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

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ADVANCED STABILITY THEORY 285

holds. Then it follows that x(t) ∈ O

, t ≥ 0, and hence, it follows from

Theorem 2.41 that the positive limit set ω(x

) of x(t), t ≥ 0, is a nonempty,

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

) as t →

∞. Now, since V (x(t)), t ≥ 0, is non increasing and bounded from below by

zero it follows that β

△

= lim

t→∞

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

since V (·) is lower s emicontinuous 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

) = D

or,

equivalently, x(t) → D

as t → ∞. Now, sup pose, ad absurdum, that

x(t), t ≥ 0, does not converge to D

or, equivalently, β > 0. Furthermore,

let y ∈ ω(x

) and let {τ

}

∞

n=0

be an increasing unbounded sequence 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∈ D

and

since y ∈ ω(x

) is arbitrary, it follows that ω(x

) ∩ D

= Ø. Next, let

△

= 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(0) ∈ D, x(0) 6∈ D

, there exists an increasing unbounded sequence {t

}

∞

n=1

such that (4.256) holds, it follows that if x(0) = x

∈ ω(x

), there exists

t > 0 s uch that V (x(t)) < V (x(0)) = γ, which is a contradiction. Hence,

ω(x

) ⊆ D

and x(t) → D

as t → ∞, establishing asymptotic stability.

Note that in the case where the function V (·) is continuously

diﬀerentiable on D in Theorem 4.29, it follows that V (x(t)) ≤ V (x(τ)), for

all t ≥ τ ≥ 0, is equivalent to

V (x)

△

= V

′

(x)f(x) ≤ 0, x ∈ D. Furthermore,

V (x) = V

′

(x)f(x) < 0, x ∈ D, x 6∈ D

, then every increasing unbounded

sequence {t

}

∞

n=0

, with t

= 0, is su ch that V (x(t

n+1

)) < V (x(t

)), n =

0, 1, . . .. Hence, the following corollary to Theorem 4.29 is immediate.

Corollary 4.7. Consider the nonlinear dynamical system (4.251) and

let D

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

⊂ D. Assume that there exists a continuously diﬀerentiable function

V : D → R such that

V (x) = 0, x ∈ D

, (4.257)

V (x) > 0, x ∈ D, x 6∈ D

, (4.258)

′

(x)f(x) ≤ 0, x ∈ D. (4.259)

Then D

is Lyapunov stable. I f, in addition,

′

(x)f(x) < 0, x ∈ D, x 6∈ D

, (4.260)

then D

is asymptotically stable.

The following theorem provides a converse to Theorem 4.29. For this

result deﬁn e the notation D

△

= {x ∈ D : dist(x, D

) < r}, r > 0, to denote

an r open neighborhood of D

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286 CHAPTER 4

Theorem 4.30. Consider the nonlinear dynamical system (4.251). If

is Lyapunov stable, then there exist D

⊆ D and a lower semicontinuous,

positive-deﬁnite (on D

) function V : D

→ R such that D

⊂

◦

, V (·)

is continuous on D

, and

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

. (4.261)

If D

is asymptotically stable, then there exists a continuous, positive-

deﬁnite function V : D

→ R such that inequality (4.261) is strictly satisﬁed.

Proof. Let ε > 0. Since D

is Lyapunov stable it follows that there

exists δ = δ(ε) > 0 such that if x

∈ D

, then s(t, x

) ∈ D

, t ≥ 0. Now,

let D

△

= {y ∈ D

: there exists t ≥ 0 and x

∈ D

such that y = s(t, x

)}.

Note that D

⊆ D

, D

is a positively invariant set, and D

⊆ D

. Hence,

⊂

◦

. Next, deﬁne V (x)

△

= sup

t≥0

dist(s(t, x), D

), x ∈ D

, and since D

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

Now, since D

is invariant, x ∈ D

implies V (x) = 0, x ∈ D

. Furth ermore,

V (x) ≥ dist(s(0, x), D

) > 0, x ∈ D

, x 6∈ D

Next, since f (·) in (4.251) is such that for every x ∈ D

, s(t, x), t ≥ 0,

is the unique solution to (4.251), it follows that s(t, x) = s(t−τ, s(τ, x)), 0 ≤

