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

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NonlinearBook10pt November 20, 2007

ADVANCED STABILITY THEORY 305

and the zero solution z(t) ≡ 0 to (4.325) is exponentially stable, then

the zero solution x(t) ≡ 0 to (4.326) is exponentially stable.

v) If D = R

, Q = R

, there exist constants ν ≥ 1, α > 0, and β > 0 such

that v : R

→ R

satisﬁes (4.328), and the zero solution z(t) ≡ 0 to

(4.325) is globally exponentially stable, then the zero solution x(t) ≡ 0

to (4.326) is globally exponentially stable.

Proof. Assume that there exist a continuously diﬀerentiable vector

function V : D → Q ∩ R

and a positive vector p ∈ R

such that v(x) =

V (x), x ∈ D, is positive deﬁnite, that is, v(0) = 0 and v(x) > 0, x 6=

0. Note that since v(x) = p

V (x) ≤ max

i=1,...,q

V (x), x ∈ D, the

function e

V (x), x ∈ D, is also positive deﬁnite. Thus, there exist r > 0

and class K functions α, β : [0, r] → R

such that B

(0) ⊂ D and

α(kxk) ≤ e

V (x) ≤ β(kxk), x ∈ B

(0). (4.329)

i) Let ε > 0 and choose 0 < ˆε < min{ε, r}. It follows from Lyapunov

stability of the nonlinear comparison system (4.325) that there exists µ =

µ(ˆε) = µ(ε) > 0 such that if kz

< µ, where k · k

denotes the absolute

sum norm, then kz(t)k

< α(ˆε), t ≥ t

. Now, choose z

= V (x

) ≥≥ 0, x

∈

D. Since V (x), x ∈ D, is continuous, the function e

V (x), x ∈ D, is also

continuous. Hence, for µ = µ(ˆε) > 0 there exists δ = δ(µ(ˆε)) = δ(ε) > 0

such that δ < ˆε, and if kx

k < δ, then e

V (x

) = e

= kz

< µ,

which implies that kz(t)k

< α(ˆε), t ≥ t

. Now, with z

= V (x

) ≥≥ 0,

∈ D, and the assumption that w(·) ∈ W, x ∈ D, it follows from (4.327)

and Corollary 4.9 that 0 ≤≤ V (x(t)) ≤≤ z(t) on any compact interval

, t

+ τ ], and hence, e

z(t) = kz(t)k

, t ∈ [t

, t

+ τ ]. Let τ > t

be s uch

that x(t) ∈ B

(0), t ∈ [t

, t

+ τ ], for all x

∈ B

(0). Thus, using (4.329), if

k < δ, then

α(kx(t)k) ≤ e

V (x(t)) ≤ e

z(t) < α(ˆε), t ∈ [t

, t

+ τ], (4.330)

which implies kx(t)k < ˆε < ε, t ∈ [t

, t

+ τ]. Now, suppose, ad absurdum,

that for some x

∈ B

(0) there exists

t > t

+ τ such that kx(

t)k = ˆε. Then,

for z

= V (x

) and the compact interval [t

t] it follows from (4.327) an d

Corollary 4.9 that V (x(

t)) ≤≤ z(

t), which implies that α(ˆε) = α(kx(

t)k) ≤

V (x(

t)) ≤ e

t) < α(ˆε). This is a contradiction, and h en ce, for a given

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

∈ B

(0), kx(t)k < ε, t ≥

, which implies Lyapunov s tability of the zero solution x(t) ≡ 0 to (4.326).

