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

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ADVANCED STABILITY THEORY 275

c > γ, for a ﬁxed γ ∈ R. Hence,

c>γ



−1

([γ, c]) ∪

γ,c



= V

−1

(γ) ∪

where

△

= ∩

c>γ

γ,c

, is s uch that V (x) < γ, x ∈

. Finally, if V (·) is

continuous, then

γ,c

= Ø, γ ∈ R, c > γ, and hence, R

= V

−1

(γ).

It is important to note that as in standard Lyapunov and invariant

set theorems involving continuously diﬀerentiable functions, Theorem 4.24

allows one to characterize the invariant set M without knowledge of the

system trajectories x(t), t ≥ 0. Similar remarks hold for Theorems 4.25–

4.27 given below.

Example 4.16. To illustrate the utility of Theorem 4.24 consider the

simple scalar nonlinear dynamical system given by

˙x(t) = −x(t)(x(t) − 1)(x(t) + 2), x(0) = x

, t ≥ 0, (4.227)

with generalized Lyapu nov function candidate

V (x) =



(x + 2)

, x < 0,

(x − 1)

, x ≥ 0.

Now, note that for all x ∈ R,

V (x)

△

= D

V (x)[−x(x − 1)(x + 2)] =



−2x(x − 1)(x + 2)

, x < 0,

−2x(x − 1)

(x + 2), x ≥ 0,

≤ 0,

which implies that V (x(t)), t ≥ 0, is nonincreasing along the system

trajectories. Next, n ote that R

= V

−1

(γ), γ ∈ R \ {4}, and R

−1

(4) ∪{0}. Since the only invariant sets in R

, γ ∈ R, for the dynamical

system (4.227) are the equilibrium points x

= −2, x

= 0, x

= 1, it

follows that M

= Ø, γ 6∈ {0, 1, 4}, M

= {−2, 1}, M

= {0}, and

= {0}, which implies that M = {−2, 0, 1}. Hence, it follows from

Theorem 4.24 that for every x

∈ R the solution to (4.227) ap proaches the

invariant set M = {−2, 0, 1} as t → ∞ which can be easily veriﬁed. △

The following corollary to Th eorem 4.24 presents suﬃcient conditions

that guarantee local asymptotic stability of the nonlinear dynamical system

(3.1).

Corollary 4.5. Consider the nonlinear dynamical system (3.1), let x(t),

t ≥ 0, denote the solution to (3.1), and let D

⊂ D with 0 ∈

◦

be a

compact invariant set with respect to (3.1). Assume that there exists a

lower semicontinuous positive-deﬁnite fu nction V : D

→ R such that V (·)

is continuous at the origin and V (x(t)) ≤ V (x(τ )), 0 ≤ τ ≤ t, for all x

∈ D

Furth ermore, assume that M

△

= ∪

γ≥0

contains no invariant set other

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276 CHAPTER 4

than the set {0}. Th en the zero solution x(t) ≡ 0 to (3.1) is asymptotically

stable and D

is a subset of the domain of attraction of (3.1).

Proof. The result f ollows as a direct consequence of Theorems 4.22

and 4.24.

Next, we specialize Theorem 4.24 to the Barbashin-Krasovskii-LaSalle

invariant set theorem wh erein V (·) is a continuously diﬀerentiable function.

Corollary 4.6. Consider the nonlinear dynamical system (3.1), assume

that D

⊂ D is a compact invariant set with respect to (3.1), and assume

that there exists a continuously d iﬀerentiable function V : D

→ R such

that V

′

(x)f(x) ≤ 0, x ∈ D

. Let R

△

= {x ∈ D

: V

′

(x)f(x) = 0} and let M

be the largest invariant set contained in R. If x

∈ D

, then x(t) → M as

t → ∞.

