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

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STABILITY THEORY FOR NONLINEAR SYSTEMS 175

− P

, t ≥ 0, (3.166)

which further implies

0 =

∞

− P

= e

− P



∞

= −(P

− P

). (3.167)

This leads to the contradiction P

= P

, and hence, there exists a unique

solution P ∈ R

n×n

satisfying (3.157).

To apply Theorem 3.16 we choose any positive-deﬁnite matrix R ∈

n×n

and solve the Lyapunov equation for P ∈ R

n×n

. If (3.157) has no

solution or has multiple solutions, th en the zero solution x(t) ≡ 0 to (3.152)

is not asymptotically stable. Alternatively, if P is a unique positive-deﬁnite

solution to (3.157), then the zero solution x(t) ≡ 0 to (3.152) is globally

asymptotically stable.

Next, we use the Barbashin -Krasovskii-LaSalle invariant set theorem

to weaken the conditions in Th eorem 3.16.

Theorem 3.17. Consider the linear dynamical sys tem (3.152). Let

R = C

C, where C ∈ R

l×n

, and assume (A, C) is observable. Then the zero

solution x(t) ≡ 0 to (3.152) is globally asymptotically stable if and only if

there exists a unique positive-deﬁnite matrix P ∈ R

n×n

satisfying

0 = A

P + P A + C

C. (3.168)

Proof. To show suﬃciency, consider the Lyapunov function candidate

V (x) = x

P x, where x ∈ R

and P satisﬁes (3.168). Thus , the correspon -

ding Lyapunov derivative is given by

′

(x)f(x) = 2x

P Ax = x

P + P A)x = −x

Cx ≤ 0, x ∈ R

(3.169)

which proves Lyapunov stability. Next, note that V

′

(x)f(x) = 0 if and only

if Cx = 0. Let R

△

= {x ∈ R

: V

′

(x)f(x) = 0} = {x ∈ R

: Cx = 0}. Now,

let M denote the largest invariant set contained in R and note that since

Cx = 0 it follows that for all x ∈ M, C ˙x = 0, C ¨x = 0, . . . , Cx

(n−1)

= 0, and

hence, for all x ∈ M, Cx = 0, CAx = 0, . . . , CA

n−1

x = 0 or, equivalently,

Ox = 0, w here O denotes the observability matrix. Next, since (A, C) is

observable it follows that Ox = 0 if and only if x = 0, and hence, M =

{0}. Now, it follows from Theorem 3.3 that x(t) → M = {0} as t → ∞,

establishing asymptotic stability. Furthermore, global asymptotic s tability

is a direct consequence of the fact that V (x) → ∞ as kxk → ∞.

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176 CHAPTER 3

To show necessity, suppose A is asymptotically stable. Then existence

and uniquen ess of a nonnegative-deﬁnite P ∈ R

n×n

satisfying (3.168) follows

as in the proof of Theorem 3.16. To show P > 0, suppose, ad absurdum,

that P is not positive deﬁnite. Then there exists x ∈ R

, x 6= 0, such that

P x = 0, which, by (3.162), implies Ce

x = 0 for all t ≥ 0. Next, let

g(t) = Ce

x and note that g(t) = 0, ˙g(t) = 0, ¨g(t) = 0, . . ., g

(n−1)

(t) =

0, t ≥ 0, or, equivalently, Ce

x = 0, CAe

x = 0, CA

x = 0,. . . ,

n−1

x = 0, t ≥ 0. Now, for t = 0, it follows that Ox = 0, which, since

(A, C) is observable, imp lies that x = 0 and, hence, leads to a contradiction.

Hence, P > 0.

Finally, we give necessary and s uﬃcient cond itions for Lyapunov

stability of the zero solution x(t) ≡ 0 to (3.152) in terms of the existence of

a positive-deﬁnite solution to the Lyapunov equation (3.157).

Theorem 3.18. Consider the linear dynamical system (3.152). The

zero solution x(t) ≡ 0 to (3.152) is Lyapunov stable if and only if there

exists a positive-deﬁnite matrix P ∈ R

n×n

and a non negative-deﬁnite matrix

R ∈ R

n×n

such that (3.157) holds.

Proof. Su ﬃciency is immediate fr om Theorem 3.1 with the Lyapunov

function candidate V (x) = x

P x. To show necessity note that if the zero

solution x(t) ≡ 0 to (3.152) is Lyapunov stable, then it follows from Theorem

3.15 that Re λ < 0, or Re λ = 0 and λ is semisimple, where λ ∈ spec(A).