ii) It follows f rom i) and the asymptotic stability of the nonlinear

comparison system (4.325) that the zero solution to (4.326) is Lyapunov

stable and there exists µ > 0 such that if kz

< µ, then lim

t→∞

z(t) = 0

for every x

∈ D. As in i), choose z

= V (x

) ≥≥ 0, x

∈ D. It follows

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306 CHAPTER 4

from Lyapunov stability of the zero solution to (4.326) and the continuity of

V : D → Q ∩ R

that there exists δ = δ(µ) > 0 such that if kx

k < δ, then

kx(t)k < r, t ≥ t

, and e

V (x

) = e

= kz

< µ. Thus, by asymptotic

stability of (4.325) for every arbitrary ε > 0 there exists T = T (ε) > t

such

that kz(t)k

< α(ε), t ≥ T . Thus, it follows from (4.327) and Corollary 4.9

that 0 ≤≤ V (x(t)) ≤≤ z(t) on any compact interval [t

, T + τ], and hence,

z(t) = kz(t)k

, t ∈ [t

, T + τ], and, by (4.329),

α(kx(t)k) ≤ e

V (x(t)) ≤ e

z(t) < α(ε), t ∈ [T, T + τ ]. (4.331)

Now, suppose, ad absurdum, that for some x

∈ B

(0), lim

t→∞

x(t) 6= 0,

that is, there exists a sequence {t

}

∞

k=1

, with t

→ ∞ as k → ∞, such th at

kx(t

)k ≥ ˆε, k ∈ Z

, for some 0 < ˆε < r. C hoose ε = ˆε and the interval

[T, T + τ] such that at least one t

∈ [T, T + τ ]. Then it f ollows from

(4.331) that α(ε) ≤ α(kx(t

)k) < α(ε), which is a contradiction. Hence,

there exists δ > 0 such that for all x

∈ B

(0), lim

t→∞

x(t) = 0 which, along

with Lyapunov stability, implies asymptotic stability of the zero solution

x(t) ≡ 0 to (4.326).

iii) Suppose D = R

, Q = R

, v : R

→ R

is a radially

unbounded function, and the n on linear comparison system (4.325) is globally

asymptotically stable. I n this case, for V : R

→ R

the in equ ality (4.329)

holds for all x ∈ R

, w here the functions α, β : R

→ R

are of class K

∞

Furth ermore, Lyapunov stability of the zero solution x(t) ≡ 0 to (4.326)

follows from i). Next, for every x

∈ R

and z

= V (x

) ∈ R

, identical

arguments as in ii) can be used to show that lim

t→∞

x(t) = 0, wh ich proves

global asymptotic stability of the zero solution x(t) ≡ 0 to (4.326).

iv) Suppose (4.328) holds. Since p ∈ R

, then

ˆαkxk

≤ e

V (x) ≤

βkxk

, x ∈ D, (4.332)

where ˆα , α/ max

i=1,...,q

} and

β , β/ min

i=1,...,q

}. It follows from the

exponential stability of the nonlinear comparison system (4.325) that there

exist positive constants γ, µ, and η such that if kz

< µ, then

kz(t)k

≤ γkz

−η(t−t

)

, t ≥ t

, (4.333)

for all x

∈ D. Let x

∈ D and z

= V (x

) ≥≥ 0. By continuity of

V : D → Q ∩ R

, there exists δ = δ(µ) > 0 such that for all x

∈ B

(0),

V (x

) = e

= kz

< µ. Furthermore, it follows from (4.327), (4.332),

(4.333), and Corollary 4.9 that, for all x

∈ B

(0), the inequality

ˆαkx(t)k

≤ e

V (x(t)) ≤ e

z(t) ≤ γkz

−η(t−t

)

≤ γ

βkx

−η(t−t

)

(4.334)

holds on any compact interval [t

, t

+τ]. This in turn implies that, for every

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ADVANCED STABILITY THEORY 307

∈ B

(0),

kx(t)k ≤

ˆα

−

(t−t

)

, t ∈ [t

, t

+ τ ]. (4.335)

Now, suppose, ad absurdum, that for some x

∈ B

(0) there exists

t > t

+ τ

such that kx(

t)k >



ˆα



−

(

t−t

)

. Then for the compact interval

t], it follows from (4.335) that kx(

t)k ≤



ˆα



−

(

t−t

)

, which is

a contradiction. Thus, inequality (4.335) holds for all t ≥ t

, establishing

exponential stability of the zero solution x(t) ≡ 0 to (4.326).

v) The proof is identical to the proof of iv).

