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

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ADVANCED STABILITY THEORY 215

case, W (x

(t), x

(t)) → 0 as t → ∞, and hence, it f ollows from (4.13) that

(t) → 0 as t → ∞, proving asymptotic stability of (4.6) and (4.7) with

respect to x

iv) Lyapunov stability uniformly in x

follows from ii). Next, let

ε > 0 and δ = δ(ε) > 0 be such that for every x

∈ B

(0), x

(t) ∈ B

(0),

t ≥ 0, (the existence of such a (δ, ε) pair follows f rom uniform Lyapunov

stability), and assume that (4.15) holds. Since (4.15) implies (4.10) it f ollows

that for every x

∈ B

(0), V (x

(t), x

(t)) is a nonincreasing function of

time and, since V (·, ·) is bounded from below, it follows from the monotone

convergence theorem (Theorem 2.10) that there exists L ≥ 0 such that

lim

t→∞

V (x

(t), x

(t)) = L. Now, suppose that for some x

∈ B

(0), ad

absurdum, L > 0 so that D

△

= {x

∈ B

(0) : V (x

, x

) ≤ L for all x

∈ R

}

is nonempty and x

(t) 6∈ D

, t ≥ 0. Thus, as in the proof of i), there exists

δ > 0 such that B

(0) ⊂ D

. Hence, it follows from (4.15) that for the given

∈ B

(0) \D

and t ≥ 0,

V (x

(t), x

(t)) = V (x

, x

) +

V (x

(s), x

(s))ds

≤ V (x

, x

) −

γ(kx

(s)k)ds

≤ V (x

, x

) −γ(

δ)t.

Letting t >

V (x

)−L

γ(

δ)

, it follows that V (x

(t), x

(t)) < L, which is a

contradiction. Hence, L = 0, and, since x

∈ B

(0) was chosen arbitrarily,

it follows that V (x

(t), x

(t)) → 0 as t → ∞ for all x

∈ B

(0). Now,

since V (x

(t), x

(t)) ≥ α(kx

(t)k) ≥ 0, it follows that α(kx

(t)k) → 0

or, equivalently, x

(t) → 0 t → ∞, establishing asymp totic stability with

respect to x

uniformly in x

v) Let δ > 0 be such that kx

k < δ. Since α(·) is a class K

∞

function,

it follows that there exists ε > 0 s uch that V (x

, x

) < α(ε). Now, (4.14)

implies that V (x

(t), x

(t)) is a nonincreasing function of time, and hence,

it follows from (4.9) that α(kx

(t)k) ≤ V (x

(t), x

(t)) ≤ V (x

, x

) < α(ε),

t ≥ 0. Hence, x

(t) ∈ B

(0), t ≥ 0. Now, the proof follows as in the proof of

iii).

vi) Let δ > 0 be such that kx

k < δ. Since α(·) is a class K

∞

function,

it follows that there exists ε > 0 such that β(δ) ≤ α(ε). Now, (4.15) implies

that V (x

(t), x

(t)) is a nonincreasing function of time, and hence, it follows

from (4.11) that α(kx

(t)k) ≤ V (x

(t), x

(t)) ≤ V (x

, x

) < β(δ) ≤ α(ε),

t ≥ 0. Hence, x

(t) ∈ B

(0), t ≥ 0. Now, the proof follows as in the proof of

iv).

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216 CHAPTER 4

vii) Let ε > 0 and B

(0) be given as in the proof of i) and let η

△

= αε

and δ =





1/p

. Now, (4.17) implies that

V (x

, x

) ≤ 0, and hence, as in

the proof of ii), it follows that for all (x

, x

) ∈ B

(0)×R

, x

(t) ∈ B

(0),

t ≥ 0. Furtherm ore, it follows from (4.16) and (4.17) that for all t ≥ 0 and

, x

) ∈ B

(0) × R

V (x

(t), x

(t)) ≤ −γkx

(t)k

≤ −

V (x

(t), x

(t)),

which implies that

V (x

(t), x

(t)) ≤ V (x

, x

−

It now follows from (4.16) that

αkx

(t)k

≤ V (x

(t), x

(t)) ≤ V (x

, x

−

≤ βkx

−

, t ≥ 0,

and h en ce,

(t)k ≤





1/p

−

βp

, t ≥ 0,

establishing exponential stability with respect to x

uniformly in x

viii) The proof follows as in vi) and vii).

