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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 775

for α(δ) = α

, β(k) = β

−k

, γ(σ) = σ

∞

k=0

−pk

< ∞. (13.28)

Hence,

V (x) =

∞

k=0

ks(k, x)k

(13.29)

is a continuous Lyapunov function candid ate for (13.1).

To show that (13.26) holds note that

V (x) =

∞

k=0

ks(k, x)k

≤

∞

k=0

kxk

−pk

− 1

kxk

, x ∈ D, (13.30)

which proves the upper bound in (13.26) with β =

−1

> 1. Next,

V (x) =

∞

k=0

ks(k, x)k

≥ ks(0, x)k

= kxk

, x ∈ D

, (13.31)

which proves the lower bound in (13.26) with α = 1.

Finally, to s how (13.27) note that w ith γ(σ) = σ

it follows from

Theorem 13.6 that V (f(x)) −V (x) = −γ(kxk) = −kxk

. Now, the result is

immediate from (13.26) by noting that

V (f(x)) = V (x) −kxk

≤ (1 −

)V (x), x ∈ D

, (13.32)

which proves (13.27) with ρ =

β−1

Next, we present a corollary to Theorem 13.7 that shows that p in

Theorem 13.7 can be taken to be equal to 2 without loss of generality.

Corollary 13.2. Assume that the zero solution x(k) ≡ 0 to (13.1) is

geometrically stable and f : D → D is continuous. Then there exist a set

⊆ D with 0 ∈

◦

, a continuous function V : D → R, and scalars α,

β > 1, and ρ > 1, such th at

αkxk

≤ V (x) ≤ βkxk

, x ∈ D

, (13.33)

ρV (f (x)) ≤ V (x), x ∈ D

. (13.34)

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776 CHAPTER 13

Proof. The proof is a direct consequence of Theorem 13.7 with p = 2.

Finally, we present a converse theorem for global geometric stability.

Theorem 13. 8. Assume that the zero solution x(k) ≡ 0 to (13.1) is

globally geometrically stable and f : R

→ R

is continuous. Then there

exist a continuous function V : R

→ R and scalars α, β > 1, and ρ > 1,

such that

αkxk

≤ V (x) ≤ βkxk

, x ∈ R

, (13.35)

ρV (f(x)) ≤ V (x), x ∈ R

. (13.36)

Proof. The proof is identical to th e proof of Theorem 13.7 by replacing

with R

and setting p = 2.

13.5 Partial Stability of Discrete-Time Nonlinear

Dynamical Systems

In this section, we present partial stability theorems for discrete-time

nonlinear dynamical systems. Speciﬁcally, consider the discrete-time

nonlinear autonomous dynamical system

(k + 1) = f

(k), x

(k)), x

(0) = x

, k ∈ Z

, (13.37)

(k + 1) = f

(k), x

(k)), x

(0) = x

, (13.38)

where x

∈ D ⊆ R

, D is an open set with 0 ∈ D, x

∈ R

, f

: D ×

→ R

is continuous and for every x

∈ R

, f

(0, x

) = 0, and f

D × R

→ R

is continuous. Note that under the above assumptions the

solution (x

(k), x

(k)) to (13.37) and (13.38) exists and is unique over Z

The following deﬁn ition introdu ces eight types of partial stability, that is,

stability with respect to x

, for the nonlinear dynamical system (13.37) and

(13.38).

