the nonlinear

dynamical system is boun ded. T his leads to the notions of Lagrange stability,

boundedness, and ultimate boundedness. These notions are closely related

to what is known in the literature as practical stability. In this section, we

present Lyapunov-like th eorems for boundedness and ultimate boundedness

of nonlinear dynamical systems.

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ADVANCED STABILITY THEORY 237

Deﬁnition 4.3. i) The nonlinear dynamical system (4.6) and (4.7) is

Lagrange stable with respect to x

if, for every x

∈ D and x

∈ R

, there

exists ε = ε(x

, x

) > 0 such that kx

(t)k < ε, t ≥ 0.

ii) The nonlinear dynamical system (4.6) and (4.7) is bounded with

re spect to x

uniformly in x

if there exists γ > 0 such that, for every

δ ∈ (0, γ), there exists ε = ε(δ) > 0 such that kx

k < δ implies kx

(t)k < ε,

t ≥ 0. The nonlinear dynamical system (4.6) and (4.7) is g lobally bounded

with respect to x

uniformly in x

if, for every δ ∈ (0, ∞), there exists

ε = ε(δ) > 0 such that kx

k < δ implies kx

(t)k < ε, t ≥ 0.

iii) T he nonlinear dynamical system (4.6) and (4.7) is ultimately boun-

ded with respect to x

uniformly in x

with bound ε if there exists γ > 0 such

that, for every δ ∈ (0, γ), there exists T = T (δ, ε) > 0 such that kx

k < δ

implies kx

(t)k < ε, t ≥ T . The nonlinear dynamical system (4.6) and (4.7)

is globally ultimately bounded with respect to x

uniformly in x

with bound

ε if, for every δ ∈ (0, ∞), there exists T = T (δ, ε) > 0 such that kx

k < δ

implies kx

(t)k < ε, t ≥ T .

Note that if a nonlinear dynamical system is globally bounded with

respect to x

uniformly in x

, th en it is Lagrange stable with respect to

. Alternatively, if a nonlinear dynamical system is (globally) bounded

with respect to x

uniformly in x

, then there exists ε > 0 such that it

is (globally) ultimately bounded with respect to x

uniformly in x

with a

bound ε. Conversely, if a nonlinear dynamical system is (globally) ultimately

bounded with respect to x

uniformly in x

with a bound ε, then it is

(globally) bou nded with respect to x

uniformly in x

. The following results

present Lyapunov-like theorems for boundedn ess and ultimate boundedness.

For these results deﬁne

V (x

, x

)

△

= V

′

, x

)f(x

, x

), where f (x

, x

)

△

, x

), f

, x

)]

and V : D × R

→ R is a given continuously

diﬀerentiable function.

Theorem 4.13. C onsider the nonlinear dyn amical system (4.6) and

(4.7). Assume that there exist a continuously diﬀerentiable function V :

D × R

→ R and class K functions α(·), β(·) such that

α(kx

k) ≤ V (x

, x

) ≤ β(kx

k), x

∈ D, x

∈ R

, (4.114)

V (x

, x

) ≤ 0, x

∈ D, kx

k ≥ µ, x

∈ R

, (4.115)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η ≥ β(µ). Then the

nonlinear dynamical system (4.6) and (4.7) is boun ded with respect to x

uniformly in x

. Furth ermore, for every δ ∈ (0, γ), x

∈ B

(0) implies that

(t)k ≤ ε, t ≥ 0, where

ε = ε(δ)

△



−1

(β(δ)), δ ∈ (µ, γ),

−1

(η), δ ∈ (0, µ],

(4.116)

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238 CHAPTER 4

and γ

△

= sup{r > 0 : B

−1

(β(r))