By setting n

= n and n

= 0, Theorem 4.1 specializes to the case

of nonlinear autonomous systems of the f orm ˙x

(t) = f

(t)). In this

case, Lyapunov (respectively, asymptotic) stability with respect to x

and

Lyapunov (respectively, asymptotic) stability with respect to x

uniformly

in x

are equivalent to the classical Lyapunov (respectively, asymptotic)

stability of nonlinear autonomous systems presented in Section 3.2. In

particular, note that it follows from Problems 3.75 and 3.76 that there exists

a continuously diﬀerentiable fu nction V : D → R such that (4.9), (4.11), and

(4.15) hold if and only if V (0) = 0, V (x

) > 0, x

6= 0, V

′

) < 0,

6= 0. In addition, if D = R

and there exist class K

∞

functions α(·), β(·)

and a continuously diﬀerentiable function V (·) such that (4.9), (4.11), and

(4.15) hold if and only if V (0) = 0, V (x

) > 0, x

6= 0, V

′

) < 0,

6= 0, and V (x

) → ∞ as kx

k → ∞. Hence, in this case, Theorem

4.1 collapses to the classical Lyapunov stability theorem for autonomous

systems given in Section 3.2.

It is important to note th at there is a key diﬀerence between the

partial stability deﬁnitions given in Deﬁnition 4.1 and the deﬁnitions of

partial stability given in [374, 448]. In particular, the partial stability

deﬁnitions given in [374, 448] require that both the initial conditions x

and x

lie in a neighborhood of the origin, whereas in Deﬁnition 4.1 x

can be arbitrary. As will be seen in the next section, this diﬀerence allows

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ADVANCED STABILITY THEORY 217

us to unify autonomous partial stability theory with time-varying stability

theory. Lyapunov (respectively, asymptotic) stability with respect to x

given in Deﬁnition 4.1 is referred to in [448] as x

-stability (respectively, x

asymptotic stability) for large x

while Lyapunov (respectively, asymptotic)

stability with respect to x

uniformly in x

given in Deﬁnition 4.1 is referred

to in [448] as x

-stability (respectively, x

-asymptotic stability) with respect

to the whole of x

. Note that if a nonlinear dynamical system is Lyapunov

(respectively, asymptotically) stable with respect to x

in the sense of

Deﬁnition 4.1, then the system is x

-stable (respectively, x

-asymptotically

stable) in the sense of the deﬁnition given in [374, 448]. Furthermore, if

there exist a continuously diﬀerentiable function V : D × R

→ R and a

class K function α(·) (respectively, class K functions β(·) and γ(·)) such

that V (0, 0) = 0 and (4.9) and (4.10) (respectively, (4.9), (4.11), and

(4.15)) hold, then the nonlinear dynamical system (4.6) and (4.7) is x

stable (respectively, uniformly asymptotically x

-stable with respect to x

)

in the sense of the deﬁnition given in [448]. It is important to note that

the condition V (0, x

) = 0, x

∈ R

, allows us to prove partial stability

in the sense of Deﬁnition 4.1. Finally, an additional diﬀerence between our

formulation of the partial stability problem and the partial stability problem

considered in [374, 448] is in the treatment of the equilibrium of (4.6) and

(4.7). Speciﬁcally, in our formulation, we require the partial equilibrium

condition f

(0, x

) = 0 for every x

∈ R

, whereas in [374,448], th e authors

require the equilibrium cond ition f

(0, 0) = 0 and f

(0, 0) = 0.

