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

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DISCRETE-TIME THEORY 785

then the nonlinear time-varying dynamical system given by (13.51) is

uniformly Lyapunov stable.

iii) If there exist a continuous function V : Z

× D → R and class K

functions α(·), γ(·) satisfying (13.53) and

∆V (k, x) ≤ −γ(kxk), (k, x) ∈ Z

× D, (13.56)

then the nonlinear time-varying dynamical system given by (13.51) is

asymptotically stable.

iv) If there exist a continuous function V : Z

× D → R and class K

functions α(·), β(·), γ(·) satisfying (13.53), (13.55), and

∆V (k, x) ≤ −γ(kxk), (k, x) ∈ Z

× D, (13.57)

then the nonlinear time-varying dynamical system given by (13.51) is

uniformly asymptotically stable.

v) If D = R

and there exist a continuous function V : Z

× R

→ R,

a class K function γ(·), class K

∞

functions α(·) satisfying (13.53) and

(13.56), then the nonlinear time-varying dynamical system given by

(13.51) is globally asymptotically s table.

vi) If D = R

and there exist a continuous function V : Z

× R

→ R, a

class K function γ(·), class K

∞

functions α(·), β(·) satisfying (13.53),

(13.55), and (13.57), then the nonlinear time-varying dynamical

system given by (13.51) is globally uniformly asymptotically stable.

vii) If there exist a continuous function V : Z

× D → R and positive

constants α, β, γ, p such that p ≥ 1 and

αkxk

≤ V (k, x) ≤ βkxk

, (k, x) ∈ Z

× D, (13.58)

∆V (k, x) ≤ −γkxk

, (k, x) ∈ Z

× D, (13.59)

then the nonlinear time-varying dynamical system given by (13.51) is

(uniformly) geometrically stable.

viii) If D = R

and there exist a continuous function V : Z

× R

→

R an d positive constants α, β, γ, p such that p ≥ 1 and (13.58) and

(13.59) hold, then the nonlinear time-varying dynamical system given

by (13.51) is globally (uniformly) geometrically stable.

Proof. Let n

= n, n

= 1, x

(k − k

) = x(k), x

(k − k

) = k,

, x

) = f(k, x), and f

, x

) = x

+1. Now, note that with κ = k−k

the solution x(k), k ≥ k

, to the nonlinear time-varying d ynamical system

(13.51) is equivalently characterized by the solution x

(κ), κ ≥ 0, to the

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786 CHAPTER 13

nonlinear autonomous dynamical system

(κ + 1) = f

(κ), x

(κ)), x

(0) = x

, κ ≥ 0,

(κ + 1) = x

(κ) + 1, x

(0) = k

Now, the result is a dir ect consequence of Theorem 13.9.

13.7 Lagrange Stability, Boundedness, and

Ultimate Boundedness

In this section, we introduce the notions of Lagrange stability, bounded-

ness, and ultimate boundedness and present Lyapunov-like theorems for

boundedness and u ltimate boundedness of nonlinear discrete-time dynamical

systems.

Deﬁnition 13.8. i) The nonlinear dy namical system (13.37) and

(13.38) is Lagrange stable with respect to x

if, for every x

∈ D and

∈ R

, there exists ε = ε(x

, x

) > 0 such that kx

(k)k < ε, k ∈ Z

ii) The n on linear dynamical system (13.37) and (13.38) is bounded

with respect to x

uniformly in x

if there exists γ > 0 such that, for every

δ ∈ (0, γ), there exists ε = ε(δ) > 0 such that kx

k < δ imp lies kx

(k)k < ε,

k ∈ Z

. The nonlinear dynamical system (13.37) and (13.38) is globally

bounded with respect to x

uniformly in x

if, for every δ ∈ (0, ∞), there

exists ε = ε(δ) > 0 su ch that kx

k < δ implies kx

(k)k < ε, k ∈ Z

iii) The nonlinear dyn amical system (13.37) and (13.38) is ultimately

bounded with respect to x

uniformly in x

with bound ε if there exists γ > 0

such that, for every δ ∈ (0, γ), there exists K = K(δ, ε) > 0 such that

k < δ implies kx

(k)k < ε, k ≥ K. The nonlinear dynamical system

(13.37) and (13.38) is globally ultimately bounded with respect to x

uniformly

in x

with bound ε if, for every δ ∈ (0, ∞), there exists K = K(δ, ε) > 0

such that kx

k < δ implies kx

(k)k < ε, k ≥ K.