(0) ⊂ D}. If, in addition, D = R

and

α(·) is a class K

∞

function, then the nonlinear dynamical system (4.6) and

(4.7) is globally bounded with respect to x

uniformly in x

and for every

∈ R

, kx

(t)k ≤ ε, t ≥ 0, where ε is given by (4.116) with δ = kx

Proof. First, let δ ∈ (0, µ] and assu me kx

k ≤ δ. If kx

(t)k ≤ µ,

t ≥ 0, then it follows from (4.114) that kx

(t)k ≤ µ ≤ α

−1

(β(µ)) ≤ α

−1

(η),

t ≥ 0. Alternatively, if there exists T > 0 such that kx

(T )k > µ, then

it follows from the continuity of x

(·) that there exists τ < T such that

(τ)k = µ and kx

(t)k ≥ µ, t ∈ [τ, T ]. Hence, it follows from (4.114) and

(4.115) that

α(kx

(T )k) ≤ V (x

(T ), x

(T )) ≤ V (x

(τ), x

(τ)) ≤ β(µ) ≤ η,

which implies that kx

(T )k ≤ α

−1

(η). Next, let δ ∈ (µ, γ) and assume

∈ B

(0) and kx

k > µ. Now, for every

t > 0 such that kx

(t)k ≥ µ,

t ∈ [0,

t], it follows from (4.114) and (4.115) that

α(kx

(t)k) ≤ V (x

(t), x

(t)) ≤ V (x

, x

) ≤ β(δ), t ≥ 0,

which implies that kx

(t)k ≤ α

−1

(β(δ)), t ∈ [0,

t]. Next, if there exists

T > 0 such that kx

(T )k ≤ µ, then it follows as in the proof of the ﬁrst

case that kx

(t)k ≤ α

−1

(η), t ≥ T . Hence, if x

∈ B

(0) \ B

(0), then

(t)k ≤ α

−1

(β(δ)), t ≥ 0. Finally, if D = R

and α(·) is a class K

∞

function it follows that β(·) is a class K

∞

function, an d hence, γ = ∞.

Hence, the n onlinear dynamical system (4.6) and (4.7) is globally bounded

with respect to x

uniformly in x

Theorem 4.14. C onsider the nonlinear dyn amical system (4.6) and

(4.7). Assume that there exist a continuously diﬀerentiable function V :

D × R

→ R and class K functions α(·), β(·) such that (4.114) holds.

Furth ermore, assume that there exists a continuous function W : D → R

such that W (x

) > 0, kx

k > µ, and

V (x

, x

) ≤ −W (x

), x

∈ D, kx

k > µ, x

∈ R

, (4.117)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η > β(µ). Then the

nonlinear dynamical system (4.6) and (4.7) is ultimately bounded with

respect to x

uniformly in x

with bound ε

△

= α

−1

(η). Furthermore,

lim sup

t→∞

(t)k ≤ α

−1

(β(µ)). If, in addition, D = R

and α(·) is a

class K

∞

function, then the nonlinear dynamical system (4.6) and (4.7) is

globally ultimately bounded with respect to x

uniformly in x

with bound

ε.

Proof. First, let δ ∈ (0, µ] an d assume kx

k ≤ δ. As in the proof of

Theorem 4.13, it follows that kx

(t)k ≤ α

−1

(η) = ε, t ≥ 0. Next, let δ ∈

(µ, γ), where γ

△

= sup{r > 0 : B

−1

(β(r))

(0) ⊂ D} and assume x

∈ B

(0)

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ADVANCED STABILITY THEORY 239

and kx

k > µ. In this case, it follows from Theorem 4.13 that kx

(t)k ≤

−1

(β(δ)), t ≥ 0. Suppose, ad absurdum, that kx

(t)k ≥ β

−1

(η), t ≥ 0,

or, equivalently, x

(t) ∈ O

△

= B

−1

(β(δ))

(0) \ B

−1

(η)

(0), t ≥ 0. Since O is

compact and W (·) is continuous and W (x

) > 0, kx

k ≥ β

−1

(η) > µ, it

follows from Theorem 2.13 that k

△

= min

∈O

W (x

) > 0 exists. Hence, it

follows from (4.117) that

V (x

(t), x

(t)) ≤ V (x

, x

) −kt, t ≥ 0, (4.118)

which implies that

α(kx

(t)k) ≤ β(kx

k) − kt ≤ β(δ) − kt, t ≥ 0. (4.119)

Now, letting t > β(δ)/k it follows that α(kx

(t)k) < 0, which is a contradic-

tion. Hence, th ere exists T = T (δ, η) > 0 such that kx

(T )k < β

−1

(η). Thus,

it follows from Theorem 4.13 that kx

(t)k ≤ α

−1

(β(β

−1

(η))) = α

−1

(η),

t ≥ T , wh ich proves that the nonlinear dynamical system (4.6) an d (4.7)

is ultimately bounded with respect to x

uniformly in x

with bound

ε = α

−1

(η). Furtherm ore, lim sup

t→∞

(t)k ≤ α

−1

(β(µ)). Finally, if

D = R

and α(·) is a class K

∞

function it follows that β(·) is a class K

∞

function, and hence, γ = ∞. Hence, the nonlinear dynamical system (4.6)

and (4.7) is globally ultimately bounded with respect to x

uniformly in x

with bound ε.