If V : D → Q∩R

satisﬁes the conditions of Theorem 4.37 we say that

V (x), x ∈ D, is a vector Lyapunov function. Note that for stability analysis

each compon ent of a vector Lyapun ov function need not be positive deﬁnite

with a negative-deﬁnite or negative-semideﬁnite time derivative along the

trajectories of (4.326). This provides more ﬂexibility in searching f or a

vector Lyapunov function as compared to a scalar Lyapunov function for

addressing the stability of nonlinear dynamical systems.

Example 4.20. Consider the nonlinear d ynamical system given by

˙x

(t) = −x

(t) −x

(t)x

(t), x

(0) = x

, t ≥ 0, (4.336)

˙x

(t) = −x

(t) + x

(t)x

(t), x

(0) = x

. (4.337)

Note that Lyapunov’s indirect method fails to yield any information on the

stability of the zero solution x(t)

△

= [x

(t), x

(t)]

≡ 0 of (4.336) and (4.337).

To examine th e stability of (4.336) and (4.337) consider the vector Lyapunov

function candidate V (x) = [v

(x), v

(x)]

, x ∈ R

, with v

(x) =

and

(x) =

. Clearly, V (0) = 0 and e

V (x), x ∈ R

, is a positive-deﬁnite

function. Next, consider the domain D

△

= {x ∈ R

: |x

| ≤ 1, |x

| ≤ c

where c

> 0, and note that

˙v

(x(t)) = x

(t)(−x

(t) −x

(t)x

(t))

≤ −x

(t) + |x

(t)x

(t)|

≤ (−2 + 2c

(x(t)), (4.338)

˙v

(x(t)) = x

(t)(−x

(t) + x

(t)x

(t))

≤ −x

(t) + |x

(t)x

(t)|

≤ 2c

(x(t)) − 8v

(x(t)), (4.339)

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308 CHAPTER 4

for all x(t) ∈ D, t ≥ 0. Thus, the comparison system (4.325) is given by

˙z

(t) = (−2 + 2c

(t), z

(0) = z

, t ≥ 0, (4.340)

˙z

(t) = 2c

(t) − 8z

(t), z

(0) = z

, (4.341)

where (z

, z

) ∈ R

×R

. Note that w(z)

△

= [(−2+ 2c

, 2c

−8z

]

∈

W, where z

△

= [z

, z

]

Next, to show stability of the zero solution z(t) ≡ 0 to the comparison

system, consider the linear Lyapunov fu nction candidate v(z) = z

+ z

, z ∈

(see Problem 3.8). Clearly, v(0) = 0 and v(z) > 0, z ∈ R

\ {0}.

Moreover,

˙v(z(t)) = 2(−1 + c

+ c

(t) −8z

(t), t ≥ 0. (4.342)

In order to ensure asymptotic stability of the zero solution z(t) ≡ 0, it

suﬃces to take c

= 0.83. In this case, it follows from Theorem 4.37 that

the zero solution x(t) ≡ 0 to (4.336) and (4.337) is asymptotically stable. △

Next, we p resent a convergence result via vector Lyapunov fun ctions

that allows us to establish asymptotic stability of the nonlinear dynamical

system (4.326).