The following results present Lyapunov-like theorems for bounded-

ness and ultimate boundedness. For these results deﬁne ∆V (x

, x

)

△

V (f (x

, x

)) − V (x

, x

), where f(x

, x

)

△

= [f

, x

) f

, x

)]

and

V : D ×R

→ R is a continuous function.

Theorem 13.12. Consider the nonlinear dynamical s ystem (13.37) and

(13.38). Ass ume there exist a continuous function V : D × R

→ R and

class K functions α(·), β(·) such that

α(kx

k) ≤ V (x

, x

) ≤ β(kx

k), x

∈ D, x

∈ R

, (13.60)

∆V (x

, x

) ≤ 0, x

∈ D, kx

k ≥ µ, x

∈ R

, (13.61)

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DISCRETE-TIME THEORY 787

where µ > 0 is such that B

−1

(β(µ))

(0) ⊂ D. Furthermore, assume that

sup

)∈B

(0)×R

V (f (x

, x

)) exists. Then the nonlinear dynamical

system (13.37) and (13.38) is bound ed with respect to x

uniformly in x

Furth ermore, for every δ ∈ (0, γ), x

∈ B

(0) implies that kx

(k)k ≤ ε,

k ∈ Z

, where

ε = ε(δ)

△

= α

−1

(max{η, β(δ)}), (13.62)

η ≥ max{β(µ), sup

)∈B

(0)×R

V (f (x

, x

))}, and γ

△

= sup{r > 0 :

−1

(β(r))

(0) ⊂ D}. If, in addition, D = R

and α(·) is a class K

∞

function,

then the nonlinear dynamical system (13.37) and (13.38) is globally bounded

with respect to x

uniformly in x

and for every x

∈ R

, kx

(k)k ≤ ε,

k ∈ Z

, where ε is given by (13.62) with δ = kx

Proof. First, let δ ∈ (0, µ] and assume kx

k ≤ δ. If kx

(k)k ≤ µ,

k ∈ Z

, then it follows from (13.60) that kx

(k)k ≤ µ ≤ α

−1

(β(µ)) ≤

−1

(η), k ∈ Z

. Alternatively, if there exists K > 0 such that kx

(K)k >

µ, then, since kx

(0)k ≤ µ, it follows that there exists κ ≤ K such that

(κ − 1)k ≤ µ and kx

(k)k > µ, k ∈ {κ, . . . , K}. Hence, it follows from

(13.60) and (13.61) that

α(kx

(K)k) ≤ V (x

(K), x

(K))

≤ V (x

(κ), x

(κ))

= V (f(x

(κ − 1), x

(κ − 1)))