Example 4.1. In this example, we use Theorem 4.1 to show that

the rotational/translational proof-mass model (4.4) and (4.5) is partially

Lyapunov stable with respect to q, ˙q, and

θ. To show this, let x

= q,

= ˙q, x

= θ, x

θ and consider the Lyapunov function candidate

V (x

, x

) =

[kx

+ (M + m)x

+ (I + me

+ 2mex

cos x

(4.20)

Note that V (x

, x

) =

˜x

P (x

)˜x, where ˜x = [x

]

and

P (x

) =



M + m me cos x

me cos x

I + me



. (4.21)

Since

2λ

min

(P (x

)) = M + m + I + me

−

(M + m − I − me

)

+ 4m

cos

, (4.22)

2λ

max

(P (x

)) = M + m + I + me

(M + m − I − me

)

+ 4m

cos

, (4.23)

it follows that α

min

≤ 2P (x

) ≤ α

max

, x

∈ R, where

min

△

= M + m + I + me

−

(M + m − I − me

)

+ 4m

, (4.24)

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218 CHAPTER 4

φ sin θ

Figure 4.4 Spherical pendulum.

max

△

= M + m + I + me

(M + m − I − me

)

+ 4m

. (4.25)

Hence, it f ollows that

min

+ x

) ≤ V (x

, x

) ≤

max

+ x

), which implies that V (·) satisﬁes (4.9) and (4.11). Now,

since

V (x

, x

) = 0, it follows from ii) of Theorem 4.1 that (4.4) and

(4.5) is Lyapunov stable with respect to x

, x

, and x

uniformly in x

Furth ermore, since the system involves a nonlinear coupling of an u ndamped

oscillator with a rotational rigid body m ode it follows that x

(t) does not

converge as t → ∞. △

Example 4.2. In this example, we apply Theorem 4.1 to a Lagrange-

Dirichlet problem involving a conservative Euler-Lagrange system with

a nonnegative-deﬁnite kinetic energy function T and a positive-deﬁnite

potential function U. Speciﬁcally, we consider the motion of the spherical

pendulum shown in Figure 4.4, where θ denotes the angular position of the

pendulum with respect to vertical z-axis, φ denotes the angular position of

the pendulum in the x-y plane, m d en otes the mass of the pendulum, L

denotes the length of the pendulum, k d en otes the torsional spring stiﬀness,

and g denotes the gravitational acceleration. Deﬁning q

△

= [θ φ]

to be the

generalized system positions and ˙q

△

= [

φ]

to be the generalized system

velocities, it follows that the governing equations of motion are given by the

Euler-Lagrange equation



∂L

∂ ˙q

(q(t), ˙q(t))



−



∂L

∂q

(q(t), ˙q(t))



= 0, q(0) = q

, ˙q(0) = ˙q

, t ≥ 0,

(4.26)

where L(q, ˙q) = T (q, ˙q) − U (q) denotes the system Lagrangian, T (q, ˙q)

△

m[(L

θ)

+ (

φL sin θ)

] d en otes the system kinetic energy, and U(q)

△

mgL(1 − cos θ) +

kφ

denotes the system potential energy. Equivalently,

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ADVANCED STABILITY THEORY 219

(4.26) can be rewritten as

θ(t) − sin θ(t) cos θ(t)

(t) + (g/L) sin θ(t) = 0,

θ(0) = θ

θ(0) =

, t ≥ 0, (4.27)

sin

θ(t)

φ(t) + 2 sin θ(t) cos θ(t)

φ(t)

θ(t) + (k/mL

)φ(t) = 0,

φ(0) = φ

φ(0) =

. (4.28)

Next, consider the Lagrange-Dirichlet energy function V (q, ˙q) = T (q,

˙q) + U(q) and note that since the system kinetic energy function T (q, ˙q)

is not positive deﬁnite in ˙q at θ such that sin θ = 0, the function V (q, ˙q)

cannot be used as a Lyapunov function candidate to analyze the stability

of the system using standard Lyapunov th eory. However, the Lagrange-

Dirichlet energy function V (q, ˙q) can be used as a valid Lyapun ov function

within p artial stability theory to gu arantee partial Lyapunov stability with

respect to [θ φ

θ]

. Speciﬁcally, let x

= [θ φ

θ]

, let x

φ, and let

α(s) = max{mgL(1 − cos(s),

} and kx

k = max{|θ|, |φ|, |

θ|}.