Theorem 4.38. Consider the nonlinear dynamical system (4.326),

assume that there exist a continuously diﬀerentiable vector function V =

, . . . , v

]

: D → Q∩R

and a positive vector p ∈ R

such that V (0) = 0,

the scalar function v : D → R

deﬁned by v(x) , p

V (x), x ∈ D, is such

that v(x) > 0, x 6= 0, and

′

(x)f(x) ≤ ≤ w(V (x)), x ∈ D, (4.343)

where w : Q → R

is continuous, w(·) ∈ W, and w(0) = 0, s uch that the

nonlinear comparison system (4.325) is Lyapunov stable. Let R

, {x ∈

D : v

′

(x)f(x) −w

(V (x)) = 0}, i = 1, . . . , q. Then there exists D

⊂ D such

that x(t) → R , ∩

i=1

as t → ∞ for all x(t

) = x

∈ D

. Moreover,

if R contains no trajectory other than the trivial trajectory, then the zero

solution x(t) ≡ 0 to (4.326) is asymptotically stable.

Proof. Since the nonlinear comparison system (4.325) is Lyapunov

stable, it follows that there exists

δ > 0 such that, if kz

δ, then the

system traj ectories z(t), t ≥ t

, of (4.325) are bounded. Furthermore, s ince

V (x), x ∈ D, is continuous, it follows that there exists δ

= δ

(

δ) > 0 such

that e

V (x

) <

δ for all x

∈ B

(0). In addition, it follows from T heorem

4.37 that the zero solution x(t) ≡ 0 to (4.326) is Lyapunov stable, and hence,

for a given ε > 0 such that B

(0) ⊂

◦

, there exists δ

= δ

(ε) > 0 such that,

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ADVANCED STABILITY THEORY 309

if x

∈ B

(0), then x(t) ∈ B

(0), t ≥ t

, where x(t), t ≥ t

, is the solution

to (4.326). Choose δ = m in{δ

, δ

} and deﬁne D

, B

(0) ⊂ D. Then for

every x

∈ D

and z

= V (x

), it follows that x(t) ∈ B

(0), t ≥ t

, and

z(t), t ≥ t

, is bounded.

Next, consider the function

(t) , v

(x(t)) −

(V (x(s)))ds, t ≥ t

, x ∈ D, i = 1, . . . , q.

It follows f rom (4.343) that

(t) = v

′

(x(t))f(x(t)) − w

(V (x(t))) ≤ 0, t ≥ t

, x

∈ D, (4.344)

which implies that W

(t), i ∈ {1, . . . , q}, is a nonincreasing function of

time, and hence, lim

t→∞

(t), i ∈ {1, . . . , q}, exists. Moreover, W

) =

(x(t

)) < ∞, i ∈ {1, . . . , q}. Now suppose, ad absurdum, that for

some initial condition x(t

) = x

∈ D

, lim

t→∞

(t) = −∞ for some

i ∈ {1, . . . , q}. Since the function v

(x), x ∈ D, is continuous on the compact

set B

(0), it follows that v

(x(t)), t ≥ t

, is uniformly bounded, and hence,

lim

t→∞

(V (x(s)))ds = ∞. Now, it follows from (4.343) and Corollary

4.9 that V (x(t)) ≤≤ z(t), t ≥ t

, for z(t

) = V (x(t

)). Note that since

∈ D

it follows that z(t), t ≥ t

, is bounded. Furthermore, since w(·) ∈ W

it follows that

(x(t)) ≤ v

(x(t

)) +

(V (x(s)))ds ≤ z

) +

(z(s))ds = z

(t)

for all t ≥ t

. Since z(t) and v

(x(t)) are bounded for all t ≥ t

it follows

that there exists M > 0 such that |

(V (x(s)))ds| < M, t ≥ t

. T his is

a contradiction, and hence, lim

t→∞

(t), i ∈ {1, . . . , q}, exists and is ﬁnite

for every x

∈ D

. Thus , for every x

∈ D

, it follows that

(s)ds =

′

(x(s))f(x(s)) − w

(V (x(s)))]ds

= W

(t) −W

), t ≥ t

, (4.345)

and hence, lim

t→∞

′

(x(s))f(x(s))−w

(V (x(s)))]ds, i ∈ {1, . . . , q}, exists

and is ﬁnite.