≤ η,

which implies that kx

(K)k ≤ α

−1

(η). Next, let δ ∈ (µ, γ) and assume

∈ B

(0) and kx

k > µ. Now, for every

k > 0 such that kx

(k)k ≥ µ,

k ∈ {0, . . . ,

k}, it follows from (13.60) and (13.61) that

α(kx

(k)k) ≤ V (x

(k), x

(k)) ≤ V (x

, x

) ≤ β(δ),

which implies that kx

(k)k ≤ α

−1

(β(δ)), k ∈ {0, . . . ,

k}. Next, if there

exists K > 0 such that kx

(K)k ≤ µ, then it follows as in the proof of

the ﬁrst case that kx

(k)k ≤ α

−1

(η), k ≥ K. Hence, if x

∈ B

(0), then

(k)k ≤ α

−1

(max{η, β(δ)}), k ∈ Z

. Finally, if D = R

and α(·) is a

class K

∞

function it follows that β(·) is a class K

∞

function, and hence,

γ = ∞. Hence, the nonlinear dynamical sys tem (13.37) and (13.38) is

globally bounded with respect to x

uniformly in x

Theorem 13.13. Consider the nonlinear dynamical s ystem (13.37) and

(13.38). Ass ume there exist a continuous function V : D × R

→ R and

class K functions α(·), β(·) such that (13.60) holds. Furthermore, assu me

that there exists a continuous function W : D → R such that W (x

) > 0,

k > µ, and

∆V (x

, x

) ≤ −W (x

), x

∈ D, kx

k > µ, x

∈ R

, (13.63)

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788 CHAPTER 13

where µ > 0 is such that B

−1

(β(µ))

(0) ⊂ D. Finally, assume that

sup

)∈B

(0)×R

V (f(x

, x

))

exists. Then the nonlinear dynamical s ystem (13.37) and (13.38) is ulti-

mately bounded with respect to x

uniformly in x

with bound ε

△

= α

−1

(η),

where η > max{β(µ), sup

)∈B

(0)×R

V (f (x

, x

))}. Fu rthermore,

lim sup

k→∞

(k)k ≤ α

−1

(η). If, in addition, D = R

and α(·) is a class

∞

function, then the non linear dynamical system (13.37) and (13.38) is

globally ultimately bounded with respect to x

uniformly in x

with bound

ε.

Proof. First, let δ ∈ (0, µ] an d assume kx

k ≤ δ. As in the proof of

Theorem 13.12, it follows that kx

(k)k ≤ α

−1

(η) = ε, k ∈ Z

. Next, let δ ∈

(µ, γ), where γ

△

= sup{r > 0 : B

−1

(β(r))

(0) ⊂ D} and assume x

∈ B

(0)

and kx

k > µ. In this case, it follows from Theorem 13.12 that kx

(k)k ≤

−1

(max{η, β(δ)}), k ∈ Z

. Suppose, ad absurdum, that kx

(k)k ≥ β

−1

(η),

k ∈ Z

, or, equivalently, x

(k) ∈ O

△

= B

−1

(max{η,β(δ)})

(0)\B

−1

(η)

(0), k ∈

. Since O is compact and W (·) is continuous and W (x

) > 0, kx

k ≥

−1

(η) > µ, it follows from Theorem 2.13 that θ

△

= min

∈O

W (x

) > 0

exists. Hence, it f ollows from (13.63) that

V (x

(k), x

(k)) ≤ V (x

, x

) −kθ, k ∈ Z

, (13.64)

which implies that

α(kx

(k)k) ≤ β(kx

k) − kθ ≤ β(δ) − kθ, k ∈ Z

. (13.65)

Now, letting k > β(δ)/θ it follows that α(kx

(k)k) < 0, which is a

contradiction. Hence, there exists K = K(δ, η) > 0 such that kx

(K)k <

−1

(η). Thus, it follows from Theorem 13.12 that kx

(k)k ≤ α

−1

(η),

k ≥ K, which proves that the nonlinear dynamical s ystem (13.37) and

(13.38) is ultimately bounded w ith respect to x

uniformly in x

with

bound ε = α

−1

(η). Furthermore, lim sup

k→∞

(k)k ≤ α

−1

(η). Finally, if

D = R

and α(·) is a class K

∞

function it follows that β(·) is a class K

∞

function, and hence, γ = ∞. Hence, th e nonlinear dynamical system (13.37)

and (13.38) is globally ultimately bounded with respect to x

uniformly in

with bound ε.