Deﬁnition 13.6. i) The nonlinear dy namical system (13.37) and

(13.38) is Lyapunov stable with respect to x

if, for every ε > 0 and x

∈ R

there exists δ = δ(ε, x

) > 0 such that kx

k < δ implies that kx

(k)k < ε

for all k ∈ Z

ii) The nonlinear dynamical system (13.37) and (13.38) is Lyapunov

stable with respect to x

uniformly in x

if, for every ε > 0, th ere exists

δ = δ(ε) > 0 s uch that kx

k < δ implies that kx

(k)k < ε for all k ∈ Z

and for all x

∈ R

iii) The nonlinear dynamical system (13.37) and (13.38) is asymptoti-

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DISCRETE-TIME THEORY 777

cally stable with respect to x

if it is Lyapunov stable with respect to x

and,

for every x

∈ R

, there exists δ = δ(x

) > 0 such that kx

k < δ implies

that lim

k→∞

(k) = 0.

iv) The nonlinear dynamical system (13.37) and (13.38) is asymptoti-

cally stable with respect to x

uniformly in x

if it is Lyapunov stable with

respect to x

uniformly in x

and there exists δ > 0 such that kx

k < δ

implies that lim

k→∞

(k) = 0 uniformly in x

and x

for all x

∈ R

v) The nonlinear dyn amical system (13.37) and (13.38) is globally

asymptotically stable with respect to x

it is Lyapunov stable with respect

to x

and lim

k→∞

(k) = 0 for all x

∈ R

and x

∈ R

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

asymptotically stable with respect to x

uniformly in x

if it is Lyapunov

stable with respect to x

uniformly in x

and lim

k→∞

(k) = 0 uniformly

in x

and x

for all x

∈ R

and x

∈ R

vii) The nonlinear dynamical system (13.37) and (13.38) is geometri-

cally stable with respect to x

uniformly in x

if there exist scalars α, β > 1,

and δ > 0 such that kx

k < δ implies that kx

(k)k ≤ αkx

kβ

−k

, k ∈ Z

for all x

∈ R

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

geometrically stable with respect to x

uniformly i n x

if there exist scalars

α, β > 1 such that kx

(k)k ≤ αkx

kβ

−k

, k ∈ Z

, for all x

∈ R

and

∈ R

Next, we present suﬃcient conditions for partial stability of the

nonlinear d ynamical system (13.37) and (13.38). For the f ollowing re-

sult deﬁne ∆V (x

, x

)

△

= V (f (x

, x

)) − V (x

, x

), where f (x

, x

)

△

, x

) f

, x

)]

, f or a given continuous function V : D × R

→ R.

Theorem 13.9. Consider the nonlinear dynamical system (13.37) and

(13.38). Then the following statements hold:

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

→ R and a class K

function α(·) such that

V (0, x

) = 0, x

∈ R

, (13.39)

α(kx

k) ≤ V (x

, x

), (x

, x

) ∈ D × R

, (13.40)

V (f (x

, x

)) ≤ V (x

, x

), (x

, x

) ∈ D × R

, (13.41)

then the nonlinear dynamical system given by (13.37) and (13.38) is

Lyapunov stable with respect to x

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778 CHAPTER 13

ii) If there exist a continuous function V : D × R

→ R and class K

functions α(·), β(·) satisfying (13.40), (13.41), and

V (x

, x

) ≤ β(kx

k), (x

, x

) ∈ D × R

, (13.42)

then the nonlinear dynamical system given by (13.37) and (13.38) is

Lyapunov s table with respect to x

uniformly in x

iii) If there exist a continuous function V : D × R

→ R and class K

functions α(·) and γ(·) satisfying (13.39), (13.40), and

∆V (x

, x

) ≤ −γ(kx

k), (x

, x

) ∈ D × R

, (13.43)

then the nonlinear dynamical system given by (13.37) and (13.38) is

asymptotically stable with respect to x

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

→ R and class K

functions α(·), β(·), γ(·) satisfying (13.40), (13.42), and

∆V (x

, x

) ≤ −γ(kx

k), (x

, x

) ∈ D × R

, (13.44)

then the nonlinear dynamical system given by (13.37) and (13.38) is

asymptotically stable with respect to x

uniformly in x

v) If D = R

and there exist a continuous function V : R

× R

→ R,

a class K function γ(·), and a class K

∞

function α(·) satisfying (13.40)

and (13.43), then the nonlinear dynamical system given by (13.37) and

(13.38) is globally asymptotically stable with respect to x

vi) If D = R

and there exist a continuous function V : R

× R

→ R,

a class K function γ(·), and class K

∞

functions α(·), β(·) satisfying

(13.40), (13.42), and (13.44), then the nonlinear d y namical system

given by (13.37) and (13.38) is globally asymptotically stable with

respect to x

uniformly in x

vii) If there exist a continuous fu nction V : D × R

→ R and positive

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

αkx

≤ V (x

, x

) ≤ βkx

, (x

, x

) ∈ D × R

,(13.45)