Hence, it follows from the real Jordan decomposition [201] that A = SJS

−1

where S ∈ R

n×n

is a nonsingular matrix and J = block−diag[J

, J

, . . . , J

]

is the real Jordan form of A. Now, w ithout loss of generality, assume that J

, . . ., J

correspond to Jordan blocks of A with Re λ < 0, and J

r+1

, . . . , J

correspond to Jordan blo cks of A with Re λ = 0 and λ semisimple. Next,

it follows fr om Theorem 3.16 that there exists a positive-deﬁnite matrix P

such that J

< 0, where J

△

= block−diag[J

, . . . , J

]. Furthermore,

since J

r+1

, . . . , J

correspond to Jordan blo cks of A with Re λ = 0 and λ

semisimple, it follows that J

= block−diag[J

r+1

, . . . , J

] is skew-symmetric,

and hence, J

= 0, which implies that there exists a nonnegative-deﬁnite

matrix

R ∈ R

n×n

such that

0 = J

P +

P J +

R, (3.170)

where

P = block−diag[P

, I]. Now, forming S

−T

(3.170)S

−1

yields (3.157)

with P = S

−T

P S

−1

and R = S

−T

−1

Next, using the results of Section 3.2 and this section we provide a

key result on linearization of nonlinear systems. Speciﬁcally, we present

Lyapunov’s indirect method to draw conclusions about local stability of an

equilibrium point of a nonlinear system by examining the s tability of the

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STABILITY THEORY FOR NONLINEAR SYSTEMS 177

linearization of the nonlinear system about the equilibrium point in question.

We begin by considering (3.1) with f(0) = 0 and assume that f : D → R

is continuously diﬀerentiable. Next, expand ing f (x) via a Taylor expansion

about the equilibrium point x = 0 it follows that (3.1) can be written as

˙x(t) = f (0) +

∂f

∂x



x=0

x(t) + g(x(t)), x(0) = x

, t ≥ 0, (3.171)

where g(x)

△

= f(x) −

∂f

∂x



x=0

x. Note that since

∂f

∂x

is continuous it follows

that lim

kxk→0

kg (x)k

kxk

= 0. This shows that in a small neighborhood of the

origin a nonlinear system can be approximated by its linearization about

the origin. The f ollowing theorem kn own as Lyapunov’s indirect method

gives suﬃ cient conditions for stability of an equilibrium point of a nonlinear

system by examining the equilibrium point of the linearized system.

Theorem 3.19 (Lyapunov’s Indirect Theorem). Let x(t) ≡ 0 be an

equilibrium point f or the nonlinear dynamical system

˙x(t) = f(x(t)), x(0) = x

, t ≥ 0, (3.172)

where f : D → R

is continuously diﬀerentiable and D is an open set with

0 ∈ D. Furthermore, let

A =

∂f

∂x



x=0

Then the following statements hold:

i) I f Re λ < 0, where λ ∈ spec(A), then the zero solution x(t) ≡ 0 to

(3.172) is exponentially stable.

ii) If there exists λ ∈ spec(A) such that Re λ > 0, then the zero solution

x(t) ≡ 0 to (3.172) is unstable.

Proof. i) If R e λ < 0, λ ∈ spec(A), A is asymptotically stable, and

hence, it follows from Th eorem 3.16 that there exists a un iqu e positive-

deﬁnite matrix P ∈ R

n×n

satisfying (3.157). Next, choose the Lyapunov

function candidate V (x) = x

P x for the non linear system (3.172) and n ote

that the Lyapunov derivative along the system trajectories of (3.172) is given

V (x) = V

′

(x)f(x)

= 2x

P [Ax + g(x)]

= x

P + P A)x + 2x

P g(x)

= −x

Rx + 2x

P g(x), x ∈ D. (3.173)

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178 CHAPTER 3

Now, noting −x

Rx ≤ −λ

min

(R)kxk

and using the Cauchy-Schwarz

inequality it follows that

V (x) ≤ −λ

min

(R)kxk

+ 2λ

max

(P )kxk

kg(x)k

, x ∈ D. (3.174)

Next, since g(x), x ∈ D, is such that lim

kxk

→0

kg (x)k

kxk

= 0 it follows that for

every γ > 0, there exists ε > 0 such that kg(x)k

< γkxk

for all x ∈ B

(0).

