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

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ADVANCED STABILITY THEORY 265

Example 4.14. Consider the nonlinear d ynamical system given by

˙x

(t) = σ

(t)) − σ

(t)), x

(0) = x

, t ≥ 0, (4.205)

˙x

(t) = σ

(t)) − σ

(t)), x

(0) = x

, (4.206)

where x

, x

∈ R, σ

(·), i, j = 1, 2, i 6= j, are Lipschitz continuous, σ

)−

) = 0 if and only if x

= x

, and (x

− x

)(σ

) − σ

)) ≤ 0,

, x

∈ R. Note that f

−1

(0) = {(x

, x

) ∈ R

: x

= x

= α, α ∈ R}. To

show that (4.205) and (4.206) is semistable, consider the Lyapunov function

candidate V (x

, x

) =

−α)

, where α ∈ R. Now, it follows

that

V (x

, x

) = (x

− α)[σ

) − σ

)] + (x

− α)[σ

) − σ

)]

= x

[σ

) −σ

)] + x

[σ

) −σ

)]

= (x

− x

)[σ

) − σ

)]

≤ 0, (x

, x

) ∈ R × R, (4.207)

which implies that x

= x

= α is Lyapunov stable.

Next, let R , {(x

, x

) ∈ R

V (x

, x

) = 0} = {(x

, x

) ∈ R

= x

= α, α ∈ R}. Since R consists of equilibrium points, it follows that

M = R. Hence, f or every x

(0), x

(0) ∈ R, (x

(t), x

(t)) → M as t → ∞.

Hence, it follows from Theorem 4.20 that x

= x

= α is semistable for all

α ∈ R. △

Finally, we provide a converse Lyapunov theorem for semistability. For

this result, recall that for a given continuous function V : D → R, the upper

right Dini derivative of V along the solution of (4.202) is deﬁned by

V (s(t, x)) , lim sup

h→0

[V (s(t + h, x)) − V (s(t, x))]. (4.208)

It is easy to see that

V (x

) = 0 for every x

∈ f

−1

(0). Also note that it

follows from (4.208) that

V (x) =

V (s(0, x)).

Theorem 4.21. Consider the nonlinear dynamical system (4.202).

Suppose (4.202) is semistable with the domain of semistability D

. Then

there exist a continuous nonnegative function V : D

→ R

and a

class K function α(·) such that i) V (x) = 0, x ∈ f

−1

(0), ii) V (x) ≥

α(dist(x, f

−1

(0))), x ∈ D

, and iii)

V (x) < 0, x ∈ D

−1

(0).

Proof. Deﬁne the function V : D

→ R

V (x) , sup

t≥0



1 + 2t

1 + t

dist(s(t, x), f

−1

(0))



, x ∈ D

. (4.209)

Note that V (·) is well deﬁned since (4.202) is semistable. Clearly, i) holds.

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266 CHAPTER 4

Furth ermore, since V (x) ≥ dist(x, f

−1

(0)), x ∈ D

, it follows that ii) holds.

To show that V (·) is continuous on D

−1

(0), deﬁne T : D

−1

(0) →

[0, ∞) by T (z) , inf{h : dist(s(t, z), f

−1

(0)) < dist(z, f

−1

(0))/2 for all t ≥

h > 0}, and denote

, {x ∈ D

: dist(s(t, x), f

−1

(0)) < ε, t ≥ 0}. (4.210)

Note that W

⊃ f

−1

(0) is open and positively invariant, and contains an

open neighborh ood of f

−1

(0). Consider z ∈ D

−1

(0) and deﬁne λ ,

dist(z, f

−1

(0)) > 0. Then it follows from semistability of (4.202) that there

exists h > 0 such that s(h, z) ∈ W

λ/2

. Cons equ ently, s(h + t, z) ∈ W

λ/2

for all t ≥ 0, and hence, it follows that T (z) is well deﬁned. Since W

λ/2

is open, there exists a neighborhood B

(s(T (z), z)) ⊂ W

λ/2

. Hence, N ,

−T (z)