τ ≤ t. Hence, f or every t, τ ≥ 0, such th at t ≥ τ ,

V (s(τ, x)) = sup

θ≥0

dist(s(θ, s(τ, x)), D

)

= sup

θ≥0

dist(s(τ + θ, x), D

)

≥ sup

θ≥t−τ

dist(s(τ + θ, x), D

)

= sup

θ≥t−τ

dist(s(θ − (t − τ), s(t, x)), D

)

= sup

θ≥0

dist(s(θ, s(t, x)), D

)

= V (s(t, x)), (4.262)

which proves (4.261). Next, let p

∈ D

. Since D

is Lyapunov stable

it follows that for every ˆε > 0 there exists

δ =

δ(ε) > 0 such that if

∈ B

), then s(t, x

) ∈ D

ˆε/2

, t ≥ 0, which implies that V (x

) =

sup

t≥0

dist(s(t, x

), D

) ≤

ˆε

. Hence, for every point p

∈ D

and ˆε > 0

there exists

δ =

δ(ˆε) such that if x

∈ B

), th en V (x

) < ˆε, establishing

that V (·) is continuous on D

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

let x ∈ D

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

△

= sup

t≥0

dist(s(t, x), D

there exists T = T (x, ˆε) > 0 such that V (x) − dist(s(T, x), D

) < ˆε. Now,

consider a sequence {x

}

∞

i=1

∈ D

such that x

→ x as i → ∞. Next, since

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ADVANCED STABILITY THEORY 287

s(t, ·) is continuous for every t ≥ 0 and dist(·, D

) : D

→ R is continuous,

it f ollows that dist(s(T, x), D

) = lim

i→∞

dist(s(T, x

), D

). Next, note that

dist(s(T, x

), D

) ≤ sup

t≥0

dist(s(t, x

), D

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

lim inf

i→∞

sup

t≥0

dist(s(t, x

), D

) ≥ lim inf

i→∞

dist(s(T, x

), D

)

= lim

i→∞

dist(s(T, x

), D

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

which implies that

V (x) < dist(s(T, x), D

) + ˆε

= lim

i→∞

dist(s(T, x

), D

) + ˆε

≤ lim inf

i→∞

sup

t≥0

dist(s(t, x

), D

) + ˆε

= lim inf

i→∞

V (x

) + ˆε. (4.263)

Now, since ˆε > 0 is arbitrary, (4.263) 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

To show the existence of a continuous, positive-deﬁnite, strictly

decreasing function along th e system trajectories of (4.251) in the case where

is asymptotically stable, consider the function V : D

→ R given by

V (x)

△

∞

sup

t≥0

dist(s(t, s(σ, x)), D

−σ

dσ, x ∈ D

, (4.264)

where D

⊆ D

is a domain of attraction of D

. Since D

is asymptotically

stable, it follows that

V (x)

△

= sup

t≥0

dist(s(t, x), D

), x ∈ D

, is continuous

on D

(see [58, p. 67]), which implies that V (·) is continuous on D

and

positive deﬁnite on D

\ D

. Next, sup pose, ad absurdum, that

V (s(t, x)) = V (s(τ, x)), (4.265)

for some 0 ≤ τ < t and x ∈ D

. Now, since

V (·) is positive deﬁnite (on

\ D

) and (4.261) holds for

V (·), (4.265) implies that

V (s(σ + t, x)) =

V (s(σ + τ, x)) for all σ ≥ 0. However, since D

is asymptotically stable

it follows that dist(s(t, x), D

) → 0 as t → ∞ for x ∈ D

, and h en ce, for

suﬃciently large σ

∗

> 0, dist(s(σ

∗

+t, x), D

) < dist(s(σ

∗

+τ, x), D

). Thus,

V (s(σ

∗

+ t, x)) <

V (s(σ

∗

+ τ, x)), which leads to a contradiction. Hence,

inequality (4.261) is strictly satisﬁed f or V (·) which completes the proof.