Hence, (3.174) implies

V (x) < −[λ

min

(R) − 2γλ

max

(P )]kxk

, x ∈ B

(0). (3.175)

Now, choosing γ ≤ λ

min

(R)/2λ

max

(P ) it follows that

V (x) < 0, x ∈ B

(0),

x 6= 0, which proves local asymptotic stability. To show local exponential

stability it need only be noted that λ

min

(P )kxk

≤ V (x) ≤ λ

max

(P )kxk

ii) First, assume that A is such that (3.158) holds. Hence, it follows

from Lemma 3.2 that

P + P A = I

(3.176)

has a unique solution with P ∈ R

n×n

. By assumption, there exists λ ∈

spec(A) such that Re λ > 0. Next, assume that the remaining eigenvalues λ

of A are such that Re λ < 0; that is, the eigenvalues of A cluster into a group

of eigenvalues in the open right half plane and a group of eigenvalues in the

open left half plane. Now, it follows from the real Jordan decomposition

[201] that A = SJS

−1

, where S ∈ R

n×n

is a nonsingular matrix and J =

block−diag[J

, J

, . . . , J

] is the real Jordan form of A. Now, without loss

of generality, let J = block−diag[J

, J

−

], where J

corresponds to a block

diagonal matrix having Jordan blocks with Re λ > 0 and J

−

corresponds to a

block diagonal matrix having Jordan blocks with Re λ < 0. Next, since −J

and J

−

are asymptotically stable, it follows from Theorem 3.16 that there

exist positive-deﬁnite matrices P

and P

−

such that −J

−P

+I = 0

and J

−

+ P

−

+ I = 0, which implies that

0 = J

P +

P J + I, (3.177)

where

P = block−diag[−P

, P

−

]. Now, forming S

−T

(3.170)S

−1

yields

(3.176) with P = S

−T

P S

−1

. Hence, the solution P to (3.176) h as at least

one positive eigenvalue. Now, using identical arguments as in i) it can be

shown that for every suﬃciently small δ > 0 there exists x

∈ D such that

k < δ, V (x

) > 0, and V

′

(x)f(x) > 0, x ∈ B

(0), x 6= 0, where ε > 0 and

V (x) = x

P x. Hence, it follows from Theorem 3.12 th at the zero solution

x(t) ≡ 0 to (3.172) is unstable. The proof of the case in which there do

not exist λ ∈ s pec(A) such that Re λ < 0 follows identically with J = J

Finally, in the case where some of the eigenvalues of A lie on the ω-axis with

at least one eigenvalue in the right half plane, the proof follows by sh ifting

the imaginary axis and using the fact that the eigenvalues of a matrix are

continuous with respect to the entries of the matrix.

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STABILITY THEORY FOR NONLINEAR SYSTEMS 179

Theorem 3.19 gives a straightforward procedure for examining the

stability of a nonlinear system by examining the stability of the linearized

system. Furthermore, it follows from the proof of Theorem 3.19 th at if the

linearized system is asymptotically stable, then we can always construct

a quadratic Lyapunov function that guarantees local exponential stability

of the nonlinear system. To maximize the domain of attraction with a

quadratic Lyapunov fu nction it is also clear from the proof of Theorem 3.19

that the larger the ratio λ

min

(R)/λ

max

(P ) the larger the possible choice of ε.

Finally, it should be noted that if x

6= 0 is an equilibrium point of (3.172),

then Theorem 3.19 holds with A =

∂f

∂x



x=0

replaced by A =

∂f

∂x



x=x

Example 3.15. Consider the dynamical system given in Example 3.2

describing the motion of a simple pendulum with viscous damping given by

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.178)

˙x

(t) = −

sin x

(t) − x

(t), x

(0) = x

. (3.179)

This model has countably inﬁnitely many equilibrium points in R

given by

, x

) = (nπ, 0), n = 0, ±1, ±2, . . .. Here, we examine the stability of the

physical equilibria given by (0, 0) and (π, 0) using Lyapunov’s linearization

method. For (3.178) and (3.179) the Jacobian matrix

∂f

∂x

is given by

∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x



0 1

−

cos x

−1



. (3.180)

Note that

∂f

∂x



x=(0,0)



0 1

−

−1



, A

∂f

∂x



x=(π,0)



0 1

−1



(3.181)

