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

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ADVANCED STABILITY THEORY 315

˙x

(t) = −x

(t) −e

−t

(t), x

(0) = x

. (4.374)

Show that the zero solution (x

(t), x

(t)) ≡ (0, 0) to (4.373) and (4.374) is

Lyapunov s table.

Problem 4.9. Consider the linear dynamical system

˙x

(t) = −a(t)x

(t) −bx

(t), x

(0) = x

, t ≥ 0, (4.375)

˙x

(t) = bx

(t) −c(t)x

(t), x

(0) = x

, (4.376)

where b ∈ R, a(t) ≥ α > 0, t ≥ 0, and c(t) ≥ β > 0, t ≥ 0. Sh ow that

the zero solution (x

(t), x

(t)) ≡ (0, 0) to (4.375) and (4.376) is globally

exponentially stable.

Problem 4.10. Consider the damped Mathieu equation given by

¨x(t) + ˙x(t) + (2 + sin t)x(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0. (4.377)

Show that the zero solution (x(t), ˙x(t)) ≡ (0, 0) to (4.377) is uniformly

Lyapunov s table.

Problem 4.11. Consider the spring-mass-damper system with a time-

varying damping coeﬃcient given by

¨x(t) + b(t) ˙x(t) + kx(t) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (4.378)

where b(t) is such that

b(t) > α > 0,

b(t) ≤ β < 2k, t ≥ 0. (4.379)

Assuming b(t) is upper bounded, show that the zero solution (x(t), ˙x(t)) ≡

(0, 0) to (4.378) is uniformly asymptotically stable. Are either or both of

the conditions in (4.379) necessary to establish u niform asymptotic stability?

(Hint: Consider the Lyapunov function candidate V (t, x) =

( ˙x + αx)

γ(t)x

, where 0 < α <

√

k and γ(t) = k − α

+ αb(t).)

Problem 4.12. Consider the nonlinear dynamical system

˙x

(t) = α(t)x

(t) + β(t)x

(t)[x

(t) + x

(t)], x

(0) = x

, t ≥ 0, (4.380)

˙x

(t) = −α(t)x

(t) + β(t)x

(t)[x

(t) + x

(t)], x

(0) = x

, (4.381)

where α(·) and β(·) are continuous f unctions. Analyze the stability of the

zero solution (x

(t), x

(t)) ≡ (0, 0) to (4.380) and (4.381) using Lyapunov’s

direct method.

Problem 4.13. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (4.382)

˙x

(t) = −sin x

(t) −a(t)x

(t), x

(0) = x

, (4.383)

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316 CHAPTER 4

where a(·) is continuously diﬀerentiable and satisﬁes 0 < α < a(t) ≤ β < ∞,

t ≥ 0, and ˙α(t) ≤ γ < 2, t ≥ 0. Show that the zero solution (x

(t), x

(t)) ≡

(0, 0) to (4.382) and (4.383) is uniformly asymptotically stable. (Hint:

Consider the Lyapunov function candidate

V (t, x) =

(α sin x

+ x

)

+ [1 + αa(t) − α

](1 −cos x

), (4.384)

and show that (4.384) is a valid candidate.)

Problem 4.14. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (4.385)

˙x

(t) = −a(t)x

(t) −e

−t

(t), x

(0) = x

, (4.386)

where a(·) is a continuous function. Analyze the stability of the zero solution

(t), x

(t)) ≡ (0, 0) to (4.385) and (4.386) by using the Lyapunov function

candidate V (t, x

, x

) = x

+ e

Problem 4. 15. Show that if V : [0, ∞) ×D ⊆ R

→ R is continuously

diﬀerentiable, lower bounded,

V (t, x) ≤ 0, t ∈ [0, ∞) × D, and

V (t, x) is

uniformly continuous in time, then

V (t, x) → 0 as t → ∞. (Hint: Use

Barbalat’s lemma to prove the result.)