Next, since f(·) is Lipschitz continuous on D and x(t) ∈ B

(0) for all

∈ D

and t ≥ t

it follows that

kx(t

) − x(t

)k =



f(x(s))ds



≤ L

kx(s)kds

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310 CHAPTER 4

≤ Lε(t

− t

), t

≥ t

, (4.346)

where L is the Lipschitz constant on D

. Thus, it follows from (4.346)

that for every γ > 0 there exists µ = µ(γ) =

Lε

such that kx(t

) −x(t

)k <

γ, |t

−t

| < µ, which shows that x(t), t ≥ t

, is uniform ly continuous. Next,

since x(t) is uniformly continuous and v

′

(x)f(x) − w

(V (x)), x ∈ D, i ∈

{1, . . . , q}, is continuous, it follows that v

′

(x)f(x) − w

(V (x)) is uniform ly

continuous on B

(0), and h en ce, v

′

(x(t))f(x(t))−w

(V (x(t))), i ∈ {1, . . . , q},

is uniformly continuous at every t ≥ t

. Hence, it follows from Barbalat’s

lemma (Lemma 4.1) that v

′

(x(t))f(x(t)) −w

(V (x(t))) → 0 as t → ∞ for all

∈ D

and i ∈ {1, . . . , q}. Repeating the above analysis for all i = 1, . . . , q,

it follows that x(t) → R = ∩

i=1

for all x

∈ D

. Finally, if R contains

no trajectory other than the trivial trajectory, then R = {0}, and hence,

x(t) → 0 as t → ∞ for all x

∈ D

, which proves asymptotic stability of the

zero solution x(t) ≡ 0 to (4.326).

Note that R = ∩

i=1

6= Ø since 0 ∈ R. Furthermore, recall that for

every bou nded solution x(t), t ≥ t

, to (4.326) with initial condition x(t

) =

, the positive limit set ω(x

) of (4.326) is a nonempty, compact, invariant,

and connected set with x(t) → ω(x

) as t → ∞. I f q = 1 and w(V (x)) ≡ 0,

then it can be shown that the Lyapunov derivative

V (x) vanishes on the

positive limit set ω(x

), x

∈ D

, so that ω(x

) ∈ R. Moreover, since ω(x

)

is a positively invariant set with respect to (4.326), it follows that for all

∈ D

, the trajectory of (4.326) converges to the largest invariant set M

contained in R. In this case, Theorem 4.38 specializes to Theorem 3.3.

If for some k ∈ {1, . . . , q}, w

(V (x)) ≡ 0 and v

′

(x)f(x) < 0, x ∈

D, x 6= 0, then R = R

= {0}. In this case, it follows from Theorem 4.38

that the zero solution x(t) ≡ 0 to (4.326) is asymptotically stable. Note

that even th ough for k ∈ {1, . . . , q} the time derivative ˙v

(x), x ∈ D, is

negative deﬁnite, the function v

(x), x ∈ D, can be nonnegative deﬁ nite, in

contrast to classical Lyapunov stability theory, to ensure asymptotic stability

of (4.326).

Next, we present a converse Lyapu nov theorem that establishes

the existence of a vector Lyapunov function for an asymptotically stable

nonlinear dynamical system.

Theorem 4.39. Consider the nonlinear dynamical system (4.326).

Assume that f : D → R

is continuously diﬀerentiable, the zero solution

x(t) ≡ 0 to (4.326) is asymptotically stable, and let δ > 0 be such that

(0) ⊂ D is contained in the domain of attraction of (4.326). Then

there exist a continuously diﬀerentiable componentwise positive-deﬁnite

vector function V = [v

, . . . , v

]

: B

(0) → R

and a continuous function

w = [w

, . . . , w

]

: R

→ R

such that V (0) = 0, w(·) ∈ W, w(0) = 0,

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ADVANCED STABILITY THEORY 311

′

(x)f(x) ≤ ≤ w(V (x)), x ∈ B

(0), and the zero solution z(t) ≡ 0 to

˙z(t) = w(z(t)), z(t

) = z

, t ≥ t

, (4.347)

where z

∈ R

, is asymptotically stable.