Next, we generalize the Barbashin-Krasovskii-LaSalle invariant set

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

In particular, we show that the system trajectories converge to a union of

largest invariant sets contained on the bou ndary of the intersections over

ﬁnite intervals of the closure of generalized Lyapunov level surfaces. For the

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288 CHAPTER 4

remainder of the results of this section deﬁne the n otation

△

c>γ

−1

([γ, c]), (4.266)

for arbitrary V : D ⊆ R

→ R and γ ∈ R, and let M

denote the largest

invariant set (with respect to (4.251)) contained in R

Theorem 4.31. Consider the nonlinear dynamical system (4.251), let

and D

be compact positively invariant sets with respect to (4.251) such

that D

⊂ D

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

corresponding to x

∈ D

. Assume that there exists a lower semicontinuous

function V : D

→ R such that

V (x) = 0, x ∈ D

, (4.267)

V (x) > 0, x ∈ D

, x 6∈ D

, (4.268)

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t. (4.269)

Furth ermore, assum e that for all x

∈ D

, x

6∈ D

, th ere 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, . . . . (4.270)

Then, either M

⊂

△

= R

\ V

−1

(γ), or M

= Ø, γ > 0. Furthermore,

if x

∈ D

, then x(t) →

△

= ∪

γ∈G

as t → ∞, where G

△

= {γ ≥ 0 :

∩ D

6= Ø}. If, in addition, D

⊂

◦

and V (·) is continuous on D

then D

is locally asymptotically stable and D

is a subset of the domain of

attraction.

Proof. Sin ce D

is a compact positively invariant s et, it follows that for

all x

∈ D

, the forward solution x(t), t ≥ 0, to (4.251) is bounded. Hence,

it follows from Theorem 2.41 that, for all x

∈ D

, ω(x

) is a nonempty,

compact, connected invariant set. Next, it follows from Theorem 4.24 and

the fact that V (·) is positive deﬁnite (with respect to D

\ D

), that for

every x

∈ D

there exists γ

≥ 0 such that ω(x

) ⊆ M

⊆ R

. Now,

given x(0) ∈ V

−1

(γ

), γ

> 0, (4.270) implies that there exists t

> 0

such that V (x(t

)) < γ

and x(t

) 6∈ V

−1

(γ

). Hence, V

−1

(γ

) ⊂ R

does not contain any invariant set. Alternatively, if x(0) ∈

, then

V (x(0)) < γ

and (4.270) implies that x(t) 6∈ V

−1

(γ

), t ≥ 0. Hence,

any invariant set contained in R

is a subset of

, w hich implies that

⊂

, γ

> 0. If ˆγ > 0 is such that ˆγ 6= γ

for all x

∈ D

then there does not exist x

∈ D

such that ω(x

) ⊆ R

ˆγ

, and hence, M

ˆγ

Ø. Now, ad absurdum, suppose D

∩ ω(x

) = Ø. Since V (·) is lower

semicontinuous it follows from Theorem 2.11 that there exists ˆx ∈ ω(x

)

such that α = V (ˆx) ≤ V (x), x ∈ ω(x

). Now, with x(0) = ˆx 6∈ D

it follows

from (4.270) that there exists an increasing unbounded sequence {t

}

∞

n=0

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ADVANCED STABILITY THEORY 289

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, there exists

q ∈ D

such that q ∈ ω(x

) ⊆ R

, which implies that R

∩ D

6= Ø.

Thus , γ

∈ G for all x

∈ D

, which further implies that ω(x

) ⊆

M. Now,

since x(t) → ω(x

) ⊆

M as t → ∞ it follows that x(t) →

M as t → ∞.

If V (·) is continuous on D

⊂

◦

, then Lyapunov stability of the com-

pact positively invariant set D

follows from Theorem 4.29. Furthermore,

from the continuity of V (·) on D

and the fact that V (x) = 0 for all x ∈ D

it follows th at G = {0} and

M ≡ M

. Hence, ω(x

) ⊆ D

for all x

∈ D

establishing local asymptotic stability of the compact positively invariant

set D

of (4.251) with a subset of the domain of attraction given by D

Finally, we present a generalized global invariant set theorem for

guaranteeing global attraction and global asymptotic stability of a compact

positively invariant set of a nonlinear dynamical system.