Now, note that α(·) is a class K function and V (x

, x

) = V (q, ˙q) ≥ α(kx

k).

Furth ermore, note that V (0, x

) = 0, x

∈ R, and

V (x

, x

) = 0. Now, it

follows from i) of Th eorem 4.1 that the Euler-Lagrange system given by

(4.27) and (4.28) is partially Lyapunov stable with respect to x

. Finally, it

can be easily shown via simulations that the E uler-Lagrange system given

by (4.27) and (4.28) is not Lyapunov stable in the standard s ense. △

Another important app lication of partial stability th eory is the

uniﬁcation it pr ovides between time-invariant stability theory and stability

theory for time-varying systems. To see this, consider the time-varying

nonlinear dynamical system given by

˙x(t) = f (t, x(t)), x(t

) = x

, t ≥ t

, (4.29)

where x(t) ∈ R

, t ≥ t

, and f : [t

, t

)×R

→ R

. Now, deﬁne x

(τ)

△

= x(t)

and x

(τ)

△

= t, where τ

△

= t − t

, and note that the solution x(t), t ≥ t

to the non linear time-varying dynamical system (4.29) can be equivalently

characterized by the solution x

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

dynamical system

˙x

(τ) = f(x

(τ), x

(τ)), x

(0) = x

, τ ≥ 0, (4.30)

˙x

(τ) = 1, x

(0) = t

, (4.31)

where ˙x

(·) and ˙x

(·) denote diﬀerentiation with respect to τ. However,

in this case, stability results for time-invariant systems do not apply to

the augmented system (4.30) and (4.31) since one of the states, namely, the

state x

representing time, is unb ou nded. However, w riting the time-varying

nonlinear system (4.29) as (4.30) and (4.31), it is clear that partial stability

theory provides a natural formulation for addressing stability theory for

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220 CHAPTER 4

autonomous and nonautonomous systems within a uniﬁed framework. The

following example demonstrates the utility of Theorem 4.1 to the stability

of time-varying systems.

Example 4.3. Consider the spring-mass-damper s ystem with time-

varying damping coeﬃcient given by

¨q(t) + c(t) ˙q(t) + kq(t) = 0, q(0) = q

, ˙q(0) = ˙q

, t ≥ 0. (4.32)

This is an interesting system to analyze since physical intuition would lead

one to surmise that if c(t) ≥ α > 0, t ≥ 0, then the zero solution (q(t), ˙q(t)) ≡

(0, 0) to (4.32) is asymptotically stable since we have constant dissipation of

energy. However, this is not the case. A simple counterexample (see [424]) is

c(t) = 2+e

, k = 1, with q(0) = 2 and ˙q(0) = −1, which gives q(t) = 1+e

−t

t ≥ 0, and hen ce, q(t) → 1 as t → ∞. This is due to the fact that dampin g

increases so fast that the system halts at q = 1.

To analyze (4.32) using Theorem 4.1, we consider c(t) = 3 + sin t and

k > 1. Now, (4.32) can be equivalently written as

˙z

(t) = z

(t), z

(0) = q

, t ≥ 0, (4.33)

˙z

(t) = −kz

(t) −c(t)z

(t), z

(0) = ˙q

, (4.34)

where z

△

= q and z

△

= ˙q. Next, let n

= 2, n

= 1, x

= [z

, z

]

= t, f

, x

) = [x

v, −x

h(x

)]

, and f

, x

) = 1, where h(x

) =

[k, c(x

)]

and v = [0, 1]

. Now, the solution (z

(t), z

(t)), t ≥ 0, to the

nonlinear time-varying dynamical s ystem (4.33) and (4.34) is equivalently

characterized by the solution x

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

dynamical system

˙x

(t) = f

(t), x

(t)), x

(0) = [q

, ˙q

]