Proof. Since the zero solution x(t) ≡ 0 to (4.326) is asymptotically

stable it follows from Theorem 3.9 that there exist a continuously diﬀeren-

tiable positive deﬁnite function ˜v : B

(0) → R

and class K functions α(·),

β(·), and γ(·) such that

α(kxk) ≤ ˜v(x) ≤ β(kxk), x ∈ B

(0), (4.348)

˜v

′

(x)f(x) ≤ −γ(kxk), x ∈ B

(0). (4.349)

Furth ermore, it follows from (4.348) and (4.349) that

˜v

′

(x)f(x) ≤ −γ ◦ β

−1

(˜v(x)), x ∈ B

(0), (4.350)

where “◦” denotes the composition operator and β

−1

: [0, β(δ)] → R

is the

inverse function of β(·), and hence, β

−1

(·) and γ◦β

−1

(·) are class K functions.

Next, deﬁne V = [v

, . . . , v

]

: B

(0) → R

such that v

(x) , ˜v(x), x ∈

(0), i = 1, . . . , q. Then it follows that V (0) = 0 and V

′

(x)f(x) ≤≤

w(V (x)), x ∈ B

(0), where w = [w

, . . . , w

]

: R

→ R

is such that

(V (x)) = −γ ◦ β

−1

(x)), x ∈ B

(0). Note that w(·) ∈ W and w(0) = 0.

To show that the zero solution z(t) ≡ 0 to (4.347) is asymptotically stable,

consider the Lyapunov function candidate ˆv(z) , e

z, z ∈ R

. Note that

ˆv(0) = 0, ˆv(z) > 0, z ∈ R

, z 6= 0, and

ˆv(z) = −

i=1

γ ◦ β

−1

) < 0, z ∈

, z 6= 0. Thus, the zero solution z(t) ≡ 0 to (4.347) is asymptotically

stable.

Next, we provide a time-varying extension of Theorem 4.37. In

particular, we consider the nonlinear time-varying dynamical system

˙x(t) = f(t, x(t)), x(t

) = x

, t ≥ t

, (4.351)

where x(t) ∈ D ⊆ R

, 0 ∈ D, f : [t

, ∞) × D → R

is such that f(·, ·) is

jointly continuous in t and x, for every t ∈ [t

, ∞), f (t, 0) = 0, and f (t, ·) is

locally Lipschitz in x uniformly in t for all t in compact subsets of [0, ∞).

Theorem 4.40. Consider the nonlinear time-varying dynamical sys-

tem (4.351). Assume that there exist a continuous ly diﬀerentiable vector

function V : [0, ∞) × D → Q ∩ R

, a positive vector p ∈ R

, and class K

functions α, β : [0, r] → R

such that V (t, 0) = 0, t ∈ [0, ∞), the s calar

function v : [0, ∞) × D → R

deﬁned by v(t, x)

△

= p

V (t, x), (t, x) ∈

[0, ∞) ×D, is such that

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

(0), B

(0) ⊆ D, (4.352)

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312 CHAPTER 4

and

∂V

∂t

(t, x) +

∂V

∂x

(t, x)f(t, x) ≤≤ w(t, V (t, x)), (t, x) ∈ [0, ∞) ×D, (4.353)

where w : [0, ∞) × Q → R

is continuous, w(t, ·) ∈ W, and w(t, 0) = 0, t ∈

[0, ∞). Then the stability pr operties of the zero solution z(t) ≡ 0 to

˙z(t) = w(t, z(t)), z(t

) = z

, t ≥ t

, (4.354)

where z

∈ Q∩R

, imply the corresponding s tability properties of the zero

solution x(t) ≡ 0 to (4.351). That is, if the zero solution z(t) ≡ 0 to (4.354)