Proof. The result follows f rom Theorem 4.24. Speciﬁcally, since

′

(x)f(x) ≤ 0, x ∈ D

, it follows that

V (x(t)) − V (x(τ)) =

′

(x(s))f(x(s))ds ≤ 0, t ≥ τ,

and hence, V (x(t)) ≤ V (x(τ)), t ≥ τ. Now, since V (·) is continuously

diﬀerentiable it follows that R

= V

−1

(γ), γ ∈ R. I n this case, it follows

from Theorem 4.24 that for every x

∈ D

there exists γ

∈ R such that

ω(x

) ⊆ M

, where M

is the largest invariant set contained in R

−1

(γ

), which implies that V (x) = γ

, x ∈ ω(x

). Hence, since M

is an invariant set it follows that for all x(0) ∈ M

, x(t) ∈ M

, t ≥ 0,

and thus,

V (x(0))

△

dV (x(t))



t=0

= V

′

(x(0))f(x(0)) = 0, which implies the

is contained in M which is the largest invariant set contained in R.

Hence, since x(t) → ω(x

) ⊆ M as t → ∞, it follows that x(t) → M as

t → ∞.

Next, we sharpen th e results of Theorem 4.24 by providing a reﬁned

construction of the invariant set M. In particular, we show that the

system traj ectories converge to a union of largest invariant sets contained in

intersections over the largest limit value of V (·) at th e origin of the closure

of generalized Lyapu nov surfaces. First, however, the following key lemma

is needed.

Lemma 4.2. Let Q ⊆ R

, let V : Q → R, and let γ

△

= lim sup

x→0

V (x). If 0 ∈ R

for some γ ∈ R, then γ ≤ γ

Proof. If 0 ∈ R

for γ ∈ R, then there exists a sequence {x

}

∞

n=0

⊂ R

such that lim

n→∞

= 0. Now, since γ

= lim sup

x→0

V (x), it follows

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ADVANCED STABILITY THEORY 277

that lim sup

n→∞

V (x

) ≤ γ

. Next, note th at x

∈ V

−1

([γ, c]), c > γ,

n = 0, 1, . . ., which implies that V (x

) ≥ γ, n = 0, 1, . . .. Thus, using the

fact that lim sup

n→∞

V (x

) ≤ γ

it follows that γ ≤ γ

Note that if in Lemma 4.2 V (·) is continuous at the origin, then γ

V (0).

Theorem 4.25. Consider the nonlinear dynamical system (3.1), let

x(t), t ≥ 0, denote the solution to (3.1), and let D

⊂ D with 0 ∈

◦

a compact invariant set with respect to (3.1). Assume that th ere exists

a lower semicontinuous positive-deﬁnite function V : D

→ R such that

V (x(t)) ≤ V (x(τ )), 0 ≤ τ ≤ t, for all x

∈ D

. Fu rthermore, assume that

for all x

∈ D

, x

6= 0, there exists an increasing unboun ded sequence

}

∞

n=0

, with t

= 0, such th at

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . . (4.228)

If x

∈ D

, then x(t) →

△

= ∪

γ∈G

as t → ∞, where G

△

= {γ ∈ [0, γ

] :

0 ∈ R

} and γ

△

= lim sup

x→0

V (x). If , in addition, V (·) is continuous at

the origin, then the zero solution x(t) ≡ 0 to (3.1) is locally asymptotically

stable and D

is a subset of the domain of attraction.

Proof. I t follows from Theorem 4.24 and the fact that V (·) is

positive deﬁ nite that, for every x

∈ D

, there exists γ

≥ 0 such that

ω(x

) ⊆ M

⊆ R

. Furthermore, since all solutions x(t), t ≥ 0, to

(3.1) are bounded, it follows from Theorem 2.41 that ω(x

) is a nonempty,

compact, invariant set. Now, ad absurdum, suppose 0 6∈ ω(x

). Since V (·) is

lower semicontinuous it follows from Theorem 2.11 that α

△

= min

x∈ω(x

)

V (x)

exists. Furthermore, there exists ˆx ∈ ω(x

) such that V (ˆx) = α. Now,

with x(0) = ˆx 6= 0 it follows from (4.228) that there exists an increasing

unbounded sequence {t

}

∞

n=0

, with t

= 0, such that V (x(t

n+1

)) < V (x(t

)),

n = 0, 1, . . ., which implies that there exists t > 0 such that V (x(t)) < α,

and hence, x(t) 6∈ ω(x

), contradicting the fact that ω(x

) is an invariant set.

Hence, 0 ∈ ω(x

) ⊆ R

, which, using Lemma 4.2, implies that γ

≤ γ

for all x

∈ D

, and which further implies that ω(x

) ⊆

M. Now, since

x(t) → ω(x

) ⊆

M as t → ∞ it follows that x(t) →

M as t → ∞.