Corollary 13.3. Consider the nonlinear dynamical system (13.37) and

(13.38). Ass ume there exist a continuous function V : D × R

→ R and

class K functions α(·), β(·) such that (13.60) holds. Furthermore, assu me

that there exists a class K function γ : D → R such that

∆V (x

, x

) ≤ −γ(kx

k) + γ(µ), x

∈ D, x

∈ R

, (13.66)

where µ > 0 is such that B

−1

(β(µ))

(0) ⊂ D. Then the nonlinear

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DISCRETE-TIME THEORY 789

dynamical system (13.37) and (13.38) is ultimately bounded with respect

to x

uniformly in x

with bound ε

△

= α

−1

(η), where η = β(µ) + γ(µ).

Furth ermore, lim sup

k→∞

(k)k ≤ α

−1

(η). If, in addition, D = R

and

α(·) is a class K

∞

function, then the nonlinear dynamical system (13.37)

and (13.38) is globally ultimately bounded with respect to x

uniformly in

with bound ε.

Proof. The result is a direct consequence of T heorem 13.13 with

W (x

) = γ(kx

k) − γ(µ) and η = β(µ) + γ(µ). Speciﬁcally, it follows from

(13.66) that for every (x

, x

) ∈ D × R

, kx

k ≤ µ,

V (f(x

, x

)) ≤ V (x

, x

) −γ(kx

k) + γ(µ)

≤ β(kx

k) − γ(kx

k) + γ(µ)

≤ β(µ) + γ(µ).

Hence, it follows that

sup

)∈B

(0)×R

V (f (x

, x

)) ≤ β(µ) + γ(µ).

Finally, note that β(µ) ≤ β(µ) + γ(µ), and hence,

η ≥ max

(

β(µ), sup

)∈B

(0)×R

V (f (x

, x

))

)

satisfying all the conditions of Theorem 13.13.

Next, we specialize Theorems 13.12 and 13.13 to nonlinear time-

varying d ynamical s ystems. The following deﬁnition is needed for these

results.

Deﬁnition 13.9. i) The nonlinear time-varying dynamical system

(13.51) is Lagrange stable if, for every x

∈ R

and k

∈ Z, there exists

ε = ε(k

, x

) > 0 such that kx(k)k < ε, k ≥ k

ii) The nonlinear time-varyin g d ynamical system (13.51) is uniformly

bounded if there exists γ > 0 such that, for every δ ∈ (0, γ), th ere exists

ε = ε(δ) > 0 such that kx

k < δ implies kx(k)k < ε, k ≥ k

. The nonlinear

time-varying dynamical system (13.51) is globally uniformly bounded if, for

every δ ∈ (0, ∞), there exists ε = ε(δ) > 0 such that kx

k < δ imp lies

kx(k)k < ε, k ≥ k

iii) The nonlinear time-varying dynamical system (13.51) is uniformly

ultimately bounded with bound ε if there exists γ > 0 s uch that, for every δ ∈

(0, γ), there exists K = K(δ, ε) > 0 such that kx

k < δ implies kx(k)k < ε,

k ≥ k

+K. Th e nonlinear time-varying dynamical system (13.51) is globally

uniformly ultimately bounded with bound ε if, for every δ ∈ (0, ∞), there

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790 CHAPTER 13

exists K = K(δ, ε) > 0 s uch that kx

k < δ implies kx(k)k < ε, k ≥ k

+ K.

For the following result deﬁne

∆V (k, x)

△

= V (k + 1, f(k, x)) − V (k, x),

where V : R × D → R is a given continuous function.

Corollary 13.4. Consider the nonlinear time-varying dynamical sys-

tem (13.51). Assume there exist a continuous function V : R × D → R and

class K functions α(·), β(·) such that

α(kxk) ≤ V (k, x) ≤ β(kxk), x ∈ D, k ∈ Z, (13.67)

∆V (k, x) ≤ 0, x ∈ D, kxk ≥ µ, k ∈ Z, (13.68)

where µ > 0 is such that B

−1

(β(µ))

(0) ⊂ D. Furthermore, assume

that sup

(k,x)∈Z×B

(0)

V (k, f (k, x)) exists. Then the nonlinear time-varying

dynamical system (13.51) is uniformly bounded. If, in addition, D = R

and α(·) is a class K

∞

function, then the nonlinear time-varying dynamical

system (13.51) is globally u niformly boun ded.