, t ≥ 0, (4.35)

˙x

(t) = 1, x

(0) = 0. (4.36)

To examine the stability of this system, consider the Lyapunov

function candidate V (x

, x

) = x

P (x

, where

P (x

) =



k + 3 + sin(x

) 1

1 1



Note that since

≤ V (x

, x

) ≤ x

, (x

, x

) ∈ R

× R, (4.37)

where



k + 2 1

1 1



, P



k + 4 1

1 1



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ADVANCED STABILITY THEORY 221

it f ollows that V (x

, x

) satisﬁes (4.16) with D = R

and p = 2. Next, since

V (x

, x

) = −2x

[R + R

)]x

≤ −2x

≤ −min{k − 1, 1}kx

, (4.38)

where R = diag[k − 1, 1] > 0 and R

) = diag[1 −

cos(x

), 1 + sin(x

)],

it follows from v) and vi) of Theorem 4.1 that the dynamical system (4.35)

and (4.36) is globally exponentially stable with respect to x

uniformly in

. △

In the case of time-invariant systems the Barbashin-Krasovskii-LaSalle

invariance theorem (Theorem 3.3) shows that bounded system tr aj ectories

of a nonlinear dynamical system approach the largest invariant set M

characterized by the set of all points in a compact set D of the state

space where the Lyapunov derivative identically vanishes. In the case of

partially stable systems, however, it is not generally clear how to deﬁne

the set M since

V (x

, x

) is a function of both x

and x

. However, if

V (x

, x

) ≤ −W (x

) ≤ 0, where W : D → R is continuous and nonnegative

deﬁnite, then a set R ⊃ M can be deﬁned as the set of points where W (x

)

identically vanishes, that is, R = {x

∈ D : W (x

) = 0}. In this case, as

shown in the next theorem, the partial system trajectories x

(t) approach

R as t tends to inﬁnity. For this r esult, the following lemma, known as

Barbalat’s lemma, is necessary.

Lemma 4.1 (Barbalat’s Lemma ) . Let σ : [0, ∞) → R be a uniformly

continuous function and s uppose that lim

t→∞

σ(s)ds exists and is ﬁnite.

Then, lim

t→∞

σ(t) = 0.

Proof. Suppose, ad absurdum, that lim sup

t→∞

|σ(t)| > 0 and let α

∈

R be such th at 0 < α

< lim sup

t→∞

|σ(t)|. In this case, for every T

> 0,

there exists t

′

≥ T

such that |σ(t

′

)| ≥ α

. Next, since σ : [0, ∞) → R is

uniformly continuous, it follows that there exists α

= α

(α

) > 0 such that

|σ(t + τ ) − σ(t)| < α

/2 for all t ≥ 0 and τ ∈ [0, α

]. Hence,

|σ(t)| = |σ(t) − σ(t

′

) + σ(t

′

≥ |σ(t

′

)| −|σ(t) − σ(t

′

> α

/2, t ∈ [t

′

, t

′

+ α

]. (4.39)

Now, since σ : [0, ∞) → R is continuous and |σ(t)| > α

/2 > 0, t ∈ [t

′

, t

′

], it follows that the sign of σ(t) is constant over t ∈ [t

′

, t

′

+ α

]. Hence,



′

+α

′

σ(t)dt



′

+α

′

|σ(t)|dt >

. (4.40)

Now, repeating the above argument it can be shown that there exists a

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222 CHAPTER 4

sequence {t

′

}

∞

i=1

such that t

′

+ α

≤ t

′

i+1

, i = 1, 2, . . ., and

′

+α

′

|σ(t)dt >

, i = 1, 2, . . .. Hence,

∞

|σ(t)|dt ≥

∞

i=1

′

+α

′

|σ(t)|dt > α

∞

i=1

= ∞, (4.41)

which is a contradiction.