is uniformly Lyapunov (respectively, uniformly asymptotically) stable, then

the zero solution x(t) ≡ 0 to (4.351) is uniformly Lyapunov (respectively,

uniformly asymptotically) stable. If, in addition, D = R

, Q = R

, and

α(·), β(·) are class K

∞

functions, then global uniform asymptotic stability

of the zero solution z(t) ≡ 0 to (4.354) implies global uniform asymptotic

stability of the zero solution x(t) ≡ 0 to (4.351). Moreover, if there exist

constants ν ≥ 1, α > 0, and β > 0 such that v : [0, ∞ ) × D → R

satisﬁes

αkxk

≤ v(t, x) ≤ βkxk

, (t, x) ∈ [0, ∞) × D, (4.355)

then exponential stability of the zero solution z(t) ≡ 0 to (4.354) implies

exponential stability of the zero solution x(t) ≡ 0 to (4.351). Finally, if

D = R

, Q = R

, there exist constants ν ≥ 1, α > 0, and β > 0 such that

v : [0, ∞) × R

→ R

satisﬁes (4.355), then global exponential stability of

the zero solution z(t) ≡ 0 to (4.354) implies global exponential stability of

the zero solution x(t) ≡ 0 to (4.351).

Proof. The pr oof is similar to the proof of Theorem 4.37 and is left

as an exercise for the reader.

Finally, to elucidate how to use the vector Lyapunov function fr ame-

work to address the problem of control design for nonlinear dyn amical

systems consider the controlled nonlinear dynamical system given by

˙x(t) = F (x(t), u(t)), x(t

) = x

, t ≥ t

, (4.356)

where x

∈ D, D ⊆ R

is an open set, 0 ∈ D, u(t) ∈ U ⊆ R

is the control

input, U is the set of all admissib le control inputs, F : D × U → R

Lipschitz continuous for all (x, u) ∈ D × U, and F (0, 0) = 0. Moreover,

assume that for every x

∈ D and u(t) ∈ U, t ≥ t

, the solution x(t) to

(4.356) is uniqu e and deﬁned f or all t ≥ t

. Now, assume there exist a

continuously diﬀerentiable vector function V : D → Q ∩ R

and a positive

vector p ∈ R

such that V (0) = 0, v(x)

△

= p

V (x), x ∈ D, is positive

deﬁnite, and

′

(x)F (x, u) ≤≤ w(V (x), u), x ∈ D, u ∈ U, (4.357)

where w : Q × U → R

is continuous. Furthermore, deﬁne the f eedback

NonlinearBook10pt November 20, 2007

ADVANCED STABILITY THEORY 313

control law φ : Q → U given by u = φ(V (x)), x ∈ D, so that φ(0) = 0 and

(4.356) is given by

˙x(t) = F (x(t), φ(V (x(t)))), x(t

) = x

, t ≥ t

. (4.358)

Now, if φ(·) is s uch that the zero solution z(t) ≡ 0 to

˙z(t) = ˜w(z(t)), z(t

) = z

, t ≥ t

, (4.359)

where ˜w(z)

△

= w(z, φ(z)), z ∈ Q, ˜w(·) ∈ W, ˜w(0) = 0, z

∈ Q ∩ R

, is

asymptotically stable then the zero solution x(t) ≡ 0 of the closed-loop

system (4.358) is asymptotically stable.

4.12 Problems

Problem 4.1. Consider the nonlinear dynamical system

˙x

(t) = f

(t), x

(t)), x

(0) = x

, t ≥ 0, (4.360)

˙x

(t) = f

(t), x

(t)), x

(0) = x

, (4.361)

where x

∈ R

, x

∈ R

, f

: R

× R

→ R

is Lipschitz continuous

on R

× R

and satisﬁes f

(0, x

) = 0 for all x

∈ R

, and f

: R

→ R

is Lipschitz continuous on R

× R

. Show that if there exists

a continuously diﬀerentiable function V : R

→ R such that

V (0) = 0, (4.362)