(s(T (z), z))) is a neighborhood of z and N ⊂ D

. Choose η > 0

such that η < λ/2 and B

(z) ⊂ N. Then, for every t > T (z) and y ∈ B

(z),

[(1 + 2t)/(1 + t)]dist(s(t, y), f

−1

(0)) ≤ 2dist(s(t, y), f

−1

(0)) ≤ λ. Therefore,

for each y ∈ B

(z),

V (z) −V (y) = sup

t≥0



1 + 2t

1 + t

dist(s(t, z), f

−1

(0))



−sup

t≥0



1 + 2t

1 + t

dist(s(t, y), f

−1

(0))



= sup

0≤t≤T (z)



1 + 2t

1 + t

dist(s(t, z), f

−1

(0))



− sup

0≤t≤T (z)



1 + 2t

1 + t

dist(s(t, y), f

−1

(0))



(4.211)

Hence,

|V (z) −V (y)|

≤ sup

0≤t≤T (z)



1 + 2t

1 + t



dist(s(t, z), f

−1

(0)) −dist(s(t, y), f

−1

(0))





≤ 2 sup

0≤t≤T (z)



dist(s(t, z), f

−1

(0)) −dist(s(t, y), f

−1

(0))



≤ 2 sup

0≤t≤T (z)

dist(s(t, z), s(t, y)), z ∈ D

−1

(0), y ∈ B

(z).

(4.212)

Now, it follows from continuous dependence of solutions s(·, ·) on system

initial conditions (Theorem 2.26) and (4.212) that V (·) is continuous on

−1

(0).

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ADVANCED STABILITY THEORY 267

To show that V (·) is continuous on f

−1

(0), consider x

∈ f

−1

(0).

Let {x

}

∞

n=1

be a sequence in D

−1

(0) that converges to x

. Since x

Lyapunov stable, it follows that x(t) ≡ x

is the u nique solution to (4.202)

with x

= x

. By continuous dependence of s olutions s(·, ·) on system initial

conditions (Theorem 2.26), s(t, x

) → s(t, x

) = x

as n → ∞, t ≥ 0.

Let ε > 0 and note that it follows from ii) of P roposition 4.8 that there

exists δ = δ(x

) > 0 such that for every solution of (4.202) in B

) there

exists

T =

T (x

, ε) > 0 s uch that s

)) ⊂ W

for all t ≥

T . Next, note

that there exists a positive integer N

such that x

∈ B

) for all n ≥ N

Now, it follows from (4.209) that

V (x

) ≤ 2 sup

0≤t≤

dist(s(t, x

), f

−1

(0)) + 2ε, n ≥ N

. (4.213)

Next, it f ollows from Lemma 3.1 of Chapter I of [179] that s(·, x

) converges

to s(·, x

) uniformly on [0,

T ]. Hence,

lim

n→∞

sup

0≤t≤

dist(s(t, x

), f

−1

(0)) = sup

0≤t≤

dist( lim

n→∞

s(t, x

), f

−1

(0))

= sup

0≤t≤

dist(x

, f

−1

(0))

= 0, (4.214)

which implies that there exists a positive integer N

= N

, ε) ≥ N

such

that sup

0≤t≤

dist(s(t, x

), f

−1

(0)) < ε for all n ≥ N

. Combining (4.213)

with the above resu lt yields V (x

) < 4ε for all n ≥ N

, w hich implies that

lim

n→∞

V (x

) = 0 = V (x

Next, we show that V (x(t)) is strictly decreasing along the solution of

(4.202) on D\f

−1

(0). Note that for every x ∈ D

−1

(0) and 0 < h ≤ 1/2

such that s(h, x) ∈ D

−1

(0), it follows from the deﬁnition of T (·) that

V (s(h, x)) is reached at some time

t such that 0 ≤

t ≤ T (x). Hence,

V (s(h, x)) = dist(s(

t + h, x), f

−1

(0))

1 + 2

1 +

= dist(s(

t + h, x), f

−1

(0))

1 + 2

t + 2h

1 +

t + h



1 −

(1 + 2

t + 2h)(1 +



≤ V (x)



1 −

2(1 + T (x))



, (4.215)

which implies that

V (x) ≤ −

V (x)(1 + T (x))

−2

< 0, x ∈ D

−1

(0), and

hence, iii) holds.