Furth ermore, since tr A

< 0, det A

> 0, tr A

< 0, an d det A

< 0,

it follows from Theorem 3.19 that the zero solution (x

(t), x

(t)) ≡ (0, 0)

to (3.178) and (3.179) is locally exponentially stable while the solution

(t), x

(t)) ≡ (π, 0) to (3.178) and (3.179) is unstable. △

Example 3.16. Consider the Van der Pol oscillator given by

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.182)

˙x

(t) = −µ(1 −x

(t))x

(t) −x

(t), x

(0) = x

, (3.183)

where µ ∈ R. Note that (3.182) and (3.183) has a un ique equilibrium at

, x

) = (0, 0). Furthermore, the Jacobian matrix

∂f

∂x

evaluated at (0, 0)

is given by

A =

∂f

∂x



x=(0,0)



0 1

−1 −µ



. (3.184)

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180 CHAPTER 3

Now, it follows from Theorem 3.19 that if µ < 0, then the zero solution

(t), x

(t)) ≡ (0, 0) to (3.182) and (3.183) is unstable and if µ > 0, then

the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.182) and (3.183) is locally

exponentially stable. △

Finally, we apply Lyapunov’s in direct method to the stabilization of a

nonlinear dynamical system. Speciﬁcally, consider the nonlinear controlled

system

˙x(t) = F (x(t), u(t)), x(0) = x

, t ≥ 0, (3.185)

where x(t) ∈ R

, u(t) ∈ R

, t ≥ 0, and F : R

× R

→ R

. Here, we seek

feedback controllers of the form u(t) = φ(x(t)), wh ere φ : R

→ R

and

φ(0) = 0, so that the zero s olution x(t) ≡ 0 of the closed-loop system given

˙x(t) = F (x(t), φ(x(t))), x(0) = x

, t ≥ 0, (3.186)

is asymptotically stable.

Theorem 3.20. Consider the nonlinear controlled system (3.185)

where F : R

× R

→ R

is continuously diﬀerentiable and F (0, 0) = 0.

Furth ermore, deﬁne

△

∂F

∂x



(x,u)=(0,0)

, B

△

∂F

∂u



(x,u)=(0,0)

, (3.187)

and assume (A, B) is stabilizable. Then, there exists a matrix K ∈ R

m×n

such that spec(A + BK) ⊂ C

−

. In addition, with the linear control law

u(t) = Kx(t), the zero solution x(t) ≡ 0 of the closed-loop nonlinear system

˙x(t) = F (x(t), Kx(t)), x(0) = x

, t ≥ 0, (3.188)

is locally exponentially stable.

Proof. First, note that the closed-loop system (3.188) has the form

˙x(t) = F (x(t), Kx(t)) = f (x(t)), x(0) = x

, t ≥ 0. (3.189)

Next, it follows that

∂f

∂x



x=0



∂F

∂x

(x, Kx) +

∂F

∂u

(x, Kx)K





x=0

= A + BK, (3.190)

and hence, since (A, B) is stabilizable, K can be chosen so that A + BK

is asymptotically stable. Now, the result is a direct consequence of i) of

Theorem 3.19.

It follows f rom Theorem 3.20 that if the linearization of (3.185) is

stabilizable, that is, (A, B) is stabilizable, then we can always construct

a Lyapun ov function for the nonlinear closed-loop system (3.186). In

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STABILITY THEORY FOR NONLINEAR SYSTEMS 181

particular, since there exists K ∈ R

m×n

such that A+ BK is asymptotically

stable, then for every R > 0 the Lyapunov equation

0 = (A + BK)

P + P (A + BK) + R, (3.191)

is guaranteed to have a unique positive-deﬁnite solution by Theorem 3.16.

Hence, the quadratic Lyapunov function V (x) = x

P x is a Lyapunov

function for the closed-loop nonlinear system (3.186) that guarantees local

asymptotic stability.