Theorem 4.2. Consider the nonlinear dynamical system (4.6) and

(4.7), and assume D × R

is a positively invariant set with respect to

(4.6) and (4.7) where f

(·, x

) is Lipschitz continuous in x

, uniformly

in x

. Furthermore, assume there exist functions V : D × R

→ R,

W, W

, W

: D → R such that V (·, ·) is continuously diﬀerentiable, W

(·)

and W

(·) are continuous and positive deﬁnite, W (·) is continuous and

nonnegative deﬁnite, and, for all (x

, x

) ∈ D × R

) ≤ V (x

, x

) ≤ W

), (4.42)

V (x

, x

) ≤ −W (x

). (4.43)

Then there exists D

⊆ D such that for all (x

, x

) ∈ D

× R

, x

(t) →

△

= {x

∈ D : W (x

) = 0} as t → ∞. If, in addition, D = R

and W

(·) is

radially unbounded, then for all (x

, x

) ∈ R

× R

, x

(t) → R

△

= {x

∈

: W (x

) = 0} as t → ∞.

Proof. Assume (4.42) and (4.43) hold. Then it follows from Theorem

4.1 that the nonlinear dynamical system given by (4.6) and (4.7) is Lyapunov

stable with respect to x

uniformly in x

. Let ε > 0 be such that B

(0) ⊂ D

and let δ = δ(ε) > 0 be such that if x

∈ B

(0), then x

(t) ∈ B

(0),

t ≥ 0. Now, since V (x

(t), x

(t)) is nonincreasing and bounded from below

by zero, it follows from the monotone convergence theorem (Theorem 2.10)

that lim

t→∞

V (x

(t), x

(t)) exists and is ﬁnite. Hence, since for every t ≥ 0,

W (x

(τ))dτ ≤ −

V (x

(τ), x

(τ))dτ = V (x

, x

) − V (x

(t), x

(t)),

it follows that lim

t→∞

W (x

(τ))dτ exists and is ﬁnite.

Next, since by (4.42) kx

(t)k ≤ ε, t ≥ 0, and, for every x

∈ R

(·, x

) is Lipschitz continuous on D uniformly in x

, it follows that

) −x

)k =



(τ), x

(τ))dτ



≤ L

(τ)kdτ

≤ L

ε(t

− t

), t

≥ t

, (4.44)

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ADVANCED STABILITY THEORY 223

where L

is th e Lipschitz constant of f

(·, ·) on {x

∈ D : kx

k ≤ ε}. Now,

for every ˆε > 0, letting δ = δ(ˆε) =

ˆε

yields

) −x

)k < ˆε, t

− t

≤ δ,

which shows that x

(·) is uniformly continuous. Next, since x

(·) is

uniformly continuous and W (·) is continuous on a compact set B

(0), it

follows that W (x

(t)) is uniform ly continuous at every t ≥ 0. Now, it

follows f rom Barbalat’s lemma (Lemma 4.1) that W (x

(t)) → 0 as t → ∞.

Finally, if, in addition, D = R

and W

(·) is radially unbounded, then, as

in the proof of iv) of Theorem 4.1, f or every x

∈ R

there exists ε, δ > 0

such that x

∈ B

(0) and x

(t) ∈ B

(0), t ≥ 0. Now, the proof follows by

repeating th e above arguments.

Theorem 4.2 shows that the partial system traj ectories x

(t) app roach

R as t tends to inﬁnity. However, since the positive limit s et of the partial

trajectory x

(t) is a subset of R, Theorem 4.2 is a weaker result than the

standard invariance principle wherein one would conclude that the partial

trajectory x

(t) approaches the largest invariant set M contained in R. This

is not tru e in general for partially stable systems since the positive limit set of

a partial traj ectory x

(t), t ≥ 0, is not an invariant set. However, in the case

where f

(·, x

) is periodic, almost periodic, or asymptotically independent of

, then an invariance principle for partially stable systems can be derived.

This result is left as an exercise for the reader.

Next, we state two converse theorems for partial stability. The proofs

of these theorems are virtually identical to the proofs of the converse

theorems given in Section 3.5 and are left as exercises for the reader.