It is important to note that a converse Lyapun ov theorem for

semistability involving a smooth (i.e., inﬁnitely diﬀerentiable) Lyapunov

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268 CHAPTER 4

function for dynamical systems w ith continuous vector ﬁelds can also be

established. For details, see [209].

4.8 Generalized Lyapunov Theorems

Lyapunov’s results, along with the Barbashin-Krasovskii-LaSalle invariance

principle, provide a powerful framework for analyzing the stability of

nonlinear dynamical systems as well as designing feedback controllers that

guarantee closed-loop system s tability. In particular, as discussed in

Sections 3.2 and 3.3, Lyapunov’s direct method can provide local and global

stability conclusions of an equilibrium point of a nonlinear dynamical system

if a continuously diﬀerentiable positive-deﬁnite function of the nonlinear

system states (Lyapunov function) can be constructed for which its time

rate of change due to perturbations in a neighborhood of the system’s

equilibrium is always negative or zero, with strict negative-deﬁniteness en-

suring asymptotic stability. Alternatively, using the Barbashin-Krasovskii-

LaSalle invariance principle the strict negative-deﬁniteness condition on the

Lyapunov derivative can be relaxed wh ile ensuring asymptotic s tability. In

particular, if a continuously diﬀerentiable fun ction deﬁned on a compact

invariant set with respect to the nonlinear d ynamical system can be

constructed whose derivative along the s ystem’s trajectories is negative

semideﬁnite, and no system trajectories can stay indeﬁnitely at points where

the fun ction’s derivative identically vanishes, then the system’s equilibrium

is asymptotically stable.

Most Lyapunov stability and invariant set theorems presented in the

literature require that the Lyapunov function candidate for a nonlinear

dynamical system be a continuously diﬀerentiable function with a negative-

deﬁnite derivative (see [178, 228, 235, 260, 445, 474] and the numerous

references therein). This is due to the fact that the majority of the

dynamical systems considered are systems possessing continuous motions,

and hence, Lyapunov theorems provide stability conditions that do not

require knowledge of the system trajectories. However, in light of the

increasingly complex nature of dynamical systems such as biological systems

[284], hybrid systems [473], sampled-data systems [176], discrete-event

systems [346], gain scheduled systems [271,299,348], constrained mechanical

systems [18], and impulsive systems [20], system discontinuities arise

naturally. Even though standard Lyapunov theory is applicable f or systems

with discontinuous system dynamics and continuous motions, it might be

simpler to construct discontinuous “Lyapunov” functions to establish system

stability. For example, in gain scheduling control it is not uncommon to

use several diﬀerent controllers designed over several ﬁxed operating points

covering the system’s operating range and to switch between them over

this range. Even th ou gh for each operating r ange one can construct a

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ADVANCED STABILITY THEORY 269

continuously diﬀerentiable Lyapunov function, to s how closed-loop sys tem

stability over the whole s ys tem operating envelope for a given switching

control strategy, a generalized Lyapunov function involving combinations of

the Lyapunov functions for each operating range can be constructed [269–

271, 299, 348]. However, in this case, it can be shown that the generalized

Lyapunov function is nonsmooth and noncontinuous [269–271, 299,348].

In this section, we develop generalized Lyapunov and invariant set the-

orems for nonlinear dynamical systems wherein all regularity assumptions on

the L yapunov fu nction and the system dynamics are removed. In particular,

local and global stability theorems are presented using generalized Lyapunov

functions that are lower semicontinuous. Furthermore, generalized invariant

set theorems are derived wherein system trajectories converge to a union

of largest invariant sets contained in intersections over ﬁ nite intervals of

the closure of generalized Lyapunov level surfaces. In the case where the

generalized Lyapunov function is taken to be a continuously diﬀerentiable

function, the results collapse to the standard Lyapunov stability and

invariant set theorems presented earlier. Lower semicontinuous Lyapunov

functions have been considered in [15] in the context of viability th eory and

diﬀerential inclusions. However, the present formulation provides invariant

set stability theorem generalizations not considered in [15].