Problem 4.16. Consider the dynamical sys tem

˙x

(t) = −x

(t) + x

(t)w(t), x

(0) = x

, t ≥ 0, (4.387)

˙x

(t) = −x

(t)w(t), x

(0) = x

, (4.388)

where w(t), t ≥ 0, is a bounded continuous disturbance. Show that x

(t) →

0 as t → ∞ and x

(t), t ≥ 0, is bounded. (Hint: Use the result in Problem

4.15.)

Problem 4. 17. Consider th e time-varying nonlinear dynamical system

(4.57). Show that if there exist a continuously diﬀerentiable function V :

[0, ∞) × D → R and class K fu nctions α(·) and γ(·) such that

V (t, 0) = 0, t ∈ [0, ∞), (4.389)

α(kxk) ≤ V (t, x), (t, x) ∈ [0, ∞) ×D, (4.390)

V (t, x) ≤ −γ(V (t, x)), (t, x) ∈ [0, ∞) × D, (4.391)

then the zero solution x(t) ≡ 0 to (4.57) is asymptotically stable. If, in

addition, D = R

and α(·) is a class K

∞

function, show that the zero

solution x(t) ≡ 0 to (4.57) is globally asymptotically stable.

Problem 4. 18. Consider th e time-varying nonlinear dynamical system

(4.57). Show that if there exist continuously diﬀerentiable functions V :

[0, ∞) × D → R and W : D → R, and class K functions α(·) and γ(·) such

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ADVANCED STABILITY THEORY 317

that W (0) = 0, W (x) > 0, x ∈ D, x 6= 0,

W (x(·)) is bounded from below or

above, and

V (t, 0) = 0, t ∈ [0, ∞), (4.392)

α(kxk) ≤ V (t, x), (t, x) ∈ [0, ∞) × D, (4.393)

V (t, x) ≤ −γ(W (x)), (t, x) ∈ [0, ∞) ×D, (4.394)

then the zero solution x(t) ≡ 0 to (4.57) is asymptotically stable. If, in

addition, D = R

and α(·) is a class K

∞

function sh ow that the zero solution

x(t) ≡ 0 to (4.57) is globally asymptotically stable.

Problem 4.19. Prove Theorem 4.10.

Problem 4.20. Prove Theorem 4.11.

Problem 4.21. Prove Theorem 4.12.

Problem 4.22. Consider the nonlinear dynamical system

˙x

(t) = −ax

(t) + bx

(t), x

(0) = x

, t ≥ 0, (4.395)

˙x

(t) = −cx

(t), x

(0) = x

, (4.396)

where a, b, c > 0. Use Proposition 4.2 to show that the zero solution

(t), x

(t)) ≡ (0, 0) of (4.395) and (4.396) is globally asymptotically stable.

Problem 4.23. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) −2x

(t) + x

(t), x

(0) = x

, t ≥ 0, (4.397)

˙x

(t) = −x

(t), x

(0) = x

. (4.398)

Use Pr oposition 4.2 to show that the zero solution (x

(t), x

(t)) ≡ (0, 0) is

globally asymptotically stable.

Problem 4.24. Consider the nonlinear dynamical system

˙x

(t) = −x

(t)x

(t), x

(0) = x

, t ≥ 0, (4.399)

˙x

(t) = −

(t) −x

(t) +

u(t), x

(0) = x

. (4.400)

Show that (4.399) and (4.400) is input-to-state stable.

Problem 4.25. Consider the nonlinear perturbed dynamical system

˙x(t) = f(t, x(t)) + g(t, x(t)), x(t

) = x

, t ≥ t

, (4.401)

where x(t) ∈ D, t ≥ 0, D ⊆ R such that 0 ∈ D, f(t, ·) : D → R

and

g(t, ·) : D → R

are Lipschitz continuous on D for all t ∈ [0, ∞), and f(·, x) :

[0, ∞) → R

and g(·, x) : [0, ∞) → R

are piecewise continuous on [0, ∞).

Assume th at the zero solution x(t) ≡ 0 to the nominal system (4.401), that

is, (4.401) with g(t, x) ≡ 0, is exponentially stable. Furthermore, assume

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318 CHAPTER 4

there exists a continuously diﬀerentiable function V : [0, ∞) × D → R such

that the conditions of Corollary 4.2 are satisﬁed for D = {x ∈ R

: kxk < ε}.