Theorem 4.3. Assume that the n onlinear dyn amical system (4.6) and

(4.7) is asymptotically stable with respect to x

uniformly in x

and f

D×R

→ R

and f

: D×R

→ R

are continuously diﬀerentiable. Let

δ > 0 be such th at B

(0) ⊂ D and x

∈ B

(0) implies that lim

t→∞

(t) = 0

uniformly in x

and x

for all x

∈ R

, and assume

∂f

∂x

is bounded on

(0) u niformly in x

. Then there exist a continuously diﬀerentiable function

V : B

(0) × R

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

α(kx

k) ≤ V (x

, x

) ≤ β(kx

k), (x

, x

) ∈ B

(0) × R

, (4.45)

V (x

, x

) ≤ −γ(kx

k), (x

, x

) ∈ B

(0) × R

. (4.46)

Theorem 4.4. Assume that the n onlinear dyn amical system (4.6) and

(4.7) is exponentially stable with respect to x

uniformly in x

and f

: D×

→ R

and f

: D×R

→ R

are continuously diﬀerentiable. Let δ > 0

be s uch that B

(0) ⊂ D and x

∈ B

(0) implies that kx

(t)k ≤ αkx

−βt

for some α, β > 0 and for all t ≥ 0 and x

∈ R

, and assume

∂f

∂x

is bounded

NonlinearBook10pt November 20, 2007

224 CHAPTER 4

on B

(0) uniformly in x

. Then, for every p > 1, there exist a continuously

diﬀerentiable function V : B

(0)×R

→ R and positive constants α, β, and

γ su ch that

αkx

≤ V (x

, x

) ≤ βkx

, (x

, x

) ∈ B

(0) ×R

, (4.47)

V (x

, x

) ≤ −γkx

, (x

, x

) ∈ B

(0) × R

. (4.48)

Finally, we close this section by addressing a partial stability notion

wherein both initial conditions x

and x

lie in a neighborhood of the

origin. For this result we modify Deﬁnition 4.1 to reﬂect the fact that the

entire initial state x

= [x

, x

]

lies in the neighborhood of the origin so

that kx

k < δ is replaced by kx

k < δ in the deﬁnition. Furthermore, for

this result we assume f

: D × R

→ R

and f

: D × R

→ R

are such

that f

(0, 0) = 0 and f

(0, 0) = 0.

Theorem 4.5. Consider the nonlinear dynamical system (4.6) and

(4.7). Then the following statements hold:

i) I f there exist a continuous ly diﬀerentiable function V : D × R

→ R

and a class K function α(·) such that V (0, 0) = 0,

α(kx

k) ≤ V (x

, x

), (x

, x

) ∈ D × R

, (4.49)

V (x

, x

) ≤ 0, (x

, x

) ∈ D × R

, (4.50)

then the nonlinear dynamical system given by (4.6) and (4.7) is

Lyapunov s table with respect to x

ii) If there exist a continuously diﬀerentiable fun ction V : D × R

→ R

and class K functions α(·), β(·) satisfying (4.49), (4.50), and

V (x

, x

) ≤ β(kxk), (x

, x

) ∈ D × R

, (4.51)

where x

△

= [x

, x

]

, then the n onlinear dynamical system given by

(4.6) and (4.7) is Lyapunov stable with respect to x

uniformly in x

iii) If there exist continuously diﬀerentiable functions V : D × R

→ R

and W : D ×R

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

that

W (x

(·), x

(·)) is bounded from below or above, (4.49) holds, and

β(kxk) ≤ W (x

, x

), (x

, x

) ∈ D × R

, (4.52)

V (x

, x

) ≤ −γ(W (x

, x

)), (x

, x

) ∈ D × R

, (4.53)

then the nonlinear dynamical system given by (4.6) and (4.7) is

asymptotically stable with respect to x

iv) If there exist a continuously diﬀerentiable function V : D × R

→ R

and class K functions α(·), β(·), γ(·) s atisfying (4.49), (4.51), and

V (x

, x

) ≤ −γ(kxk), (x

, x

) ∈ D × R

, (4.54)