To p resent the main results of this section recall that a continuous

function x : I

→ D is said to be a solution to (3.1) on the interval I

⊆ R

if x(t) satisﬁes (3.1) for all t ∈ I

. Furthermore, u nless otherwise stated,

in this section we do not assume any regularity conditions on the system

dynamics f(·). However, we do assume that f (·) is such that the s olution

x(t), t ≥ 0, to (3.1) is well deﬁned on the time interval I

= [0, ∞). That

is, we assume that for every y ∈ D there exists a unique solution x(·) of

(3.1) deﬁned on [0, ∞) satisfying x(0) = y. Furthermore, we assume that

all the solutions x(t), t ≥ 0, to (3.1) are continuous f unctions of the initial

conditions x

∈ D.

The following result presents suﬃcient conditions for Lyapunov and

asymptotic stability of a nonlinear dynamical system, wherein the assump-

tion of continuous diﬀerentiability on the Lyapunov function with a negative-

deﬁnite derivative is relaxed.

Theorem 4.22. 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 : D → R such that V (·) is continuous at

the origin and

V (0) = 0, (4.216)

V (x) > 0, x ∈ D, x 6= 0, (4.217)

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270 CHAPTER 4

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t. (4.218)

Then the zero solution x(t) ≡ 0 to (3.1) is Lyapunov stable. If, in addition,

there exists an increasing unboun ded sequence {t

}

∞

n=0

, with t

= 0, such

that

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . , (4.219)

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

Proof. To show Lyapunov stability, let ε > 0 be such that B

(0) ⊆ D.

Since ∂B

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

from Theorem 2.11 that there exists α = min

x∈∂B

(0)

V (x). Note that α > 0

since 0 6∈ ∂B

(0) an d V (x) > 0, x ∈ D, x 6= 0. Next, since V (0) = 0 and V (·)

is continuous at the origin it follows that there exists δ ∈ (0, ε] such that

V (x) < α, x ∈ B

(0). Now, it follows from (4.218) th at for all x(0) ∈ B

(0),

V (x(t)) ≤ V (x(0)) < α, t ≥ 0,

which, since V (x) ≥ α, x ∈ ∂B

(0), implies that x(t) 6∈ ∂B

(0), t ≥ 0.

Hence, for all ε > 0 such that B

(0) ⊆ D there exists δ > 0 such that if

kx(0)k < δ, then kx(t)k < ε, t ≥ 0, which proves Lyapu nov stability.

To prove asymptotic stability let x

∈ B

(0) and suppose th ere exists

an increasing unb ounded sequ en ce {t

}

∞

n=0

, with t

= 0, such that (4.219)

holds. Then it follows that x(t) ∈ B

(0), t ≥ 0, and hen ce, it follows from

Theorem 2.41 that the positive limit set ω(x

) of x(t), t ≥ 0, is a nonempty,

compact, invariant connected set. Furthermore, x(t) → ω(x

) as t → ∞.

Now, since V (x(t)), t ≥ 0, is nonincreasing and bounded from below by zero

it follows that β

△

= lim

t→∞

V (x(t)) ≥ 0 is well deﬁned . Furthermore, since

V (·) is lower semicontinuous it can be shown that V (y) ≤ β, y ∈ ω(x

and hence, since V (·) is nonnegative, β = 0 if and only if ω(x

) = {0} or,

equivalently, x(t) → 0 as t → ∞. Now, suppose, ad absurdum, that x(t),

t ≥ 0, does not converge to zero or, equivalently, β > 0. Furthermore, let

y ∈ ω(x

) and let {τ

}

∞

n=0

be an increasing unbounded s equ en ce such that

lim

n→∞

x(τ

) = y. Since {V (x(τ

))}

∞

n=0

is a lower bounded nonincreasing

sequence, lim

n→∞

V (x(τ

)) exists and is equal to β. Hence, y 6= 0, which

implies that 0 6∈ ω(x

). Next, let γ

△

= min

x∈ω(x

)

V (x) > 0 and let x

∈

ω(x

) be such that V (x

) = γ. Now, since ω(x

) is an invariant set it follows

that for all x(0) ∈ ω(x

), x(t) ∈ ω(x

), t ≥ 0, and hence, V (x(t)) ≥ γ, t ≥ 0.