Proof. The result is a direct consequence of Theorem 13.12. Specif-

ically, let n

= n, n

= 1, x

(k − k

) = x(k), x

(k − k

) = k, f

, x

) =

f(k, x), and f

, x

) = x

+1. Now, note that with κ = k−k

, the solution

x(k), k ≥ k

, to the nonlinear time-varying dynamical system (13.51) is

equivalently characterized by the solution x

(κ), κ ≥ 0, to the nonlinear

autonomous dynamical system

(κ + 1) = f

(κ), x

(κ)), x

(0) = x

, κ ≥ 0,

(κ + 1) = x

(κ) + 1, x

(0) = k

Now, the result is a dir ect consequence of Theorem 13.12.

Corollary 13.5. Consider the nonlinear time-varying dynamical sys-

tem (13.51). Assume there exist a continuous function V : R × D → R and

class K functions α(·), β(·) such that (13.67) holds. Furthermore, assu me

that there exists a continuous function W : D → R such that W (x) > 0,

kxk > µ, and

∆V (k, x) ≤ −W (x), x ∈ D, kxk > µ, k ∈ Z, (13.69)

where µ > 0 is such that B

−1

(β(µ))

(0) ⊂ D. Finally, assume that

sup

(k,x)∈Z×B

(0)

V (k, f (k, x)) exists. Then the nonlinear time-varying dy-

namical system (13.51) is uniformly u ltimately bounded with bound ε

△

−1

(η) where η ≥ max{β(µ), sup

(k,x)∈Z×B

µ(0)

V (k, f (k, x))}. Furthermore,

lim sup

k→∞

kx(k)k ≤ α

−1

(η). If, in add ition, D = R

and α(·) is a class

∞

function, then the nonlinear time-varying d ynamical system (13.51) is

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DISCRETE-TIME THEORY 791

globally uniformly ultimately bounded with bound ε.

Proof. The result is a direct consequence of Theorem 13.13 using

similar arguments as in the proof of Corollary 13.4 and, hence, is omitted.

Finally, we specialize Corollaries 13.4 and 13.5 to nonlinear time-

invariant dynamical systems. For these resu lts we need the f ollowing

specialization of Deﬁnition 13.9.

Deﬁnition 13.10. i) The nonlinear dynamical s ys tem (13.1) is La-

grange stable if, for every x

∈ R

, there exists ε = ε(x

) > 0 such that

kx(k)k < ε, k ∈ Z

ii) The nonlinear dynamical system (13.1) is bounded if there exists

γ > 0 such that, for every δ ∈ (0, γ), there exists ε = ε(δ) > 0 such that

k < δ implies kx(k)k < ε, k ∈ Z

. Th e nonlinear dynamical system

(13.1) is globally bounded if, for every δ ∈ (0, ∞), there exists ε = ε(δ) > 0

such that kx

k < δ implies kx(k)k < ε, k ∈ Z

iii) The nonlinear dynamical system (13.1) is ultimately bounded with

bound ε if there exists γ > 0 such that, for every δ ∈ (0, γ), there exists

K = K(δ, ε) > 0 such that kx

k < δ implies kx(k)k < ε, k ≥ K. The

nonlinear d ynamical system (13.1) is globally ultimately bounded with bound

ε if, for every δ ∈ (0, ∞), there exists K = K(δ, ε) > 0 such that kx

k < δ

implies kx(k)k < ε, k ≥ K.

Corollary 13.6. Consider the nonlinear dynamical system (13.1).