∆V (x

, x

) ≤ −γkx

, (x

, x

) ∈ D × R

,(13.46)

then the nonlinear dynamical system given by (13.37) and (13.38) is

geometrically stable with respect to x

uniformly in x

viii) If D = R

and there exist a continuous function V : R

× R

→

R an d positive constants α, β, γ, p such that p ≥ 1 and (13.45) and

(13.46) hold, then the nonlinear dynamical system given by (13.37) and

(13.38) is globally geometrically stable with respect to x

uniformly in

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DISCRETE-TIME THEORY 779

Proof. i) Let x

∈ R

, let ε > 0 be such that B

(0)

△

= {x

∈ R

k < ε} ⊂ D, deﬁne η

△

= α(ε), and deﬁne D

△

= {x

∈ B

(0) : V (x

, x

) <

η}. Since V (·, ·) is continuous and V (0, x

) = 0, x

∈ R

, there exists

δ = δ(ε, x

) > 0 such that V (x

, x

) < η, x

∈ B

(0). Hence B

(0) ⊂ D

Next, since ∆V (x

, x

) ≤ 0 it follows that V (x

(k), x

(k)) is a nonincreasing

function of k, and hence, for every x

∈ B

(0) it follows that

α(kx

(k)k) ≤ V (x

(k), x

(k)) ≤ V (x

, x

) < η = α(ε).

Thus , for every x

∈ B

(0), x

(k) ∈ B

(0), k ∈ Z

, establishing Lyapu nov

stability with r espect to x

ii) Let ε > 0 and let B

(0) and η be given as in the proof of i). Now,

let δ = δ(ε) > 0 be such that β(δ) = α(ε). Hence, it follows from (13.42)

that for all (x

, x

) ∈ B

(0) ×R

α(kx

(k)k) ≤ V (x

(k), x

(k)) ≤ V (x

, x

) < β(δ) = α(ε),

and h en ce, x

(k) ∈ B

(0), k ∈ Z

iii) Lyapunov stability follows from i). To show asymptotic stability

suppose, ad absurdum, that kx

(k)k 9 0 as k → ∞ or, equivalently,

lim sup

k→∞

(k)k > 0. In addition, s uppose lim inf

k→∞

(k)k > 0,

which implies that there exist constants K > 0 and θ > 0 such that

(k)k ≥ θ, k ≥ K. Then it follows from (13.43) that V (x

(k), x

(k)) →

−∞ as k → ∞, which contradicts (13.40), and hence, lim inf

k→∞

(k)k =

0. Now, since lim sup

k→∞

(k)k > 0 and lim inf

k→∞

(k)k = 0 it follows

that there exist an increasing sequen ce {k

}

∞

i=1

and a constant θ > 0 such

that

θ < kx

)k, i = 1, 2, . . . . (13.47)

Hence,

i=1

γ(kx

)k) > nγ(θ). Now, using (13.43), it follows that

V (x

), x

)) = V (x

, x

) +

k=1

∆V (x

(k), x

(k))

≤ V (x

, x

) +

i=1

∆V (x

), x

))

≤ V (x

, x

) −

i=1

γ(kx

)k)

≤ V (x

, x

) − nγ(θ). (13.48)