However, since for all x

∈ D there exists an increasing unbounded sequence

}

∞

n=1

such that (4.219) holds, it follows that if x(0) = x

∈ ω(x

), there

exists t > 0 such that V (x(t)) < V (x(0)) = γ, which is a contradiction.

Hence, x(t) → 0 as t → ∞, establishing asymptotic stability.

A lower semicontinuous function V (·), with V (·) being continuous at

the origin, satisfying (4.216) and (4.217) is called a generalized Lyapunov

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ADVANCED STABILITY THEORY 271

function candidate for the nonlinear dynamical system (3.1). If, additionally,

V (·) satisﬁes (4.218), V (·) is called a generalized Lyapunov function for

the nonlinear dynamical system (3.1). Note that if the function V (·) is

continuously diﬀerentiable on D in Theorem 4.22 and V

′

(x)f(x) ≤ 0, x ∈ D

(respectively, V

′

(x)f(x) < 0, x ∈ D, x 6= 0), then V (x(t)) is a nonincreasing

function of time (respectively, th ere exists an increasing unbounded sequence

}

∞

n=0

, with t

= 0, such that V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . .). In

this case, Theorem 4.22 specializes to Theorem 3.1.

Next, we provide a partial convers e to Theorem 3.1. Speciﬁcally, we

show that if the nonlinear dynamical system (3.1) is Lyapunov stable, then

there exists a generalized Lyapunov function for (3.1).

Theorem 4.23. C onsider the nonlinear dyn amical system (3.1) and

let s(t, x

), t ≥ 0, denote the solution to (3.1) with initial condition x

Assume that th e zero solution x(t) ≡ 0 to (3.1) is Lyapunov stable. Then

there exists a lower semicontinuous function V : D

→ R, where D

⊆ D,

such that 0 ∈

◦

, V (·) is continuous at the origin, and

V (0) = 0, (4.220)

V (x) > 0, x ∈ D

, x 6= 0, (4.221)

V (s(t, x)) ≤ V (s(τ, x)), 0 ≤ τ ≤ t, x ∈ D

. (4.222)

Proof. Let ε > 0. Since the zero solution x(t) ≡ 0 to (3.1) is Lyapunov

stable it follows that there exists δ > 0 such that if x

∈ B

(0), then s(t, x

) ∈

(0), t ≥ 0. Now, let D

= {y ∈ B

(0) : there exists t ≥ 0 and x

∈ B

(0)

such that y = s(t, x

)}, that is, D

= ∪

t≥0

(0)). Note that D

⊆ B

(0),

is positively invariant, and B

(0) ⊆ D

. Hence, 0 ∈

◦

. Next, deﬁne

V (x)

△

= s up

t≥0

ks(t, x)k, x ∈ D

, and since D

is positively invariant and

bounded it follows that V (·) is well deﬁned on D

. Now, x = 0 imp lies

s(t, x) ≡ 0, and hence, V (0) = 0. Furthermore, V (x) ≥ ks(0, x)k = kxk > 0,

x ∈ D

, x 6= 0.

Next, since f (·) in (3.1) is such that for every x ∈ D

, s(t, x), t ≥ 0,

is the unique solution to (3.1), it follows that s(t, x) = s(t − τ, s(τ, x)),

0 ≤ τ ≤ t. Hence, for every t, τ ≥ 0, such that t ≥ τ ,

V (s(τ, x)) = sup

θ≥0

ks(θ, s(τ, x))k

= sup

θ≥0

ks(τ + θ, x)k

≥ sup

θ≥t−τ

ks(τ + θ, x)k

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272 CHAPTER 4

= sup

θ≥t−τ

ks(θ − (t − τ), s(t, x))k

= sup

θ≥0

ks(θ, s(t, x))k

= V (s(t, x)), (4.223)

which proves (4.222). Next, since the zero solution x(t) ≡ 0 to (3.1) is

Lyapunov s table it follows that for every ˆε > 0 there exists

δ > 0 such that

if x

∈ B

(0), then s(t, x

) ∈ B

ˆε/2

(0), t ≥ 0, which implies that V (x

) =

sup

t≥0

ks(t, x

)k ≤ ˆε/2. Hence, f or every ˆε > 0 there exists

δ > 0 such that

if x

∈ B

(0), then V (x

) < ˆε, establishing that V (·) is continuous at the

origin.