A similar approach can be used to design dynamic ou tp ut feedback

controllers for the nonlinear s y s tem (3.185). Speciﬁcally, cons ider the

nonlinear dynamical system

˙x(t) = F (x(t), u(t)), x(0) = x

, t ≥ 0, (3.192)

y(t) = h(x(t), u(t)), (3.193)

where x(t) ∈ R

, u(t) ∈ R

, y(t) ∈ R

, t ≥ 0, F : R

× R

→ R

is such

that F (0, 0) = 0, and h : R

× R

→ R

is such that h(0, 0) = 0. Next,

linearizing (3.192) and (3.193) about x = 0 and u = 0 yields

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (3.194)

y(t) = Cx(t) + Du(t), (3.195)

where A and B are given by (3.187) and

△

∂h

∂x



(x,u)=(0,0)

, D

△

∂h

∂u



(x,u)=(0,0)

. (3.196)

Assuming (A, B) is stabilizable and (A, C) is detectable, then we can design

a full-order dynamic compensator of the form

˙x

(t) = A

(t) + B

y(t), x

(0) = x

, t ≥ 0, (3.197)

u(t) = C

(t), (3.198)

where x

(t) ∈ R

, t ≥ 0, A

∈ R

n×n

, B

∈ R

n×l

, and C

∈ R

m×n

, such that

with A

= A + BC

− B

C − B

A =



A BC

C A

+ B



(3.199)

is asymptotically stable. The asymptotic stability of

A follows from the

fact that (A, B) is stabilizable and (A, C) is detectable, and hence, there

exist B

∈ R

n×l

and C

∈ R

m×n

such that A + BC

and A − B

C are

asymptotically stable.

Now, applying the controller (3.197) and (3.198) to the nonlinear

system (3.192) and (3.193) gives the closed-loop system

˙x(t) = F (x(t), C

(t)), x(0) = x

, t ≥ 0, (3.200)

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182 CHAPTER 3

˙x

(t) = A

(t) + B

h(x, C

(t)), x

(0) = x

. (3.201)

Note that since F (0, 0) = 0 and h(0, 0) = 0, (x, x

) = (0, 0) is an equilibrium

point of (3.200) and (3.201). Furthermore, linearizing (3.200) and (3.201)

about x = 0 and x

= 0 yields

˜x(t) =

A˜x(t), ˜x(0) = ˜x

, t ≥ 0, (3.202)

where ˜x

△

= [x

, x

]

and

A is given by (3.199). Hence, choosing the

compensator triple (A

, B

, C

) such that the zero solution ˜x(t) ≡ 0 to

(3.202) is asymptotically stable, it follows from Th eorem 3.19 that the

nonlinear closed-loop system (3.200) and (3.201) is locally exponentially

stable. Furthermore, a Lyapunov function guaranteeing local exponential

stability of (3.200) and (3.201) is given by V (˜x) = ˜x

P ˜x, where

P is the

2n × 2n positive deﬁnite solution to the Lyapunov equation

0 =

P +

A +

R, (3.203)

where

R is any 2n × 2n positive-deﬁnite matrix.

3.8 Problems

Problem 3.1. Give an example of a function that is positive deﬁnite

but n ot radially unbounded. Alternatively, given an example of a function

that is radially unbounded but not positive deﬁnite.

Problem 3.2. Let V : D ⊆ R

→ R be such that V (·) is two-times

continuously diﬀerentiable, V (0) = 0, V

′

(0) = 0, and V

′′

(0) > 0. Show that

there exists an open set D

⊂ D such that 0 ∈ D

and V (x) > 0, x ∈ D

x 6= 0.

Problem 3.3. Let V : R

→ R be such that V (·) is continuously

diﬀerentiable, V (0) = 0, V (x) > 0, x ∈ D ⊂ R

, x 6= 0, and kV

′

(x)k > 0,

x ∈ R

, x 6= 0. Show that V (·) is globally positive deﬁnite with a unique

global minimum at x = 0.

Problem 3.4. Consider the dynamical system (3.1) with f(x) = Ax,

where A ∈ R

n×n

. Show that th e following statements are equivalent:

i) The zero solution x(t) ≡ 0 to (3.1) is Lyapunov stable.

ii) For every initial condition x

∈ R

, x(t), t ≥ 0, is bounded.

iii) If λ ∈ spec(A), then either R e λ < 0, or both R e λ = 0 and λ is

semisimple.

iv) There exists α > 0 such that ke

k < α, t ≥ 0.

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STABILITY THEORY FOR NONLINEAR SYSTEMS 183

Problem 3.5. Consider the dynamical system (3.1) with f (x) = Ax,

where A ∈ R

n×n

. Show that th e following statements are equivalent:

i) The zero solution x(t) ≡ 0 to (3.1) is globally asymptotically stable.

ii) For every initial condition x

∈ R

, lim

t→∞

x(t) = 0.

iii) If λ ∈ spec(A), then Re λ < 0.

iv) lim

t→∞

= 0.