Theorem 4.32. Consider the nonlinear dynamical system (4.251) with

D = R

and let x(t), t ≥ 0, denote the solution to (4.251) corresponding

to x

∈ R

. Assume that there exists a compact positively invariant set D

with respect to (4.251) and a lower semicontinuous function V : R

→ R

such that

V (x) = 0, x ∈ D

, (4.271)

V (x) > 0, x ∈ R

, x 6∈ D

, (4.272)

V (x(t)) ≤ V (x(τ)), 0 ≤ τ ≤ t, (4.273)

V (x) → ∞ as kxk → ∞. (4.274)

Then for all x

∈ R

, x(t) → M

△

= ∪

γ≥0

, as t → ∞. If, in addition,

for all x

∈ R

, x

6∈ D

, there exists an increasing unbounded sequence

}

∞

n=0

, with t

= 0, such th at

V (x(t

n+1

)) < V (x(t

)), n = 0, 1, . . . , (4.275)

then, either M

⊂

△

= R

\ V

−1

(γ), or M

= Ø, γ > 0. Furthermore,

x(t) →

△

= ∪

γ∈G

as t → ∞, where G

△

= {γ ≥ 0 : R

∩ D

6= Ø}.

Finally, if V (·) is continuous on D

then the compact positively invariant

set D

of (4.251) is globally asymptotically stable.

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 kxk> r or, equivalently,

−1

([0, β]) ⊆ {x : kxk ≤ r}, 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

△

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290 CHAPTER 4

V (x

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

function, it follows that V

−1

([0, β

]) is closed an d, s ince V (x(t)), t ≥ 0, is

nonincreasing, V

−1

([0, β

]) is positively invariant. Hence, for every x

∈

, V

−1

([0, β

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

−1

([0, β

]) it f ollows from Theorem 4.24 that there exists γ

∈ [0, β

]

such that ω(x

) ⊆ M

⊂

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

If, in addition, for all x

∈ R

, x

6∈ D

, there exists an increasing unbounded

sequence {t

}

∞

n=0

, with t

= 0, such that (4.275) holds, then it follows from

Theorem 4.31 that x(t) →

M as t → ∞.

Finally, if V (·) is continuous on D

then Lyapunov stability follows as

in the proof of Theorem 4.31. Fu rthermore, in this case, G = {0}, which

implies that

M = M

. Hence, ω(x

) ⊆ D

, establishing global asymptotic

stability of the compact positively invariant set D

of (4.251).

4.10 Poincar´e Maps and S tability of Periodic Orbits

Poincar´e’s theorem [358] provides a powerful tool in analyzing the stability

properties of periodic orbits and limit cycles of n-dimensional dynamical

systems in the case where the trajectory of the system can be relatively easily

integrated. Speciﬁcally, Poincar´e’s theorem provides necessary and suﬃcient

conditions for stability of periodic orbits based on the stability properties of a

ﬁxed point of a discrete-time dynamical system constructed from a Poincar´e

return map. In particular, for a given candidate periodic trajectory, an (n −

1)-dimensional hyperplane is constructed that is transversal to the periodic

trajectory and which deﬁnes th e Poincar´e return map. Trajectories starting

on the hyperplane which are suﬃciently close to a point on the periodic

orbit will intersect the hyp erplane after a time approximately equal to the

period of the periodic orbit. This mapping traces the system trajectory

from a point on the hyperplane to its next corresponding intersection with

the hyperplane. Hence, the Poincar´e retur n map can be used to establish

a relationship between the stability properties of a dynamical sys tem with

periodic solutions and the stability properties of an equilibrium point of

an (n − 1)-dimensional discrete-time system. In this section, using the

notions of Lyapunov and asymptotic stability of sets developed in S ection

4.9, we construct lower semicontinuous Lyapunov functions to provide a

Lyapunov f unction proof of Poincar´e’s th eorem. To begin, we introduce

the notions of Lyapunov and asymptotic stability of a periodic orbit of the

nonlinear dynamical system (4.251). For this deﬁnition recall the deﬁnitions

of periodic solutions and periodic orbits of (4.251) given in Deﬁnition 2.53.

Deﬁnition 4.12. A periodic orbit O of (4.251) is Lyapunov stable if,

for all ε > 0, there exists δ = δ(ε) > 0 such that if dist(x

, O) < δ, then

dist(s(t, x

), O) < ε, t ≥ 0. A periodic orbit O of (4.251) is asymptotically

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ADVANCED STABILITY THEORY 291

stable if it is Lyapunov stable and there exists δ > 0 such that if dist(x

, O) <

δ, then dist(s(t, x

), O) → 0 as t → ∞.