Finally, suppose g(t, x) satisﬁes

kg(t, x)k ≤ δ <

µε, (t, x) ∈ [0, ∞) × D, (4.402)

where α

, α

, and µ are positive constants with µ < 1. Show that,

for all kx(t

)k <

ε,

kx(t)k ≤ σkx(t

)ke

−γ(t−t

)

, t ≤ t < t

+ T, (4.403)

and

kx(t)k ≤ β, t ≥ t

+ T, (4.404)

where T > 0 is a ﬁnite time, σ

△

, γ

△

(1−µ)α

2α

, and β

△

Problem 4.26. Consider th e nonlinear dynamical system (4.177) with

D = {x ∈ R : |x| < 1} and f : D → R given by

f(x) =



−x(ln |x|)

, x ∈ D\{0},

0, x = 0.

(4.405)

Show that th is system is ﬁnite-time stable with settling-time function

T (x) =



−

ln |x|

, x ∈ D\{0},

0, x = 0.

(4.406)

Problem 4. 27. Consider th e nonlinear time-varying dynamical system

(4.57) where f(·, ·) is continuous on [0, ∞) ×D and (4.57) possesses unique

solutions in forward time for all x

∈ D and t

∈ [0, ∞). The zero s olution

x(t) ≡ 0 to (4.57) is ﬁnite-time stable if th e origin is Lyapunov stable and

there exists an open neighborhood N ⊆ D of the origin and a function

T : [0, ∞) × N\{0} → (0, ∞), called a settling-time function, su ch that for

every x

∈ N\{0}, s(·, t

, x

) : [t

, T (t

, x

)) → N\{0} and s(t, t

, x

) → 0

as t → T (t

, x

). The zero solution to (4.57) is uniformly ﬁnite-time stable if

the origin is uniformly Lyapu nov stable and s(t, t

, x

) → 0 as t → T (t

, x

)

for every x

∈ N\{0}. The zero solution is globally uniformly ﬁnite-time

stable if it is uniformly ﬁnite-time stable with D = N = R

. Show that the

following statements hold:

i) I f there exist a continuously diﬀerentiable function V : [0, ∞)×D → R,

class K function α(·), a continuous function c : [0, ∞) → R

with

c(t) > 0 for almost all t ∈ [0, ∞), a real number λ ∈ (0, 1), and an

open neighborhood M ⊆ D of the origin such that

V (t, 0) = 0, t ∈ [0, ∞), (4.407)

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ADVANCED STABILITY THEORY 319

α(kxk) ≤ V (t, x), x ∈ M, t ∈ [0, ∞), (4.408)

V (t, x) ≤ −c(t)(V (t, x))

, x ∈ M, t ∈ [0, ∞), (4.409)

then the zero solution x(t) ≡ 0 to (4.57) is ﬁnite-time stable.

ii) If there exist a continuously diﬀerentiable function V : [0, ∞)×D → R,

class K functions α(·) and β(·), a continuous function c : [0, ∞) → R

with c(t) > 0 for almost all t ∈ [0, ∞), a real number λ ∈ (0, 1), and an

open neighborhood M ⊆ D of the origin such that (4.408) and (4.409)

hold, and

V (t, x) ≤ β(kxk), x ∈ M, t ∈ [0, ∞), (4.410)

then the zero solution x(t) ≡ 0 to (4.57) is uniformly ﬁnite-time stable.

iii) If D = M = R

and there exist a continuously diﬀerentiable function

V : [0, ∞) × D → R, class K

∞

functions α(·) and β(·), a continuous

function c : [0, ∞) → R

with c(t) > 0 for almost all t ∈ [0, ∞), a

real number λ ∈ (0, 1), and an open n eighborhood M ⊆ D of the

origin su ch that (4.408)–(4.410) hold, then the zero solution x(t) ≡ 0

to (4.57) is globally uniformly ﬁnite-time stable.