Finally, to show that V (·) is lower semicontinuous everywhere on D

let x ∈ D

and let ˆε > 0, and note that since V (x) = sup

t≥0

ks(t, x)k there

exists T = T (x, ˆε) > 0 s uch that V (x) − ks(T, x)k < ˆε. Now, consider

a sequen ce {x

}

∞

i=1

∈ D

such that x

→ x as i → ∞. Next, since by

assumption s(t, ·) is continuous for every t ≥ 0 and k · k : D

→ R is

continuous, it follows that ks(T, x)k = lim

i→∞

ks(T, x

)k. Next, note that

ks(T, x

)k ≤ sup

t≥0

ks(t, x

)k, i = 1, 2, . . ., and hence,

lim in f

i→∞

sup

t≥0

ks(t, x

)k ≥ lim inf

i→∞

ks(T, x

)k = lim

i→∞

ks(T, x

)k, i = 1, 2, . . . ,

which implies that

V (x) < ks(T, x)k + ˆε

= lim

i→∞

ks(T, x

)k + ˆε

≤ lim inf

i→∞

sup

t≥0

ks(t, x

)k + ˆε

= lim inf

i→∞

V (x

) + ˆε. (4.224)

Now, since ˆε > 0 is arbitrary, (4.224) implies that V (x) ≤ lim inf

i→∞

V (x

Thus , since {x

}

∞

i=1

is an arbitrary sequence converging to x, it follows that

V (·) is lower semicontinuous on D

In the following example we show that there exist Lyapunov stable

systems with continuously diﬀerentiable vector ﬁelds that do not posses

continuous Lyapunov functions.

Example 4.15. Consider the scalar nonlinear dynamical system

˙x(t) = x

(t) sin



x(t)



, x(0) = x

, t ≥ 0. (4.225)

Note that the s et of equilibria for the dynamical system (4.225) is charac-

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ADVANCED STABILITY THEORY 273

terized by

△

= {0} ∪



±1

2π

±1

3π

, . . .



Next, let ε > 0 an d let δ =

nπ

> 0, where n ∈ {1, 2, . . .}, such that

nπ

< ε. Now, f or every x

∈ [0, δ), since ˙x(t) ≥ 0, t ≥ 0, it follows that

x(t) ≥ x

, t ≥ 0. Furtherm ore, s ince there exists at least one equilibrium

point x

∈ [x

, δ] it follows that 0 ≤ x(t) ≤ x

≤ δ < ε, t ≥ 0. Similarly, for

every x

∈ (−δ, 0], since ˙x(t) ≥ 0, t ≥ 0, it follows that x(t) ≥ −δ, t ≥ 0,

and since there exists at least one equilibrium point x

∈ [x

, 0], it follows

that −ε < −δ < x

≤ x(t) ≤ x

≤ 0, t ≥ 0. Hence, for every ε > 0 there

exists δ > 0 such that if |x

| < δ, then |x(t)| < ε or, equivalently, the zero

solution x(t) ≡ 0 to (4.225) is Lyapunov stable.

Next, we show that there does not exist a continuous Lyapunov

function that proves Lyapunov stability of the zero solution x(t) ≡ 0 to

(4.225). To see this, let D ⊂ R be an open interval such that 0 ∈ D

and suppose, ad absurdum, there exists a continuous function V : D → R

such that V (0) = 0, V (x) > 0, x ∈ D, x 6= 0, and V (x(t)), t ≥ 0, is

a n onincreasing function of time. Now, let n ∈ {1, 2, . . .} be such that

nπ

∈ D. Next, note that, for every x

∈ (

(n+1)π

nπ

), x(t) →

nπ

as t → ∞.

Now, since V (x(t)), t ≥ 0, is nonincreasing and V (·) is continuous on D

it follows that V (x

) ≥ lim

t→∞

V (x(t)) = V (lim

t→∞

x(t)) = V (

nπ

) > 0.