Hence, for large enough n the right-hand side of (13.48) becomes negative,

which contradicts (13.40). Hence, x

(k) → 0 as k → ∞, proving asymptotic

stability of (13.37) and (13.38) with respect to x

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780 CHAPTER 13

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

(k) ∈ B

(0),

k ∈ Z

, (the existence of such a (δ, ε) pair follows from uniform Lyapu nov

stability) and assume that (13.44) holds. Since (13.44) implies (13.41)

it follows that f or every x

∈ B

(0), V (x

(k), x

(k)) is a nonincreasing

function of time and, since V (·, ·) is bounded from below, it f ollows from

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

such that lim

k→∞

V (x

(k), x

(k)) = L. Now, suppose for some x

∈ B

(0),

ad absurdum, th at L > 0 so that D

△

= {x

∈ B

(0) : V (x

, x

) ≤

L for all x

∈ R

} is nonempty and x

(k) 6∈ D

, k ∈ Z

. Thus, as in

the proof of i), there exists

δ > 0 such that B

(0) ⊂ D

. Hence, it follows

from (13.44) that for the given x

∈ B

(0)\D

and k ∈ Z

V (x

(k), x

(k)) = V (x

, x

) +

k−1

i=0

∆V (x

(i), x

(i))

≤ V (x

, x

) −

k−1

i=0

γ(kx

(i)k)

≤ V (x

, x

) − γ(

δ)k.

Letting k >

V (x

)−L

γ(

δ)

, it follows that V (x

(k), x

(k)) < L, w hich is a

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

∈ B

(0) was chosen arbitrarily,

it follows that V (x

(k), x

(k)) → 0 as k → ∞ for all x

∈ B

(0). Now,

since V (x

(k), x

(k)) ≥ α(kx

(k)k) ≥ 0 it follows that α(kx

(k)k) → 0

or, equivalently, x

(k) → 0 k → ∞, establishing asymptotic 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 such th at V (x

, x

) < α(ε). Now, (13.43)

implies that V (x

(k), x

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

it follows that α(kx

(k)k) ≤ V (x

(k), x

(k)) ≤ V (x

, x

) < α(ε), k ∈ Z

Hence, x

(k) ∈ B

(0), k ∈ Z

. 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, (13.44)

implies that V (x

(k), x

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

it follows from (13.42) that α(kx

(k)k) ≤ V (x

(k), x

(k)) ≤ V (x

, x

) <

β(δ) < α(ε), k ∈ Z

. Hence, x

(k) ∈ B

(0), k ∈ Z

. Now, the proof follows

as in the proof of iv).

vii) Let ε > 0 be given as in the proof of i) and let η

△

= αε and

δ =

. Now, (13.46) implies that ∆V (x

, x

) ≤ 0, and hence, it follows that

V (x

(k), x

(k)) is a nonincreasing function of time. Hence, as in the proof

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DISCRETE-TIME THEORY 781

of ii), it follows that for all (x

, x

) ∈ B

(0) ×R

, x

(k) ∈ B

(0), k ∈ Z

Furth ermore, it follows from (13.45) and (13.46) that for all k ∈ Z

and

, x

) ∈ B

× R

∆V (x

(k), x

(k)) ≤ −γkx

(k)k

≤ −

V (x

(k), x

(k)),

which implies that

V (x

(k), x

(k)) ≤ V (x

, x

)(1 −

)

It now follows from (13.45) that

αkx

(k)k

≤ V (x

(k), x

(k))

≤ V (x

, x

)(1 −

)

≤ βkx

(1 −

)

, k ∈ Z

and h en ce,

(k)k ≤





1/p

k(1 −

)

, k ∈ Z

establishing geometric 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 13.9 specializes to the case

of nonlinear autonomous systems of the form x

(k) = f

(k)). 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 13.2. In

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

a continuous function V : D → R such that (13.40), (13.42), and (13.44) hold

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

) > 0, x

6= 0, and V (f

)) < V (x

), x

6= 0.

In addition, if D = R

and there exist class K

∞

functions α(·), β(·) and

a continuously diﬀerentiable function V (·) such that (13.40), (13.42), and

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

) > 0, x

6= 0, V (f

)) < V (x

6= 0, and V (x

) → ∞ as kx

k → ∞. Hence, in this case, Theorem

13.9 collapses to the classical Lyapunov stability theorem for autonomous

systems given in Section 13.2.