Finally, if V (·) is continuous at th e origin then Lyapunov stability

follows from Theorem 4.22. Furthermore, in this case, γ

= V (0) = 0,

which implies that

M ≡ {0}. Hence, x(t) → 0 as t → ∞ for all x

∈

, establishing local asymptotic stability with a subset of the domain of

attraction given by D

In all the above results we explicitly assumed that there exists a

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278 CHAPTER 4

compact invariant set D

⊂ D of (3.1). Next, we provide a result that

does n ot require the existence of such a compact invariant D

Theorem 4.26. C onsider the nonlinear dyn amical system (3.1) and

let x(t), t ≥ 0, denote the solution to (3.1). Assume that there exists a

lower semicontinuous function V : R

→ R such that

V (0) = 0, (4.229)

V (x) > 0, x ∈ R

, x 6= 0, (4.230)

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t. (4.231)

Then all solutions x(t), t ≥ 0, to (3.1) that are bounded approach M

△

∪

γ≥0

as t → ∞. If, in addition, for all x

∈ R

, x

6= 0, there exists an

increasing unbounded sequence {t

}

∞

n=0

, with t

= 0, such th at

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . , (4.232)

then all solutions x(t), t ≥ 0, to (3.1) that are boun ded approach

△

∪

γ∈G

, where G

△

= {γ ∈ [0, γ

] : 0 ∈ R

} and γ

△

= lim sup

x→0

V (x).

Proof. Let x

∈ R

be such that the trajectory x(t), t ≥ 0, is bounded.

Now, th e proof is a direct consequence of Theorems 4.24 and 4.25 with

= O

Next, we present a generalized global invariant set theorem for

guaranteeing global attraction and global asymptotic stability of a nonlinear

dynamical system.

Theorem 4. 27. Consider the nonlinear dynamical sys tem (3.1) with

D = R

and let x(t), t ≥ 0, denote the solution to (3.1). Assume that there

exists a lower semicontinuous function V : R

→ R such that

V (0) = 0, (4.233)

V (x) > 0, x ∈ R

, x 6= 0, (4.234)

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t, (4.235)

V (x) → ∞ as kxk → ∞. (4.236)

Then for all x

∈ R

, x(t) → M

△

= ∪

γ≥0

as t → ∞. If, in addition, for

all x

∈ R

, x

6= 0, there exists an increasing unbounded sequence {t

}

∞

n=0

with t

= 0, such th at

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . , (4.237)

then x(t) →

△

= ∪

γ∈G

as t → ∞, where G

△

= {γ ∈ [0, γ

] : 0 ∈ R

}

and γ

△

= lim sup

x→0

V (x). Finally, if V (·) is continuous at the origin, then

the zero solution x(t) ≡ 0 to (3.1) is globally asymptotically stable.

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ADVANCED STABILITY THEORY 279

Proof. Note that since V (x) → ∞ as kxk → ∞ it follows that for

every β > 0 there exists r > 0 such that V (x) > β for all x 6∈ B

(0) or,

equivalently, V

−1

([0, β]) ⊆ B

(0), which implies that V

−1

([0, β]) is bounded

for all β > 0. Hence, for all x

∈ R

, V

−1

([0, β

]) is bounded, where β

△

V (x

). Furthermore, since V (·) is a positive-deﬁnite lower semicontinuous

function it f ollows that V

−1

([0, β

]) is closed, and since V (x(t)), t ≥ 0, is

nonincreasing it follows that V

−1

([0, β

]) is an invariant set. Hence, for

every x

∈ R

, V

−1

([0, β

]) is a compact invariant set. Now, with D

−1

([0, β

]) it follows from Theorem 4.24 that there exists 0 ≤ γ

≤ β

such that ω(x

) ⊆ M

⊆ R

, which implies that x(t) → M as t → ∞.

If, in addition, for all x

∈ R

, x

6= 0, there exists an increasing unbounded

sequence {t

}

∞

n=0

, with t

= 0, such that (4.237) holds, then it follows from

Theorem 4.25 that x(t) →

M as t → ∞.