Problem 4. 28. Consider th e nonlinear time-varying dynamical system

(4.57). Assume that the zero solution x(t) ≡ 0 to (4.57) is ﬁnite-time

stable and let N ⊆ D and T : [0, ∞) × N\{0} → (0, ∞) be deﬁned

as in Problem 4.27. Show that, for every t

∈ [0, ∞) and x

∈ N,

there exists a unique solution s(t, t

, x

), t ≥ t

, to (4.57) such that

s(t, t

, x

) ∈ N, t ∈ [t

, T (t

, x

)), and s(t, t

, x

) = 0 for all t ≥ T (t

, x

where T (t

, 0)

△

= t

Problem 4. 29. Consider th e nonlinear time-varying dynamical system

(4.57). Assume that the zero solution x(t) ≡ 0 to (4.57) is ﬁnite-time s table

and let N ⊆ D and T : [0, ∞) × N\{0} → (0, ∞) be deﬁned as in Problem

4.27. S how that the following statements hold:

i) I f t

∈ [0, ∞), t ≥ t

, and x ∈ N, then the settling-time function

T (t, s(t, t

, x

)) = max{T (t

, x), t}.

ii) T (·, ·) is jointly continuous on [0, ∞) × N if and only if T (·, ·) is

continuous at (t, 0) for all t ≥ 0.

Problem 4. 30. Consider th e nonlinear time-varying dynamical system

(4.57). Let λ ∈ (0, 1), let N be as in P roblem 4.27, and assume th at there

exists a class K function µ : [0, r] → [0, ∞), where r > 0, such that B

(0) ⊆

N and

kf(t, x)k ≤ µ(kxk), t ∈ [0, ∞), x ∈ B

(0). (4.411)

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320 CHAPTER 4

Show th at if the zero solution x(t) ≡ 0 to (4.57) is uniformly ﬁnite-time

stable and the settling-time function T (·, ·) is continuous at (t, 0) for all

t ≥ 0, then there exist a class K function α(·), a positive constant c > 0, a

continuous function V : [0, ∞) × N → R, and a neighborhood M ⊆ N of

the origin such that

α(kxk) ≤ V (t, x), (t, x) ∈ [0, ∞) ×M, (4.412)

V (t, x) ≤ −c(V (t, x))

, (t, x) ∈ [0, ∞) ×M. (4.413)

(Hint: Consider the Lyapunov function candidate V (t, x) = [T (t, x)−t]

1−λ

Problem 4. 31. Consider th e nonlinear time-varying dynamical system

(4.57) and assume f(t, ·) is Lipschitz continuous in x on D uniformly in t f or

all t in compact subsets of [0, ∞). Furtherm ore, assume that the Lipschitz

constant of f(t, ·) is bounded for all t ≥ 0 with maximum Lipschitz constant

L > 0 over D. Show that if there exist a function V : [0, ∞) × D → R and

a ﬁnite time T > 0 such that

α(kxk) ≤ V (t, x) ≤ β(kxk), (t, x) ∈ [0, ∞) × D, (4.414)

V (t + T, x(t + T )) − V (t, x) ≤ −γ(kxk) < 0, (t, x) ∈ [0, ∞) × D, (4.415)

where α(·), β(·), and γ(·) are class K functions, then the zero solution x(t) ≡

0 to (4.57) is uniformly asymptotically stable. (Hint: Show that (4.415)

implies that there exists an increasing unbounded sequence {t

}

∞

n=0

, with

= 0, such th at T ≥ t

n+1

− t

> 0, n = 0, 1, . . ., and

V (t

n+1

, x(t

n+1

)) − V (t

, x(t

)) ≤ −γ(kx(t

)k), x(t

) ∈ D, n = 0, 1, . . . .

(4.416)

Here, the function V (·, ·) does not s atisfy any regularity assumptions.)

Problem 4.32. Consider the nonlinear dynamical sy s tem in polar

co ordinates given by

˙r(t) = −r(t)sign[r

(t) − 1]|r

(t) −1|

, r(0) = r

, t ≥ 0, (4.417)

θ(t) = −sign[r

(t) − 1]|r

(t) −1|

, θ(0) = θ

, (4.418)

where α, β ∈ R. Sh ow that the set of equilibria of (4.417) and (4.418) consists

of the origin (r, θ) = (0, 0) and the unit circle C = {(r, θ) ∈ R ×R : r

= 1}.