Hence, since x

∈ (

(n+1)π

nπ

) is arbitrary it follows from the continuity of

V (·) that V (

(n+1)π

) ≥ V (

nπ

) > 0. Repeating these arguments iteratively it

follows that

V (

Nπ

) ≥ V (

nπ

), N ∈ {n + 1, n + 2, . . .},

and h en ce, it follows from the continuity of V (·) that

V (0) = V



lim

N→∞

Nπ



= lim

N→∞

V (

Nπ

)

≥ V (

nπ

)

> 0,

which is a contradiction. Hence, there does not exist a continuous Lyapunov

function that proves Lyapunov stability of the zero solution x(t) ≡ 0 to

(4.225). △

Next, we generalize the invariant set stability theorems of Section 3.3

to the case in which the function V (·) is lower semicontinuous. For the

NonlinearBook10pt November 20, 2007

274 CHAPTER 4

remainder of the results of this section we deﬁne the notation

△

c>γ

−1

([γ, c]), (4.226)

for arbitrary V : D ⊆ R

→ R and γ ∈ R, and let M

denote the largest

invariant set (with respect to (3.1)) contained in R

Theorem 4.24. Consider the nonlinear dynamical system (3.1), let

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

⊂ D be a compact

invariant set with respect to (3.1). Assum e that there exists a lower

semicontinuous function V : D

→ R such that V (x(t)) ≤ V (x(τ)),

0 ≤ τ ≤ t, for all x

∈ D

. If x

∈ D

, then x(t) → M

△

= ∪

γ∈R

t → ∞.

Proof. Let x(t), t ≥ 0, be the solution to (3.1) with x

∈ D

. Since

V (·) is lower s emicontinuous on the compact set D

, there exists β ∈ R

such that V (x) ≥ β, x ∈ D

. Hence, since V (x(t)), t ≥ 0, is n onincreasing,

△

= lim

t→∞

V (x(t)), x

∈ D

, exists. Now, for every p ∈ ω(x

) there

exists an increasing unbounded sequence {t

}

∞

n=0

, with t

= 0, such that

x(t

) → p as n → ∞. Next, since V (x(t

)), n ≥ 0, is nonincreasing

it follows that for all N ≥ 0, γ

≤ V (x(t

)) ≤ V (x(t

)), n ≥ N, or,

equivalently, since D

is invariant, x(t

) ∈ V

−1

([γ

, V (x(t

))]), n ≥ N.

Now, since lim

n→∞

x(t

) = p it follows that p ∈ V

−1

([γ

, V (x(t

))]), n ≥ 0.

Furth ermore, since lim

n→∞

V (x(t

)) = γ

it follows that for every c > γ

there exists n ≥ 0 such that γ

≤ V (x(t

)) ≤ c, which implies that for

every c > γ

, p ∈ V

−1

([γ

, c]). Hence, p ∈ R

, which implies that

ω(x

) ⊆ R

. Now, since D

is compact and invariant it follows that the

solution x(t), t ≥ 0, to (3.1) is bounded for all x

∈ D

, and hence, it follows

from Theorem 2.41 that ω(x

) is a nonempty compact invariant set which

further implies th at ω(x

) is a subset of the largest invariant set contained

in R

, that is, ω(x

) ⊆ M

. Hence, for all x

∈ D

, ω(x

) ⊆ M. Finally,

since x(t) → ω(x

) as t → ∞ it follows that x(t) → M as t → ∞ .

If in Theorem 4.24 M contains no invariant set other than the set

{0}, then the zero solution x(t) ≡ 0 to (3.1) is attractive and D

is a subset

of the domain of attraction. Fu rthermore, note that if V : D

→ R is a

lower semicontinuous function such that all the conditions of Theorem 4.24

are s atisﬁed, then for every x

∈ D

there exists γ

≤ V (x

) such that

ω(x

) ⊆ M

⊆ M. In addition, since V

−1

([γ, c]) = {x ∈ D

: V (x) ≥

γ} ∩{x ∈ D

: V (x) ≤ c} and {x ∈ D

: V (x) ≤ c} is a closed set, it follows

that

γ,c

⊂ {x ∈ D

: V (x) < γ}, where

γ,c

△

= V

−1

([γ, c]) \ V

−1

([γ, c]),