Finally, if V (·) is continuous at the origin, then Lyapunov stability

follows from Theorem 4.22. Furthermore, in this case, γ

= V (0) = 0, which

implies that

M ≡ {0}. Hence, x(t) → 0 as t → ∞, establishing global

asymptotic stability.

If in Theorems 4.25 and 4.27 th e function V (·) is continuously

diﬀerentiable on D

and R

, respectively, and V

′

(x)f(x) < 0, x ∈ R

x 6= 0, then every increasing u nbounded sequence {t

}

∞

n=0

, with t

= 0, is

such that V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . .. In this case Theorems 4.25

and 4.27 specialize to the standard Lyapunov stability theorems for local

and global asymptotic stability, respectively, presented in Section 3.2.

Example 4.17. To illustrate the generalized s tability theorems pre-

sented in this section we consider the stability analysis of a n on linear

dynamical system given by

˙x

(t) = σ(x

(t)) + u(t), x

(0) = x

, t ≥ 0, (4.238)

˙x

(t) = σ(x

(t)) + u(t), x

(0) = x

, (4.239)

˙x

(t) = σ(x

(t)) + u(t), x

(0) = x

, (4.240)

where σ : R → R is given by σ(y) = −3my

− y

, m > 0, and

u =



0, x ∈ D

−σ(αm), x 6∈ D

(4.241)

where x

△

= [x

]

, α > 1, and D

△

= {x ∈ R

: x

≥ 0, x

≥

0}. Next, consider the radially unbounded generalized Lyapunov function

candidate given by

V (x) =



i=1

, x ∈ D

i=1

− αm)

, x 6∈ D

(4.242)

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280 CHAPTER 4

and n ote that V (x) is continuous on R

\∂D

Next, note that for i = {1, 2, 3},

−

△

= lim

→0

−

V (x) = α

j=1,6=i

− αm)

△

= lim

→0

V (x) =

j=1,6=i

, q

△

= V (x)|

j=1,6=i

and hence, q

− q

= 0 and q

−

− q

j=1,6=i

−

αm)

, which

implies that V (x) is lower semicontinuous on ∂D

, and hence, V (·) is lower

semicontinuous on R

. Now, note that

V (x)

△

= D

V (x)f (x)



−

i=1

(3mx

+ x

), x ∈ D

−2

i=1

− αm)[3m(x

− αm)

+ x

− α

], x 6∈ D







−

i=1

(3mx

+ x

), x ∈ D

−2

i=1

− αm)

[(x

m(α+3)

)

(α + 3)(α −1)],

x 6∈ D

(4.243)

where f : R

→ R

denotes the right-hand side of (4.238)–(4.240). Next,

(4.243) implies that V (x(t)), t ≥ 0, is nonincreasing along the system

trajectories. Furthermore, since

V (x) < 0, x ∈ R

, x 6= 0, it follows that for

all (x

, x

) ∈ R

\ {0} there exists an increasing unbounded sequence

}

∞

n=0

, with t

= 0, such that V (x(t

j+1

)) < V (x(t

)), j = 0, 1, . . ., so

that all the conditions of Theorem 4.27 are satisﬁed. Hence, it follows from

Theorem 4.27 that x(t) →

M as t → ∞, where

M is deﬁned as in Theorem

4.27 with G = {0, 3α

} and γ

= lim sup

x→0

V (x) = 3α

Next, note that R

= {0} and

3α

(

x ∈ D

i=1

= 9α

)

∪{0}. (4.244)

Now, it follows th at M

= {0}, γ ∈ G = {0, 3α

}, which implies that

M = {0}. Hence, it follows from Theorem 4.27 that x(t) → 0 as t → ∞.

Finally, it can be shown that the nonlinear dynamical system (4.238)–(4.240)

is Lyapunov stable so that the asymptotic stability can be established. △

Next, we remove the lower semicontinuity assumption on the function

V (·) and show that the zero solution x(t) ≡ 0 to (3.1) is asymptotically

stable if there simp ly exists an increasing unbounded sequence {t

}

∞

n=0

, with

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ADVANCED STABILITY THEORY 281

= 0, such that for T ≥ t

n+1

− t

> 0, n ∈ Z

, and all x(0) ∈ D,

V (x(T )) − V (x(0)) ≤ −γ(kx(0)k) < 0, where γ(·) is a class K function.