To proceed, we assume that for the point p ∈ D, the dynamical system

(4.251) has a periodic solution s(t, p), t ≥ 0, with period T > 0 that generates

the periodic orbit O

△

= {x ∈ D : x = s(t, p), 0 ≤ t ≤ T }. Note th at

O is a compact invariant set. Fur th ermore, we assume that there exists

a continuously diﬀerentiable function X : D → R such that the (n − 1)-

dimensional hyperplane deﬁned by H

△

= {x ∈ D : X(x) = 0} contains the

point x = p and X

′

(p) 6= 0. In addition, we assume that the hyperplane

H is not tangent to the periodic orbit O at x = p, that is, X

′

(p)f(p) 6= 0.

Next, deﬁne the local section S ⊂ H such that p ∈ S, X

′

(x) 6= 0, x ∈ S, and

no trajectory of (4.251) starting in S is tangent to H, that is, X

′

(x)f(x) 6=

0, x ∈ S. Note that a trajectory s(t, p) will intersect S at p in T seconds.

Furth ermore, let

△

= {x ∈ S : there exists ˆτ > 0 such that s(ˆτ , x) ∈ S

and s(t, x) 6∈ S, 0 < t < ˆτ}, (4.276)

and let τ : U → R

be deﬁned by

τ(x)

△

= {ˆτ > 0 : s(ˆτ, x) ∈ S and s(t, x) 6∈ S, 0 < t < ˆτ}. (4.277)

Finally, deﬁne the Poincar´e return map P : U → S by

P (x)

△

= s(τ (x), x), x ∈ U. (4.278)

Figure 4.9 gives a visualization of the Poincar´e retur n map construction.

Next, deﬁne D

△

= {x ∈ D : there exists τ(x) > 0 such that s(τ(x),

x) ∈ S} and note that, for every x ∈ O, there exists δ = δ(x) > 0 such

that B

(x) ⊂ D

, and hence, O ⊂

◦

. S imilarly, deﬁne O

△

= {x ∈ D

s(τ(x), x) ∈ S

} and U

△

= {x ∈ S

: s(τ(x), x) ∈ S

}, where S

△

= B

(p)∩S,

α > 0, and O ⊂

◦

⊆ D

. T he function τ : D

→ R

deﬁnes the minimum

time required for the trajectory s(t, x), x ∈ D

, to return to the local section

S. Note that τ (x) > 0, x ∈ U. The following lemma shows that τ (·) is

continuous on

◦

\H.

Lemma 4.4. Consider the nonlinear dynamical system (4.251). As-

sume that the point p ∈ D

generates the periodic orbit O

△

= {x ∈ D

: x =

s(t, p), 0 ≤ t ≤ T }, where s(t, p), t ≥ 0, is the periodic solution with period

T ≡ τ(p). Then the function τ :

◦

→ R

is continuous on

◦

\H.

Proof. Let ε > 0 and x ∈

◦

\H. Note that x

∗

△

= s(τ(x), x) ∈ S, and

hence, X

′

∗

) 6= 0 and X

′

∗

)f(x

∗

) 6= 0. Now, since s(·, x) is continuous in

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292 CHAPTER 4

P (x)

Figure 4.9 Visualization of the Poincar´e return map.

t, [0, t

] is a compact interval it follows from the d eﬁ nition of τ(·) that there

exists

t > 0 such that for every t

∈ (

t, τ(x)),

σ(t

)

△

= inf

0≤t≤t

dist(s(t, x), S) > 0. (4.279)

Next, for suﬃciently small ˆε > 0, deﬁne t

△

= τ(x) +

ˆε

and x

△

= s(t

, x).

Since X

′

∗

) 6= 0 and X

′

∗

)f(x

∗

) 6= 0, it follows that dist(x

, S) > 0. Now,

deﬁne t

△

= τ(x) −

ˆε

. Then it follows from the continuous dependence of

solutions to (4.251) in time and initial data that there exists δ > 0 such

that for all y ∈ B

(x), sup

≤t≤t

ks(t, y) − s(t, x)k < min{dist(x

, S), σ(t

)}.