Assume there exist a continuous function V : D → R and class K functions

α(·) and β(·) such that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ D, (13.70)

V (f (x)) ≤ V (x), x ∈ D, kxk ≥ µ, (13.71)

where µ > 0 is such that B

−1

(η)

(0) ⊂ D, with η ≥ β(µ). Then the nonlinear

dynamical system (13.1) is bounded. If, in addition, D = R

and V (x) →

∞ as kxk → ∞, then the nonlinear d ynamical system (13.1) is globally

bounded.

Proof. The resu lt is a direct consequence of Corollary 13.4.

Corollary 13.7. Consider the nonlinear dynamical system (13.1).

Assume there exist a continuous function V : R × D → R and class K

functions α(·) and β(·) such that (13.70) holds an d

V (f (x)) < V (x), x ∈ D, kxk > µ, (13.72)

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792 CHAPTER 13

where µ > 0 is such that B

−1

(η)

(0) ⊂ D w ith η > β(µ). Then the nonlinear

dynamical system (13.1) is ultimately bounded with bound ε

△

= α

−1

(η).

Furth ermore, lim sup

k→∞

kx(k)k ≤ α

−1

(η). If, in addition, D = R

and

V (x) → ∞ as kxk → ∞, then the nonlinear dynamical system (13.1) is

globally ultimately bounded with bound ε.

Proof. The resu lt is a direct consequence of Corollary 13.5.

13.8 Stability Theory via Vector Lyapunov Functions

In this section, we consider the method of vector Lyapunov functions for

stability analysis of discrete-time nonlinear dynamical systems. To develop

the theory of vector Lyapunov fun ctions, we ﬁrst introduce some results

on vector diﬀerence inequ alities and the vector comparison principle for

discrete-time systems. Speciﬁcally, consider the discrete-time nonlinear

dynamical system given by

z(k + 1) = w(z(k)), z(k

) = z

, k ∈ I

, (13.73)

where z(k) ∈ Q ⊆ R

, k ∈ I

, is the system state vector, I

⊆ T ⊆ Z is

the maximal interval of existence of a solution z(k) to (13.73), Q is an open

set, 0 ∈ Q, and w : Q → R

is a continuous function on Q.

Theorem 13.14. Consider the discrete-time nonlinear dynamical sys-

tem (13.73). Assume that the function w : Q → R

is continuous and w(·)

is of class W

. If there exists a continuous vector function V : I

→ Q su ch

that

V (k + 1) − V (k) ≤≤ w(V (k)), k ∈ I

, (13.74)

then

V (k

) ≤≤ z

, z

∈ Q, (13.75)

implies

V (k) ≤≤ z(k), k ∈ I

, (13.76)

where z(k), k ∈ I

, is the solution to (13.73).

Proof. It follows from (13.73), (13.74), (13.75), and the fact that

w(·) ∈ W

that

V (k

+ 1) = V (k

) + V (k

+ 1) − V (k

)

≤≤ V (k

) + w(V (k

))

≤≤ z

+ w(z

)

= z(k

+ 1).

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DISCRETE-TIME THEORY 793

Hence, V (k

+ 1) ≤≤ z(k

+ 1). R ecur s ively repeating this procedure for all

∈ I

yields (13.76).

Next, consider the discrete-time nonlinear dynamical system given by

x(k + 1) = f(x(k)), x(k

) = x

, k ∈ I

, (13.77)

where x(k) ∈ D ⊆ R

, k ∈ I

, is th e system state vector, I

⊂ Z is the

maximal interval of existence of a solution x(k) to (13.77), D is an open set,

0 ∈ D, and f(·) is continuous on D. The following result is a direct corollary

of Theorem 13.14.