In addition, show that all solutions of (4.417) and (4.418) starting from

nonzero initial conditions that are not on the unit circle C approach the

unit circle, and hence, all solutions are bounded, and, for every choice of α

and β, all solutions converge to the set of equilibria. However, show that if

α ≥ β + 1, then the dynamical sys tem (4.417) and (4.418) is not convergent.

(Hint: Use (4.417) and (4.418) to obtain

dθ

= r|r

− 1|

α−β

Problem 4.33. Consider the nonlinear dynamical system

˙x

(t) = −4x

(t) + x

(t)[1 −

(t) −x

(t)], x

(0) = x

, t ≥ 0, (4.419)

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ADVANCED STABILITY THEORY 321

˙x

(t) = x

(t) + x

(t)[1 −

(t) −x

(t)], x

(0) = x

. (4.420)

Show that (4.419) and (4.420) has a limit cycle that lies in the ellipse E =

{(x

, x

) ∈ R

+ x

= 1}. Using Poincar´e maps, examine the stability

of this limit cycle.

Problem 4.34. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −4x

(t) −

(t) − x

(t)x

(t), x

(0) = x

, t ≥ 0, (4.421)

˙x

(t) = x

(t) + x

(t) −

(t)x

(t) −x

(t), x

(0) = x

, (4.422)

˙x

(t) = x

(t), x

(0) = x

. (4.423)

Show that (4.421)–(4.423) has a periodic orbit given by x(t) = [2 cos 2t,

2 sin 2t, 0]

. Using Poincar´e maps examine the stability of this orbit.

Problem 4.35. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (4.424)

˙x

(t) = x

(t) + x

(t)x

(t), x

(0) = x

, (4.425)

˙x

(t) = −x

(t)[x

(t) + x

(t)], x

(0) = x

. (4.426)

Show that (4.424)–(4.426) has a periodic orbit given by x(t) = [cos t, sin t,

. Using Poincar´e maps examine the stability of this orbit. (Hint: Use

V (x

, x

) = x

+ x

to show that the traj ectories of (4.424)–(4.426)

lie on spheres x

+ x

= k

, k ∈ R.)

Problem 4.36. Consider the nonlinear dynamical system

˙x

(t) = −2x

(t) + 4x

[4 − 4x

(t) − x

(t) + x

(t)], x

(0) = x

, t ≥ 0,

(4.427)

˙x

(t) = 8x

(t) + 4x

(t)[4 −4x

(t) − x

(t) + x

(t)], x

(0) = x

, (4.428)

˙x

(t) = x

(t)[x

(t) − 16], x

(0) = x

. (4.429)

Show that (4.427)–(4.429) has a periodic orbit given by x(t) = [cos 4t,

2 sin 4t, 0]

. Using Poincar´e maps examine the stability of this orbit.

Problem 4.37. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t)[1 − x

(t) − x

(t)][16 − x

(t) −x

(t)],

(0) = x

, t ≥ 0, (4.430)

˙x

(t) = x

(t) + x

(t)[1 −x

(t) −x

(t)][16 − x

(t) − x

(t)],

(0) = x

, (4.431)

˙x

(t) = x

(t), x

(0) = x

. (4.432)

Show that (4.430)–(4.432) has two periodic orbits. Using Poincar´e maps

examine the stability of these orbits.

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322 CHAPTER 4

Problem 4.38. Prove Theorem 4.40.

Problem 4.39. Consider the nonlinear dynamical system (4.177).