V (x

) > 0, x

∈ R

, x

6= 0, (4.363)

′

, x

) ≤ 0, (x

, x

) ∈ R

× R

, (4.364)

then the system (4.360) and (4.361) is Lyapunov stable with respect to x

If, in addition, there exists a class K

∞

function γ : [0, ∞) → [0, ∞) such

that

′

, x

) ≤ −γ(kx

k), (x

, x

) ∈ R

× R

, (4.365)

show that (4.360) and (4.361) is asymptotically stable with respect to x

Finally, if V (·) is radially unbounded and satisﬁes (4.362), (4.363), and

(4.365) show that (4.360) and (4.361) is globally asymptotically stable with

respect to x

Problem 4.2. Consider the nonlinear dynamical system representing

a rigid spacecraft given by

˙x

(t) = I

(t)x

(t) + α

(t), x

(0) = x

, t ≥ 0, (4.366)

˙x

(t) = I

(t)x

(t) + α

(t), x

(0) = x

, (4.367)

˙x

(t) = I

(t)x

(t), x

(0) = x

, (4.368)

where I

= (I

− I

)/I

, I

= (I

− I

)/I

, I

= (I

− I

)/I

, I

and I

are the pr incipal moments of inertia of the spacecraft such that

NonlinearBook10pt November 20, 2007

314 CHAPTER 4

> I

> 0, and α

< 0 and α

< 0 reﬂect dissipation in the x

and

co ordinates of the spacecraft. Show that (4.366)–(4.368) is exponentially

stable with respect to (x

, x

) uniformly in x

. (Hint: Use the Lyapunov

function candidate V (x

, x

) =

(−I

+ I

).)

Problem 4.3. Analyze the partial stability conditions given in Prob-

lem 4.1 and Theorem 4.1 and comment on their interrelationship. Are the

conditions equivalent? Is one set of conditions implied by the other? Are

the partial stability conditions in Problem 4.1 and Theorem 4.1 a special

case of the stability conditions with respect to compact positively invariant

sets d iscussed in Section 4.9? Explain your answers.

Problem 4.4. Consider the nonlinear dynamical system

˙x

(t) = α

(t) − βx

(t)x

(t) cos x

(t), x

(0) = x

, t ≥ 0, (4.369)

˙x

(t) = α

(t) + βx

(t) cos x

(t), x

(0) = x

, (4.370)

˙x

(t) = 2θ

−θ

− β



(t)

− 2x

(t)



sin x

(t), x

(0) = x

, (4.371)

representing a time-averaged, two-mode thermoacoustic combustion model

where α

< 0, α

< 0, and β, θ

, θ

∈ R. Show that (4.369)–(4.371) is

globally asymptotically stable with respect to [x

]

. (Hint: Use Problem

4.1 with the partial Lyapunov function candidate V (x

, x

) =

Problem 4.5. Show that the zero s olution x(t) ≡ 0 to (4.57) is

uniformly Lyapunov stable if and only if there exists a class K function α(·)

and a positive constant δ, independent of t

, such that kx(t)k ≤ α(kx(t

)k)

for all t ≥ t

and kx(t

)k < δ.

Problem 4.6. Show that the zero s olution x(t) ≡ 0 to (4.57) is

uniformly asymptotically stable if and only if there exists a class K L

function β(·, ·) and a positive constant δ, independent of t

, such that

kx(t)k ≤ β(kx(t

)k, t − t

) for all t ≥ t

and kx(t

)k < δ. Additionally,

show that if x(t

) ∈ R

, then the above equivalence holds for global uniform

asymptotic stability.

Problem 4.7. Consider the scalar linear equation

˙x(t) = −

t + 1

x(t), x(t

) = x

, t ≥ t

. (4.372)

Show that the zero solution x(t) ≡ 0 to (4.372) is uniformly Lyapunov s table

and globally asymptotically stable, but not uniform ly asymptotically stable.

Problem 4.8. Consider the linear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (4.373)