However, for the remainder of the results of th is section we assu me that f(·)

is Lipschitz continuous on D with maximum Lipschitz constant L > 0 over

D. The following lemma is needed for this result.

Lemma 4.3. Consider the nonlinear dynamical system (3.1). Assume

f : D → R

is Lipschitz continuous on D with maximum Lipschitz constant

L > 0 over D. If x

∈ B

(0), δ = εe

−LT

, where ε > 0 and T > 0, then

x(t) ∈ B

(0) for all t ∈ [0, T ].

Proof. It f ollows from (3.1) that

x(t) = x(0) +

f(x(s))ds, t ≥ 0. (4.245)

Now, u sing the fact that f : D → R

is Lipschitz continuous on D, it follows

that

kx(t)k ≤ kx

k +

kf(x(s))kds, t ≥ 0

≤ kx

k +

Lkx(s)kds, (4.246)

where L > 0 is the maximum Lipschitz constant over D. Using Lemma 2.2,

(4.246) implies

kx(t)k ≤ kx

, t ≥ 0. (4.247)

Next, let x

∈ B

(0) ⊂ B

(0) and let τ > 0 be such that x(t) ∈ B

(0),

t ∈ [0, τ), and kx(τ )k = ε. In this case, it follows from (4.247) with t = τ

that kx(τ)k ≤ kx

Lτ

. Now, since by assumption kx(τ)k = ε, kx

k < δ,

and δ = εe

−LT

, it follows that ε < εe

L(τ−T )

, and hence, τ > T . Hence, since

x(t) ∈ B

(0), t ∈ [0, τ ), and τ > T it follows that x(t) ∈ B

(0), t ∈ [0, T ].

Theorem 4.28. C onsider the nonlinear dyn amical system (3.1) and

let x(t), t ≥ 0, denote the solution to (3.1). Assume there exist a function

V : D → R, an in creasing unbounded sequence {t

}

∞

n=0

, with t

= 0, and

T > 0 such that 0 < t

n+1

−t

≤ T , n = 0, 1, . . . , and

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ D, (4.248)

V (x(t

n+1

)) − V (x(t

)) ≤ 0, x(t

) ∈ D, n = 0, 1, . . . , (4.249)

where α(·) and β(·) are class K functions deﬁned on [0, ε) for all ε > 0. Then

the zero solution x(t) ≡ 0 to (3.1) is L yapunov stable. If, in addition, there

exists a class K function γ(·) deﬁ ned on [0, ∞) such that

V (x(t

n+1

)) − V (x(t

)) ≤ −γ(kx(t

)k) < 0, x(t

) ∈ D, n = 0, 1, . . . ,

(4.250)

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282 CHAPTER 4

then the zero solution x(t) ≡ 0 to (3.1) is asymptotically stable.

Proof. To show Lyapunov stability, deﬁne δ

△

= εe

−LT

and d eﬁ ne

△

= β

−1

(α(δ)), where L denotes the maximum Lips chitz constant over

D for the nonlinear dynamical system (3.1). Now, for all x

∈ B

(0) it

follows that β(kx

k) < α(δ), and hence, (4.248) implies V (x

) < α(δ). Next,

(4.249) implies that V (x(t

)) ≤ V (x

) < α(δ), and hence, (4.248) implies

α(kx(t

)k) ≤ V (x(t

)) < α(δ) so that x(t

) ∈ B

(0). Now, using (4.249)

with n = 2 it follows that V (x(t

)) ≤ V (x(t

)) < α(δ), and hence, repeating

this procedure, x(t

) ∈ B

(0), n = 0, 1, . . .. Since {t

}

∞

n=0

is an increasing

sequence with t

= 0 and t

→ ∞ as n → ∞, there exists an integer ˆn s uch

that t

ˆn

≤ t ≤ t

ˆn+1

, t ≥ 0. Now, u sing the fact that x(t

) ∈ B

(0), it follows

that kx(t

ˆn

)k < δ = εe

−LT

. Hence, Lemma 4.3 implies that kx(t)k < ε,

t ≥ 0, and hence, for all ε > 0 there exists η = β

−1

(α(δ)) > 0 such th at if

kx(0)k < η, then kx(t)k < ε, t ≥ 0, which proves Lyapunov stability.