Hence, for all y ∈ B

(x), it follows that t

< τ(y) < t

. Now, taking

ˆε < ε, it follows that |τ(y) −τ (x)| < ε, establishing the continuity of τ(·) at

x ∈

◦

\H.

Finally, deﬁne the discrete-time dynamical system given by

z(k + 1) = P (z(k)), z(0) ∈ U, k ∈ Z

. (4.280)

Clearly x = p is a ﬁxed point of (4.280) since T = τ(p), an d h en ce, p = P (p).

Since Poincar´e’s theorem provides necessary and suﬃcient conditions for

stability of periodic orbits based on the stability properties of a ﬁxed

point of the discrete-time dynamical sy s tem (4.280), stability notions of

discrete-time systems are required. Chapter 13 develops stability theory

for discrete-time s y s tems. Rather than embarking on a lengthy discussion

of discrete-time stability theory, here we give two key necessary results

for developing Poincar´e’s theorem. For th ese results, the deﬁnitions of

Lyapunov and asymptotic stability of a discrete-time system are analogous

to their continuous-time counterparts and are given in Chapter 13 (see

Deﬁnition 13.1). First, we note that if ρ(P

′

(z(p))) < 1, where ρ(·)

denotes the spectral radius, then the ﬁxed point x = p of th e nonlinear

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ADVANCED STABILITY THEORY 293

discrete-time dynamical system (4.280) is asymptotically stable. This is

Lyapunov’s indirect method for nonlinear discrete-time systems. For d etails,

see Problem 13.10. Second, the following theorem is a direct application of

the standard discrete-time Lyapunov stability theorem (see Theorems 13.2

and 13.6) for general nonlinear dynamical systems to the dynamical system

(4.280).

Theorem 4.33. The equilibrium solution z(k) ≡ p to (4.280) is

Lyapunov (respectively, asymptotically) stable if and only if there exist

a scalar α > 0 and a lower semicontinuous (respectively, continuous)

function V : S

→ R such that V (·) is continuous at x = p, V (p) = 0,

V (x) > 0, x ∈ S

, x 6= p, and V (P (x)) − V (x) ≤ 0, x ∈ U

(respectively,

V (P (x)) −V (x) < 0, x ∈ U

, x 6= p).

Next, we present Poincar´e’s stability th eorem.

Theorem 4.34. Consider the nonlinear dynamical system (4.251) with

the Poincar´e map deﬁned by (4.278). Assume that the point p ∈ D generates

the periodic orbit O

△

= {x ∈ D : x = s(t, p), 0 ≤ t ≤ T }, where s(t, p), t ≥ 0,

is the perio dic solution with period T ≡ τ(p). Then the following statements

hold:

i) p ∈ D is a Lyapunov stable ﬁxed point of (4.280) if and only if the

periodic orbit O generated by p is Lyapunov stable.

ii) p ∈ D is an asymptotically stable ﬁxed point of (4.280) if and only if

the periodic orbit O generated by p is asymptotically stable.