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

invariance theorem (Theorem 13.3) shows that bounded system trajectories

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 on how to deﬁne

the set M since ∆V (x

, x

) is a function of both x

and x

. However,

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782 CHAPTER 13

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 s et 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

(k) approach R as k tends to inﬁnity. For this result, the following lemma

is necessary.

Lemma 13.3. Let σ : Z

→ R

and suppose that

∞

k=0

σ(k) exists

and is ﬁnite. Then, lim

k→∞

σ(k) = 0.

Proof. Suppose, ad absurdum, that lim sup

k→∞

σ(k) > 0 and let

∈ R be s uch that 0 < α

< lim sup

k→∞

σ(k). In this case, for every

> 0, there exists k

≥ K

such that σ(k

) ≥ α

. Next, let K

> k

and using a similar argument as above it follows that there exists k

≥ K

such that σ(k

) ≥ α

. Hence, there exists an increasing unbounded sequence

}

∞

i=1

such that σ(k

) ≥ α

, and hence,

∞

i=1

σ(k

) ≥

∞

i=1

= ∞, which

is a contradiction.

Theorem 13.10. Consider the nonlinear dynamical system given by

(13.37) and (13.38) and assume D × R

is a positively invariant set with

respect to (13.37) and (13.38). Furthermore, assume there exist functions

V : D × R

→ R, W, W

, W

: D → R s uch that V (·, ·) is continuous,

(·) and W

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

and n on negative deﬁnite, and, f or all (x

, x

) ∈ D × R

) ≤ V (x

, x

) ≤ W

), (13.49)

∆V (x

, x

) ≤ −W (x

). (13.50)

Then there exists D

⊆ D such that for all (x

, x

) ∈ D

× R

, x

(k) →

△

= {x

∈ D : W (x

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

and W

(·)

is radially unboun ded, then for all (x

, x

) ∈ R

× R

, x

(k) → R

△

∈ R

: W (x

) = 0} as k → ∞.

Proof. Assume (13.49) and (13.50) hold. Then it follows from

Theorem 13.9 that the nonlinear dyn amical system given by (13.37) and

(13.38) is Lyapunov stable with respect to x

uniformly in x

. Let ε > 0

be su ch that B

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

∈ B

(0),

then x

(k) ∈ B

(0), k ∈ Z

. Now, s ince V (x

(k), x

(k)) is monotonically

nonincreasing and bounded from below by zero, it follows from the monotone

convergence theorem (Theorem 2.10) that lim

k→∞

V (x

(k), x

(k)) exists

and is ﬁnite. Hence, since for every k ∈ Z

k−1

κ=0

W (x

(κ)) ≤ −

k−1

κ=0

∆V (x

(κ), x

(κ)) = V (x

, x

) −V (x

(k), x

(k)),

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DISCRETE-TIME THEORY 783

it follows that

∞

κ=0

W (x

(κ)) exists and is ﬁnite. Hence, it follows from

Lemma 13.3 that W (x

(k)) → 0 as k → ∞. Finally, if, in addition, D = R

and W

(·) is radially unbounded, then, as in the proof of iv) of T heorem

13.9, for every x

∈ R

there exist ε, δ > 0 such that x

∈ B

(0) and

(k) ∈ B

(0), k ∈ Z

. Now, the proof follows by repeating the above

arguments.

Theorem 13.10 shows that the partial system trajectories x

(k)

approach R as k tends to inﬁnity. However, since the positive limit set

of the partial trajectory x

(k) is a subset of R, Theorem 13.10 is a weaker

result than the standard invariance principle, wherein one would conclude

that the partial trajectory x

(k) approaches the largest invariant set M

contained in R. This is not true in general for partially stable systems since

the positive limit set of a partial trajectory x

(k), k ∈ Z

, is not an invariant

set.