Next, we specialize Theorems 4.13 and 4.14 to nonlinear time-varying

dynamical systems. The following deﬁnition is needed for these results.

Deﬁnition 4.4. i) The nonlinear time-varying dynamical system (4.57)

is Lagrange stable if, for every x

∈ R

and t

∈ R, there exists ε =

ε(t

, x

) > 0 such that kx(t)k < ε, t ≥ t

ii) The n on linear time-varying dynamical system (4.57) is uniformly

bounded if there exists γ > 0 such that, for every δ ∈ (0, γ), th ere exists

ε = ε(δ) > 0 such that kx

k < δ implies kx(t)k < ε, t ≥ t

. The nonlinear

time-varying dynamical system (4.57) is g lobally uniformly bounded if, for

every δ ∈ (0, ∞), there exists ε = ε(δ) > 0 such that kx

k < δ imp lies

kx(t)k < ε, t ≥ t

iii) The nonlinear time-varying dynamical system (4.57) is uniformly

ultimately bounded with bound ε if there exists γ > 0 s uch that, for every δ ∈

(0, γ), th ere exists T = T (δ, ε) > 0 such that kx

k < δ implies kx(t)k < ε,

t ≥ t

+ T . The nonlinear time-varying dynamical system (4.57) is globally

uniformly ultimately bounded with bound ε if, for every δ ∈ (0, ∞), there

exists T = T (δ, ε) > 0 such that kx

k < δ implies kx(t)k < ε, t ≥ t

+ T .

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240 CHAPTER 4

For the following result deﬁne

V (t, x)

△

∂V

∂t

∂V

∂x

(t, x)f(t, x),

where V : R × D → R is a given continuously diﬀerentiable function.

Corollary 4.1. Consider the nonlinear time-varying dynamical system

(4.57). Assume that there exist a continuously diﬀerentiable function V :

R ×D → R and class K functions α(·), β(·) such that

α(kxk) ≤ V (t, x) ≤ β(kxk), x ∈ D, t ∈ R, (4.120)

V (t, x) ≤ 0, x ∈ D, kxk ≥ µ, t ∈ R, (4.121)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η ≥ β(µ). Then the

nonlinear time-varying dynamical system (4.57) is un iformly boun ded. If,

in addition, D = R

and α(·) is a class K

∞

function, then the nonlinear

time-varying dynamical system (4.57) is globally uniformly bounded.

Proof. The result is a direct consequence of Theorem 4.13. Speciﬁ-

cally, let n

= n, n

= 1, x

(t−t

) = x(t), x

(t−t

) = t, f

, x

) = f(t, x),

and f

, x

) = 1. Now, note that with τ = t − t

, the solution x(t),

t ≥ t

, to the nonlinear time-varying dynamical system (4.57) is equivalently

characterized by the solution x

(τ), τ ≥ 0, to the nonlinear autonomous

dynamical system

˙x

(τ) = f

(τ), x

(τ)), x

(0) = x

, τ ≥ 0,

˙x

(τ) = 1, x

(0) = t

where ˙x

(·) and ˙x

(·) denote diﬀerentiation with respect to τ . Now, the

result is a direct consequence of Theorem 4.13.

Corollary 4.2. Consider the nonlinear time-varying dynamical system

(4.57). Assu me that there exist a continuously diﬀerentiable fun ction V : R×

D → R and class K functions α(·), β(·) such that (4.120) holds. Furthermore,

assume th at there exists a continuous function W : D → R such that W (x) >

0, kxk > µ, and

V (t, x) ≤ −W (x), x ∈ D, kxk > µ, t ∈ R, (4.122)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η > β(µ). Then

the nonlinear time-varying dynamical system (4.57) is uniformly ultimately

bounded with bound ε

△

= α

−1

(η). Furthermore, lim su p

t→∞

kx(t)k ≤

−1

(β(µ)). If , in addition, D = R

and α(·) is a class K

∞

function, then

the nonlinear time-varying dynamical system (4.57) is globally uniformly

ultimately bounded with bound ε.