Problem 3.6. Consider the dynamical system (3.1) with f (x) = Ax,

where A ∈ R

n×n

. Show that if there exist a positive-deﬁnite matrix P ∈

n×n

and a scalar α > 0 such that

0 = (A + αI

)

P + P (A + αI

) + R, (3.204)

where R is a positive-deﬁnite matrix, then the eigenvalues of A have real

part less than −α.

Problem 3. 7. Let A ∈ R

n×n

. A is essentially nonnegative if A

(i,j)

≥ 0,

i, j = 1, . . . , n, i 6= j. Consider the d ynamical system (3.1) with f(x) = Ax,

where A ∈ R

n×n

. Show that R

is an invariant set with respect to (3.1) if

and only if A is essentially nonnegative.

Problem 3.8. The nonlinear dynamical system (3.1) is nonnegative if

for every x(0) ∈ R

, the solution x(t), t ≥ 0, to (3.1) is nonnegative, that

is, x(t) ≥≥ 0, t ≥ 0. The equilibrium solution x(t) ≡ x

of a nonnegative

dynamical system is Lyapunov stable if, for every ε > 0, there exists δ =

δ(ε) > 0 such that if x

∈ B

) ∩ R

, then x(t) ∈ B

) ∩ R

, t ≥ 0.

The equilibrium solution x(t) ≡ x

of a nonnegative dynamical system is

asymptotically stable if it is L yapunov stable and there exists δ > 0 such

that if x

∈ B

) ∩ R

, then lim

t→∞

x(t) = x

. Consider the dynamical

system (3.1) with f (x) = Ax, where A ∈ R

n×n

is essentially nonn egative

(see Problem 3.7). Show that the following statements hold:

i) I f there exist vectors p, r ∈ R

such that p >> 0 and r ≥≥ 0 satisfy

0 = A

p + r, (3.205)

then A is Lyapunov stable.

ii) If there exist vectors p, r ∈ R

such th at p ≥≥ 0 and r ≥≥ 0 satisfy

(3.205) and (A, r

) is obs ervable, then p >> 0 and A is asymp totically

stable.

Furth ermore, show that the f ollowing statements are equ ivalent:

NonlinearBook10pt November 20, 2007

184 CHAPTER 3

iii) A is asymptotically stable.

iv) There exist vectors p, r ∈ R

such that p >> 0 and r >> 0 satisfy

(3.205).

v) There exist vectors p, r ∈ R

such that p ≥≥ 0 and r >> 0 satisfy

(3.205).

vi) For every r ∈ R

such that r >> 0, there exists p ∈ R

such that

p >> 0 satisﬁes (3.205).

(Hint: Use the Lyapunov function candidate V (x) = p

x in your analysis

and show that th e Lyapunov stability theorem of Section 3.2 and the

invariant set theorems of Section 3.3 can be used directly for nonnegative

systems with the required suﬃcient conditions veriﬁed on R

Problem 3.9. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.206)

˙x

(t) = −k sin x

(t), x

(0) = x

, (3.207)

where k > 0. Show that the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.206)

and (3.207) is L yapunov stable.

Problem 3.10. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + αx

(t)[x

(t) + x

(t)], x

(0) = x

, t ≥ 0, (3.208)

˙x

(t) = −x

(t) + αx

(t)[x

(t) + x

(t)], x

(0) = x

, (3.209)

where α < 0. Show that the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.208)

and (3.209) is globally asymptotically stable.

Problem 3.11. Consider the nonlinear dynamical system

˙x

(t) = [x

(t) −α

(t)][x

(t) + x

(t) −1], x

(0) = x

, t ≥ 0, (3.210)

˙x

(t) = [α

(t) + x

(t)][x

(t) + x

(t) −1], x

(0) = x

, (3.211)

where α

> 0 and α

> 0. Show th at the zero solution (x

(t), x

(t)) ≡ (0, 0)

to (3.210) and (3.211) is asymptotically stable. Is the result global? If not,

give a characterization for the domain of attraction.

Problem 3.12. Consider the nonlinear dynamical system

˙x

(t) = −2x

(t) −x

(t), x

(0) = x

, t ≥ 0, (3.212)

˙x

(t) = −x

(t) −x

(t), x

(0) = x

. (3.213)

Using the quadratic Lyapunov function candidate V (x

, x

) = x

+ x

show that the zero s olution (x

(t), x

(t)) ≡ (0, 0) to (3.212) and (3.213)

is asymptotically stable. Find the maximum domain of attraction D

for