To show asymptotic stability, let 0 < ˆη < η(ε) = β

−1

(α(εe

−LT

)) and

let x

∈ B

ˆη

(0) so that V (x

) ≤ β(kx

k) < β(ˆη). Now, suppose, ad absurdum,

that kx(t

)k ≥ l > 0, n = 0, 1, . . .. In this case, it follows from (4.250) that

V (x(t

n+1

)) − V (x(t

)) ≤ −γ(kx(t

)k) ≤ −γ(l), n = 0, 1, . . . ,

and hence, V (x(t

n+1

)) − V (x

) ≤ −(n + 1)γ(l), n = 0, 1, . . .. Now, since

0 ≤ α(kxk) ≤ V (x), x ∈ D, it follows th at

0 ≤ V (x(t

n+1

)) ≤ V (x

) −(n + 1)γ(l) < β(ˆη) − (n + 1)γ(l),

for all x

∈ B

ˆη

(0) and n = 0, 1, . . .. Since β(·) and γ(·) are class K functions

it follows that there exists an integer ˆn such that β(ˆη) − (ˆn + 1)γ(l) < 0,

which leads to a contradiction, and hence, kx(t

ˆn

)k < l. Thus , it follows f rom

(4.248) that V (x(t

ˆn

)) ≤ β(kx(t

ˆn

)k) < β(l). Next, deﬁne

△

= α

−1

(β(l)) so

that V (x(t

ˆn

)) < β(l) = α(

l). Now, it follows from (4.248) and (4.250) that

α(kx(t

∗

)k) ≤ V (x(t

∗

)) < α(

l), n

∗

≥ ˆn, an d hence, kx(t

∗

)k <

l. Letting

t ≥ t

ˆn

and choosing n

∗

≥ ˆn such that t

∗

≤ t ≤ t

∗

, it follows from

Lemma 4.3 that x(t) ∈ B

ˆε

(0), ˆε

△

−LT

, t ≥ t

ˆn

. Hence, for x

∈ B

ˆη

(0),

0 < ˆη < η(ε), there exists t

ˆn

such that x(t) ∈ B

ˆε

(0), ˆε > 0, t > t

ˆn

. Now,

letting ˆn → ∞ so that t

ˆn

→ ∞, it follows that ˆε → 0, and hence, x(t) → 0

as t → ∞, which proves asymptotic stability.

The results of this section provide suﬃcient conditions for Lyapunov

and asymptotic stability of the nonlinear dynamical system (3.1) with no

regularity assumptions on the function V (·). It is important to note that

even though th e stability conditions appearing in Theorems 4.24–4.27 are

system trajectory dependent, the invariant set M can be characterized

without knowledge of the system trajectories. Furthermore, as shown in

Example 4.16, these theorems allow for a systematic way of constructing

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ADVANCED STABILITY THEORY 283

system Lyapunov functions by piecing together a collection of functions.

Alternatively, V (·) in Theorem 4.28 is not requ ir ed to be continuous,

diﬀerentiable, or decreasing. In the case where V (·) is continuously

diﬀerentiable, condition (4.250) implies that the time derivative of V (·) along

the trajectories of the n onlinear dynamical system (3.1) may have negative

and positive values. This allows the consideration of Lyapunov function

candidates for proving asymptotic stability for nonlinear dynamical systems

that might otherwise be excluded as valid Lyapunov function candidates

using classical Lyapunov stability theory.

4.9 Lyapunov and Asymptotic Stability of Sets

In this section, we extend the results of Section 4.8 to address stability and

attraction of dynamical systems with respect to compact positively invariant

sets. These results are u s ed in the next s ection to provide necessary and

suﬃcient conditions for stability of periodic orb its and limit cycles. We

begin by considering the nonlinear dynamical system

˙x(t) = f (x(t)), x(0) = x

, t ∈ I

, (4.251)

where x(t) ∈ D ⊆ R

, t ∈ I

, is the system state vector, D is an open

set, f : D → R

, and I

= [0, τ

), 0 < τ

≤ ∞, is the maximal interval

of existence for the solution x(·) of (4.251). We assume th at the dynamics

f(·) are such that the solution s(t, x

) to (4.251) is unique for every initial

condition in D and jointly continuous in t and x

. A suﬃcient condition

ensuring this is Lipschitz continuity of f (·). Furthermore, we assum e that

all solutions to (4.251) are bounded over I

, and hence, by Corollary 2.5

can be extended to inﬁnity. T he following d eﬁ nition introduces three types

of stability notions as well as attraction of (4.251) with respect to a compact

positively invariant set for I

= [0, ∞).