Assume that there exist a continuously diﬀerentiable vector function V :

D → Q ∩R

, where Q ⊂ R

and 0 ∈ Q, and a positive vector p ∈ R

such

that V (0) = 0, the scalar function p

V (x), x ∈ D, is positive deﬁn ite, and

′

(x)f(x) ≤ ≤ w(V (x)), x ∈ D, (4.433)

where w : Q → R

is continuous, w(·) ∈ W, and w(0) = 0. In addition,

assume that the vector comparison system

˙z(t) = w(z(t)), z(0) = z

, t ∈ I

, (4.434)

has a unique solution in forward time z(t), t ∈ I

, and there exist a

continuously diﬀerentiable function v : Q → R, real numbers c > 0 and

α ∈ (0, 1), and a neighborhood M ⊆ Q of the origin such that v(·) is

positive deﬁnite and

′

(z)w(z) ≤ −c(v(z))

, z ∈ M. (4.435)

Show that the zero solution x(t) ≡ 0 to (4.177) is ﬁnite-time stable.

Moreover, if N is as in Deﬁnition 4.7 and T : N → [0, ∞) is the settling-time

function, show that

T (x

) ≤

c(1 −α)

(v(V (x

)))

1−α

, x

∈ N, (4.436)

and T (·) is continuous on N. Finally, show that if D = R

, v(·) is radially

unbounded, and (4.433) and (4.435) hold on R

, then the zero solution

x(t) ≡ 0 to (4.177) is globally ﬁnite-time stable.

Problem 4.40. Consider the nonlinear dynamical system (4.177).

Assume there exist a continuously diﬀerentiable vector function V : D →

Q ∩ R

, where Q ⊂ R

and 0 ∈ Q, and a positive vector p ∈ R

such that

V (0) = 0, the scalar function p

V (x), x ∈ D, is positive deﬁn ite, and

′

(x)f(x) ≤ ≤ W (V (x))

{α}

, x ∈ D, (4.437)

where α ∈ (0, 1), W ∈ R

q×q

is essentially n onnegative (see Problem 3.7) and

Hurwitz, and (V (x))

{α}

, [(V

(x))

, . . . , (V

(x))

]

. Show that the zero

solution x(t) ≡ 0 to (4.177) is ﬁnite-time stable. (Hint: Use Problems 3.8

and 4.39.)

4.13 Notes and References

The concept of partial stability is due to Rumyantsev [374] with a thorough

treatment given by Vorotnikov [448]. See also Rouche, Habets, and Laloy

[368] and Chellaboina and Haddad [87]. The concept of input-to-state

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ADVANCED STABILITY THEORY 323

stability was introduced by Sontag [405] with key resu lts given in Sontag

[407] and Sontag and Wang [409]. A r igorous foundation for the theory

of ﬁnite-time stability was ﬁrst given by Bhat and Bernstein [55]. The

presentation here is adopted from Bhat and Bernstein [55]. Semistability

was ﬁrst introduced by Campbell and Rose [81] for linear s ystems, and

applied to matrix second-order system by Bernstein and Bhat [46]. Bh at

and Bernstein [54,56,57] also consider semistability of nonlinear systems, and

give s everal stability results for systems h aving a continuum of equilibria

based on nontangency and arc length of trajectories. Semistability was

also add ressed by Hui, Haddad, and Bhat [209] for consensus p rotocols in

nonlinear dynamical networks.

The generalized Lyapunov and invariant set theorems predicated on

lower semicontinuous Lyapunov fun ctions presented in Section 4.8 are due

to Chellaboina, Leonessa, and Haddad [91] while Theorem 4.28 is due to

Aeyels and Peuteman [3]. Lyapunov and asymptotic stability of sets were

ﬁrst introd uced by Zubov [481] and further developed by Yoshizawa [474]

and Bhatia and Szeg¨o [58]. The treatment here is adopted from Leonessa,

Haddad, and Chellab oina [272,273]. Poincar´e maps and stability of periodic

orbits are due to Poincar´e [358] with a Lyapunov function proof given

by Haddad, Nersesov, and Chellaboina [175]. Vector Lyapunov functions

were ﬁrst introduced by Bellman [38] and Matrosov [308] with further

developments given by Lakshmikantham, Matrosov, and Sivasundaram [255]

and Nersesov and Haddad [334]. See also Siljak [400,402], Lakshmikantham

and Leela [253], and L akshmikantham, Leela, and Martynyuk [254].

NonlinearBook10pt November 20, 2007