Proof. The proof is a direct consequence of Theorem 4.14 using similar

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ADVANCED STABILITY THEORY 241

arguments as in the proof of Corollary 4.1 and, hence, is omitted.

Finally, we specialize Corollaries 4.1 and 4.2 to nonlinear time-

invariant dynamical systems. For these resu lts we need the f ollowing

specialization of Deﬁnition 4.4.

Deﬁnition 4.5. i) The nonlinear dynamical system (3.1) is Lagrange

stable if, for every x

∈ R

, there exists ε = ε(x

) > 0 such that kx(t)k < ε,

t ≥ 0.

ii) The nonlinear dynamical system (3.1) is bounded if there exists

γ > 0 such that, for every δ ∈ (0, γ), there exists ε = ε(δ) > 0 such that

k < δ implies kx(t)k < ε, t ≥ 0. The nonlinear dynamical system (3.1)

is globally bounded if, for every δ ∈ (0, ∞), there exists ε = ε(δ) > 0 such

that kx

k < δ implies kx(t)k < ε, t ≥ 0.

iii) The nonlinear dynamical system (3.1) is ultimately bounded with

bound ε if there exists γ > 0 such that, for every δ ∈ (0, γ), there exists

T = T (δ, ε) > 0 such that kx

k < δ implies kx(t)k < ε, t ≥ T . The

nonlinear dynamical system (3.1) is globally ultimately bounded with bound

ε if, for every δ ∈ (0, ∞), there exists T = T (δ, ε) > 0 s uch that kx

k < δ

implies kx(t)k < ε, t ≥ T .

Corollary 4.3. Consider th e nonlinear dynamical system (3.1). As-

sume that there exist a continuously diﬀerentiable function V : D → R and

class K functions α(·) and β(·) such that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ D, (4.123)

′

(x)f(x) ≤ 0, x ∈ D, kxk ≥ µ, (4.124)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η ≥ β(µ). Then the

nonlinear dynamical system (3.1) is bounded. If, in addition, D = R

and

V (x) → ∞ as kxk → ∞, then the nonlinear dynamical system (3.1) is

globally bounded.

Proof. The resu lt is a direct consequence of Corollary 4.1.

Corollary 4.4. Consider th e nonlinear dynamical system (3.1). As-

sume that there exist a continuously diﬀerentiable function V : D → R and

class K functions α(·) and β(·) such that (4.123) holds and

′

(x)f(x) < 0, x ∈ D, kxk > µ, (4.125)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D for some η > β(µ). Then the

nonlinear dynamical system (3.1) is ultimately bounded with bound ε

△

−1

(η). Furthermore, lim sup

t→∞

kx(t)k ≤ α

−1

(β(µ)). If, in addition, D =

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242 CHAPTER 4

and V (x) → ∞ as kxk → ∞, then the nonlinear dynamical system (3.1)

is globally ultimately bounded with boun d ε.

Proof. The proof is a direct consequence of Corollary 4.2.

Corollaries 4.3 and 4.4 present Lyapunov-like theorems for bounded-

ness and u ltimate boundedness of a nonlinear dynamical system. To further

elucidate these results, consider the nonlinear dynamical system (3.1) and

assume that th ere exist a positive-deﬁnite, radially unbounded, continuously

diﬀerentiable function V : R

→ R and scalar µ > 0 s uch that

V (x) ≤ 0, x ∈ R

, kxk ≥ µ, t ≥ 0. (4.126)

Furth ermore, let α(·) and β(·) be class K

∞

functions su ch that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ R

. (4.127)

In this case, it can be shown that the set D

△

= {x ∈ R

: V (x) ≤ β(µ)}

is positively invariant. To see this, su ppose, ad absurdum, that there exists

a trajectory x(t), t ≥ 0, such that x(0) ∈ D

and x(T ) 6∈ D

for some

T > 0. Now, note that if x ∈ B

(0), then V (x) ≤ β(kxk) ≤ β(µ). Next,

since x(t), t ≥ 0, is continuous it follows that there exists

t > 0 such that

V (x(

t)) = β(µ) and x(t) 6∈ D

, t ∈ (

t, T ], and hence, kx(t)k > µ, t ∈ (

t, T ].

Now, it follows from (4.126) that

β(µ) < V (x(T )) = V (x(

t)) +

V (x(t))dt ≤ V (x(

t) = β(µ),

which is a contradiction. Hence, if x(0) ∈ D

, then x(t) ∈ D

, t ≥ 0.