13.6 Stability Theory for Discrete-Time Nonlinear

Time-Varying Systems

In this section, we use the results of Section 13.5 to extend Lyapunov’s direct

method to nonlinear time-varying systems thereby providing a uniﬁcation

between partial stability theory for au tonomous systems and stability th eory

for time-varying systems. Speciﬁcally, we consider the nonlinear time-

varying dynamical system

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

) = x

, k ≥ k

, (13.51)

where x(k) ∈ D, k ≥ k

, D ⊆ R

is an open set such that 0 ∈ D,

f : {k

, . . . , k

} × D → R

is su ch that f(·, ·) is continuous and, for every

k ∈ {k

, . . . , k

}, f(k, 0) = 0. Note that under the above assumptions the

solution x(k), k ≥ k

, to (13.51) exists and is unique over the interval

, . . . , k

}. The following deﬁnition provides eight types of stability for

the nonlinear time-varying dynamical system (13.51).

Deﬁnition 13.7. i) The nonlinear time-varying dynamical system

(13.51) is Lyapunov stable if, for every ε > 0 and k

∈ Z

, there exists

δ = δ(ε, k

) > 0 such that kx

k < δ implies that kx(k)k < ε for all k ≥ k

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

Lyapunov stable if, for every ε > 0, there exists δ = δ(ε) > 0 such that

k < δ implies that kx(k)k < ε for all k ≥ k

and for all k

∈ Z

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

totically stable if it is Lyapunov stable and, for every k

∈ Z

, there exists

δ = δ(k

) > 0 such that kx

k < δ implies that lim

k→∞

x(k) = 0.

NonlinearBook10pt November 20, 2007

784 CHAPTER 13

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

asymptotically stable if it is uniformly Lyapunov stable and there exists δ > 0

such that kx

k < δ implies that lim

k→∞

x(k) = 0 uniformly in k

and x

for

all k

∈ Z

v) The nonlinear time-varying dynamical system (13.51) is globally

asymptotically stable if it is Lyapunov stable and lim

k→∞

x(k) = 0 for all

∈ R

and k

∈ Z

vi) The nonlinear time-varying dynamical system (13.51) is globally

uniformly asymptotically stable if it is uniformly Lyapunov stable and

lim

k→∞

x(k) = 0 uniformly in k

and x

for all x

∈ R

and k

∈ Z

vii) The nonlinear time-varying dynamical sy s tem (13.51) is (uni-

formly) geometrically stable if there exist scalars α, δ > 0 and β > 1, su ch

that kx

k < δ implies that kx(k)k ≤ αkx

kβ

−k

, k ≥ k

and k

∈ Z

viii) The n on linear time-varying dynamical system (13.51) is globally

(uniformly) geometrically stable if there exist scalars α > 0 and β > 1, such

that kx(k)k ≤ αkx

kβ

−k

, k ≥ k

, f or all x

∈ R

and k

∈ Z

Next, using Theorem 13.9 we pr esent suﬃcient conditions for stability

of the nonlinear time-varying dynamical system (13.51). For the following

result deﬁne

∆V (k, x)

△

= V (k + 1, f(k, x)) −V (k, x)

for a given continuous fun ction V : Z

× D → R.

Theorem 13.11. Consider the nonlinear time-varying dynamical sys-

tem (13.51). Then the following statements hold:

i) I f th ere exist a continuous function V : Z

× D → R and a class K

function α(·) such that

V (k, 0) = 0, k ∈ Z

, (13.52)

α(kxk) ≤ V (k, x), (k, x) ∈ Z

× D, (13.53)

∆V (k, x) ≤ 0, (k, x) ∈ Z

× D, (13.54)

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

Lyapunov s table.

ii) If there exist a continuous function V : Z

× D → R and class K

functions α(·), β(·) satisfying (13.53) and (13.54), and

V (k, x) ≤ β(kxk), (k, x) ∈ Z

× D, (13.55)