Corollary 13.8. Consid er the discrete-time nonlinear dynamical sys-

tem (13.77). Assu me there exists a continuous vector function V : D →

Q ⊆ R

such that

V (f (x)) −V (x) ≤≤ w(V (x)), x ∈ D, (13.78)

where w : Q → R

is a continuous function, w(·) ∈ W

, and the equation

z(k + 1) = w(z(k)), z(k

) = z

, k ∈ I

, (13.79)

has a uniqu e solution z(k), k ∈ I

. If {k

, . . . , k

+ τ } ⊆ I

∩ I

, then

V (x

) ≤≤ z

, z

∈ Q, (13.80)

implies

V (x(k)) ≤≤ z(k), k ∈ {k

, . . . , k

+ τ }. (13.81)

Proof. For every x

∈ D, the solution x(k), k ∈ I

, to (13.77) is well

deﬁned. With η(k)

△

= V (x(k)), k ∈ I

, it follows from (13.78) th at

η(k + 1) −η(k) ≤ ≤ w(η(k)), k ∈ I

. (13.82)

Moreover, if {k

, . . . , k

+τ } ⊆ I

∩I

, then it follows from Theorem 13.14

that V (x

) = η(k

) ≤≤ z

implies

V (x(k)) = η(k) ≤≤ z(k), k ∈ {k

, . . . , k

+ τ }, (13.83)

which establishes the resu lt.

Note that if the solutions to (13.77) and (13.79) are globally deﬁned

for all x

∈ D and z

∈ Q, then the result of Corollary 13.8 holds for every

k ≥ k

. For the remaind er of this section we assume that the solutions to

the systems (13.77) and (13.79) are deﬁned for all k ≥ k

. Fu rthermore,

consider the comparison discrete-time nonlinear dynamical system given by

z(k + 1) = w(z(k)), z(k

) = z

, k ≥ k

, (13.84)

NonlinearBook10pt November 20, 2007

794 CHAPTER 13

and the nonlinear d ynamical system

x(k + 1) = f (x(k)), x(k

) = x

, (13.85)

where z

∈ Q ⊆ R

, 0 ∈ Q, x

∈ D, w : Q → R

is continuous, w(·) ∈ W

w(0) = 0, f : D → D is continuous on D, and f(0) = 0. Note that since

w(·) ∈ W

and w(0) = 0, then for every x ∈ D and z ∈ Q ∩ R

it follows

that w(z) ≥≥ 0, which implies that for every z

∈ Q ∩ R

the solution

z(k), k ≥ k

, remains in R

(see Problem 13.17).

Theorem 13.15. Consider the discrete-time nonlinear dynamical sys-

tem (13.77). Assume that there exist a continuous function 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

V (f (x)) −V (x) ≤≤ w(V (x)), x ∈ D, (13.86)

where w : Q → R

is continuous, w(·) ∈ W

, and w(0) = 0. Then the

following statements hold:

i) If the zero solution z(k) ≡ 0 to (13.84) is Lyapunov stable, then the

zero solution x(k) ≡ 0 to (13.77) is Lyapunov stable.

ii) If the zero solution z(k) ≡ 0 to (13.84) is asymptotically stable,

then the zero solution x(k) ≡ 0 to (13.77) is asymptotically stable.

iii) If D = R

, Q = R

, v : R

→ R

is positive deﬁnite and

radially unbounded, and the zero solution z(k) ≡ 0 to (13.84) is globally

asymptotically stable, then the zero solution x(k) ≡ 0 to (13.77) is globally

asymptotically stable.

iv) If th ere exist constants ν ≥ 1, α > 0, and β > 0 such that v : D →

satisﬁes

αkxk

≤ v(x) ≤ βkxk

, x ∈ D, (13.87)

and the zero solution z(k) ≡ 0 to (13.84) is geometrically stable, then the

zero solution x(k) ≡ 0 to (13.77) is geometrically stable.

v) If D = R

, Q = R

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

such that v : D → R

satisﬁes (13.87), and the zero solution z(k) ≡ 0 to

(13.84) is globally geometrically stable, then the zero solution x(k) ≡ 0 to

(13.77) is globally geometrically stable.

Proof. Assume there exist a continuous function V : D → Q ∩ R

and a positive vector p ∈ R

such that v(x) = p

V (x), x ∈ D, is positive