Similarly, for δ > µ, if x(0) ∈ B

(0), then it can be s hown that x(t) ∈ D

△

{x ∈ R

: V (x) ≤ β(δ)}. Next, if x ∈ D

, then α(kxk) ≤ V (x) ≤ β(δ),

which implies that x ∈ B

(0), where ε

△

= α

−1

(β(δ)). Hence, if x(0) ∈ B

(0),

then x(t) ∈ B

(0), t ≥ 0 (see Figure 4.6).

If (4.126) is replaced by

V (x) < 0, x ∈ R

, kxk > µ, t ≥ 0, (4.128)

then, using identical arguments as above, it can be shown that if x(0) ∈

(0), then x(t) ∈ D

(0), t ≥ 0. Furthermore, it can be shown that for

every η > µ, the trajectory starting in B

(0) enters D

△

= {x ∈ R

: V (x) ≤

β(η)} in ﬁnite time. Hence, the trajectory either enters D

in ﬁnite time

or approaches D

as t → ∞, which, sin ce D

is positively invariant, implies

that the trajectory ultimately enters B

(0), where ε

△

= α

−1

(β(µ)) (see Figure

4.7).

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ADVANCED STABILITY THEORY 243

(0)

Figure 4.6 Visualizations of the sets B

(0) ⊂ D

⊂ B

(0) ⊂ D

⊂ D

and a bounded

trajectory.

Example 4.9. Consider the nonlinear d ynamical system

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, t ≥ 0, (4.129)

˙x

(t) = −x

(t) + u, x

(0) = x

, (4.130)

where u ∈ R. To show that (4.129) and (4.130) is globally bounded consider

the radially unbounded, positive deﬁnite function V (x

, x

) =

k > 0, and note that

V (x

, x

) = −x

(−x

+ x

) + kx

(−x

+ u)

= −x

+ x

− k(1 − ε)x

− kεx

+ kx

u, ε ∈ (0, 1),

≤ −x

+ x

− k(1 − ε)x

, |x

| ≥

|u|

≤ −









, |x

| ≥

|u|

, (4.131)

where



1 −1/2

−1/2 k(1 − ε)



Now, choosing k ≥ 1/4(1−ε) ensures that R

≥ 0, and hen ce,

V (x

, x

) ≤ 0,

| ≥

|u|

Next, for |x

| ≤

|u|

, it follows that

V (x

, x

) ≤ −x

+ x

− kx

+ k|u|

/ε

NonlinearBook10pt November 20, 2007

244 CHAPTER 4

(0)

Figure 4.7 Visualizations of the sets B

(0) ⊂ D

⊂ B

(0) ⊂ B

(0) ⊂ D

and an ultimate

bounded t rajectory.

= −(1 −ε)x

+ x

−kx

− εx

+ k|u|

/ε

≤ −









, |x

| ≥ k

1/2

|u|

, (4.132)

where



1 −ε −1/2

−1/2 k



Note that with k ≥ 1/4(1 −ε), R

≥ 0, and hence,

V (x

, x

) ≤ 0, |x

| ≤

|u|

Next, let δ = min{λ

min

), λ

min

)} and note that (4.131) and (4.132)

imply

V (x

, x

) ≤ −δ(x

+ x

), kxk

∞

≥ k

1/2

|u|

/ε

, and h en ce, it follows

from Corollary 4.3 that (4.129) and (4.130) is globally bounded.

Next, to show that (4.129) and (4.130) is globally ultimately bounded,

choose k > 1/4(1 − ε) so that δ > 0. Now, with W (x) = δ(x

+ x

α(θ) = min

kxk

∞

= θ

V (x), β(θ) = max

kxk

∞

= θ

V (x), and η > β(k

1/2

|u|

/ε

it follows from Corollary 4.4 that (4.129) and (4.130) is globally ultimately

bounded with bound α

−1

(η). Next, n ote that α(θ) = min{

} and

β(θ) =

, which implies that

−1

(β(k

1/2

|u|

/ε

)) = max{γ

1/2

(|u|), (γ(|u|)/k)

1/4

where γ(|u|)

△

k|u|

|u|

. Now, it f ollows from Corollary 4.4 that

lim sup

t→∞

kx(t)k

∞

≤ max{γ

1/2

(|u|), (γ(|u|)/k)

1/4

}, which implies that,