Deﬁnition 4.10. Let D

⊂ D be a compact positively invariant set

for the nonlinear dynamical system (4.251). D

is Lyapunov stable if, for

every open neighborhood O

⊆ D of D

, there exists an open neighborhood

⊆ O

of D

such that x(t) ∈ O

, t ≥ 0, for all x

∈ O

. Equivalently,

is Lyapunov stable if, for all ε > 0, there exists δ = δ(ε) > 0 such that if

dist(x

, D

) < δ, then dist(s(t, x

), D

) < ε, t ≥ 0. D

is attractive if there

exists an open neighborhood O

⊆ D of D

such that ω(x

) ⊆ D

for all

∈ O

. D

is asymptotically stable if it is Lyapunov stable and attractive.

Equivalently, D

is asymptotically stable if D

is Lyapunov stable and there

exists ε > 0 such that if dist(x

, D

) < ε, then dist(s(t, x

), D

) → 0 as

t → ∞. D

is globally asymptotically stable if it is Lyapunov stable and

ω(x

) ⊆ D

for all x

∈ R

. Finally, D

is unstable if it is not Lyapu nov

stable.

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284 CHAPTER 4

Next, we give a set theoretic deﬁnition involving the domain, or region,

of attraction of the compact positively invariant set D

of (4.251).

Deﬁnition 4.11. Suppose the compact positively invariant set D

⊂ D

of (4.251) is attractive. Then the domain of attraction D

of D

is deﬁned

△

= {x

∈ D : ω(x

) ⊆ D

}. (4.252)

The following result gives suﬃcient conditions for Lyapunov and

asymptotic stability of a compact positively invariant set with respect to

the nonlinear dynamical system (4.251).

Theorem 4.29. Consider the nonlinear dynamical system (4.251), let

be a compact positively invariant set with respect to (4.251) such that

⊂ D, and let x(t), t ≥ 0, denote the solution to (4.251) with x

∈ D.

Assume that there exists a lower semicontinuous function V : D → R such

that V (·) is continuous on D

and

V (x) = 0, x ∈ D

, (4.253)

V (x) > 0, x ∈ D, x 6∈ D

, (4.254)

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t. (4.255)

Then D

is Lyapunov stable. If, in add ition, for all x

∈ D, x

6∈ D

, there

exists an increasing unbounded sequence {t

}

∞

n=0

, with t

= 0, such th at

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . , (4.256)

then D

is asymptotically stable.

Proof. Let O

⊆ D be a bounded open neighborhood of D

. Since

∂O

is compact and V (x), x ∈ D, is lower semicontinuous, it follows fr om

Theorem 2.11 that there exists α = min

x∈∂O

V (x). Note th at α > 0 since

∩ ∂O

= Ø and V (x) > 0, x ∈ D, x 6∈ D

. Next, using the facts that

V (x) = 0, x ∈ D

, and V (·) is continuous on D

, it follows that the set

△

= {x ∈ O

: V (x) < α}

◦

is not empty. Now, it follows from (4.255) that

for all x(0) ∈ O

V (x(t)) ≤ V (x(0)) < α, t ≥ 0,

which, since V (x) ≥ α, x ∈ ∂O

, implies that x(t) 6∈ ∂O

, t ≥ 0. Hence, f or

every open neighborhood O

⊆ D of D

, there exists an open neighborhood

⊆ O

of D

such that, if x(0) ∈ O

, then x(t) ∈ O

, t ≥ 0, which proves

Lyapunov s tability of the compact positively invariant set D

of (4.251).

To prove asymptotic stability let x

∈ O

and sup pose there exists

an increasing unb ounded sequ en ce {t

}

∞

n=0

, with t

= 0, such that (4.256)