Proof. i) To s how necessity, assume that x = p is a Lyapunov stable

ﬁxed point of (4.280). Then it follows from Theorem 4.33 that for suﬃciently

small α > 0 there exists a lower semicontinuous function V

: S

→ R such

that V

(·) is continuous at x = p, V

(p) = 0, V

(x) > 0, x ∈ S

, x 6= p, and

(P (x))−V

(x) ≤ 0, x ∈ U

. Next, deﬁne a function V : O

→ R such that

V (x) = V

(s(τ(x), x)), x ∈ O

. It follows from the deﬁnition of τ (·), Lemma

4.4, and the j oint continuity of solutions of (4.251) that V (·) is a lower

semicontinuous function on O

and V (x) = 0, x ∈ O, V (x) > 0, x ∈ O

\O,

where O ⊂

◦

. Alternatively, it f ollows from the Lyapunov stability of the

ﬁxed point x = p of (4.280) that for ε > 0 such th at B

(p) ∩ S ⊂ U

there exists δ = δ(ε) > 0 such that z(k) ∈ B

(p) ∩ S, k ∈ Z

, for all

z(0) ∈ B

(p) ∩ S, where z(k) ∈ N satisﬁes (4.280). Now, deﬁne O

△

= {x ∈

: s(τ(x), x) ∈ B

(p) ∩ S} and note that O ⊂

◦

. Hence, V (s(t, x)) ≤

V (s(τ, x)), 0 ≤ τ ≤ t, for every x ∈ O

⊆ O

. Now, to show that V (·) is

continuous on O let p

∈ O be such that p

6= p and consider any arbitrary

NonlinearBook10pt November 20, 2007

294 CHAPTER 4

sequence {x

}

∞

n=0

∈ O

such that lim

n→∞

= p

. Then, since τ(·) is

continuous on D

\ H, it follows that lim

n→∞

τ(x

) = τ(p

) and, by joint

continuity of solutions of (4.251), s(τ(x

), x

) → s(τ(p

), p

) = p as n →

∞. Now, since V

(·) is continuous at x = p, it follows that lim

n→∞

V (x

) =

lim

n→∞

(s(τ(x

), x

)) = V

(p) = 0, which, since {x

}

∞

n=0

is arbitrary,

implies continuity of V (·) at any point p

∈ O, p

6= p.

Next, we show the continuity of V (·) at x = p. Note, that V (·) is

not necessarily continuous at every point of S but x = p. Consider any

arbitrary sequence {x

}

∞

n=0

∈ O

such that lim

n→∞

= p. For this

sequence we have one of the following three cases: either lim

n→∞

τ(x

) = 0,

lim

n→∞

τ(x

) = T , or there exist su bsequences {x

}

∞

k=0

and {x

}

∞

m=0

such that {x

}

∞

k=0

∪ {x

}

∞

m=0

= {x

}

∞

n=0

, x

→ p, τ(x

) → 0, as

k → ∞, and x

→ p, τ(x

) → T , as m → ∞. We assume the latter

case, sin ce the analysis for the ﬁrst two cases follows immediately from the

arguments for p

∈ O, p

6= p, presented above. The characterization

of both subsequences and joint continuity of solutions of (4.251) yield

s(τ(x

), x

) → s(0, p) = p and s(τ(x

), x

) → s(T, p) = p, as k → ∞,

and m → ∞, respectively. Now, sin ce V

(·) is continuous at x = p, it

follows that lim

k→∞

V (x

) = lim

k→∞

(s(τ(x

), x

)) = V

(p) = 0 and

lim

m→∞

V (x

) = lim

m→∞

(s(τ(x

), x

)) = V

(p) = 0, and thus,

lim

n→∞

V (x

) = V (p) = 0, which implies that V (·) is continuous at x = p,

and hence, V (·) is continuous on O. Finally, since all th e assump tions of

Theorem 4.29 hold, the periodic orbit O is Lyapunov stable.

To show suﬃ ciency, assume that the periodic orbit O generated by the

point p ∈ D is Lyapunov s table. Th en it follows from the Theorem 4.30 that

there exists a lower semicontinuous, positive-deﬁnite (on D

\ O) function

V : D

→ R such that (4.261) is satisﬁed. Now, for suﬃciently small α > 0,

construct a function V

: S

→ R such that V

(x) = V (x), x ∈ S

. Thus,

in this case the suﬃcient conditions of Theorem 4.33 are satisﬁed for V

(·),

which implies that the point x = p is a Lyapunov stable ﬁxed point of

(4.280).

ii) To show necessity, assume that x = p is an asymptotically stable

ﬁxed point of (4.280). Then it follows from T heorem 4.33 that there exists

a continuous function V

: S

→ R such that V

(p) = 0, V

(x) > 0, x ∈

, x 6= p, and V

(P (x)) − V

(x) < 0, x ∈ U

, x 6= p. Next, as in i),

construct the lower semicontinuous function V : O

→ R such that V (x) =

(s(τ(x), x)), x ∈ O

, V (x) = 0, x ∈ O, V (x) > 0, x ∈ O

, x 6∈ O,

V (s(t, x)) ≤ V (s(τ, x)), 0 ≤ τ ≤ t, for every x ∈ O

⊆ O

, and V (·)

is continuous on O. Furthermore, for every x ∈ O

deﬁne an increasing

unbounded sequence {t

}

∞

n=0

such that t

= 0 and t

= τ(z(k − 1)), k =

1, 2, . . ., where z(·) satisﬁes (4.280) with z(0) = x. Th en it follows from