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

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STABILITY THEORY FOR NONLINEAR SYSTEMS 195

exists δ > 0 such that if x

∈ B

(x), then lim

t→∞

s(t, x

) exists and s(t, x

t ≥ 0, converges to a Lyapunov stable equilibrium point. Th e nonlinear

dynamical system (3.1) is semistable if every equilibrium point of (3.1) is

semistable. Suppose the orbit O

of (3.1) is bounded for all x ∈ D and

assume that there exists a continuously diﬀerentiable function V : D → R

such that

′

(x)f(x) ≤ 0, x ∈ D. (3.289)

Let M denote the largest invariant set contained in R

△

= {x ∈ D : V

′

(x)f(x)

= 0}. Show that if M ⊆ {x ∈ D : f(x) = 0 and x is Lyapunov stable},

then (3.1) is semistable. (Hint: First show that if the positive orbit

of (3.1) contains a Lyapunov stable equilibr ium point y, then y =

lim

t→∞

s(t, x) for all x ∈ D

, where D

⊆ D is a positively invariant set

with respect to (3.1).)

Problem 3.45. Consider the dynamical system (3.1) with f(x) = Ax,

where A ∈ R

. Show that if there exists n × n matrices P ≥ 0 and R ≥ 0

such that

0 = A

P + P A + R, (3.290)













n−1













= N(A), (3.291)

then the zero s olution x(t) ≡ 0 to (3.1) is semistable (see Problem 3.44).

(Hint: First show that N(P ) ⊆ N(A) ⊆ N(R) and N(A) ∩R(A) = {0}.)

Problem 3.46. Consider the dynamical system (3.1) with f(x) = Ax,

where A ∈ R

n×n

. Using the results of Problem 3.44 show that the following

statements are equ ivalent:

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

ii) For every initial condition x

∈ R

, lim

t→∞

x(t) exists.

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

semisimple.

iv) lim

t→∞

exists.

Problem 3.47. Consider the nonlinear dynamical system

˙x

(t) = −2x

(t), x

(0) = x

, t ≥ 0, (3.292)

˙x

(t) = −2x

(t) + 2x

(t)x

, x

(0) = x

. (3.293)

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196 CHAPTER 3

Use the variable gradient m ethod to construct a Lyapunov function for

(3.292) and (3.293). Is the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.292)

and (3.293) globally asymptotically stable?

Problem 3.48. Consider the nonlinear dynamical system

˙x

(t) =

−6x

(t)

[1 + x

(t)]

+ 2x

(t), x

(0) = x

, t ≥ 0, (3.294)

˙x

(t) = −2x

(t) −

(t)

[1 + x

(t)]

, x

(0) = x

. (3.295)

Use the variable gradient m ethod to construct a Lyapunov function for

(3.294) and (3.295). Is the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.294)

and (3.295) globally asymptotically stable?

Problem 3.49. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, t ≥ 0, (3.296)

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, (3.297)

˙x

(t) = αx

(t) − x

(t), x

(0) = x

. (3.298)

Using Lyapun ov’s indirect method , investigate the stability of the zero

solution (x

(t), x

(t)) ≡ (0, 0, 0) to (3.296)–(3.298) for α > 0, α < 0,

and α = 0.

Problem 3.50. Consider the nonlinear Lienard system given in Prob-

lem 3.18 with g(x) = x and f(0) > 0. Use Lyapun ov’s indirect theorem

(Theorem 3.19) to draw conclusions about the local stability of the zero

solution (x(t), ˙x(t)) ≡ (0, 0) to (3.221).

Problem 3.51. Consider the nonlinear dynamical system

˙x

(t) = f

(t)) + f

(t)), x

(0) = x

, t ≥ 0, (3.299)

˙x

(t) = x

(t) + ax

(t), x

(0) = x

, (3.300)

where f

(0) = f

(0) = 0, a ∈ R, and f

, f

are continuously diﬀerentiable

functions. Using Krasovskii’s theorem (Theorem 3.6) ﬁn d conditions on

′

) and f

′

) for all (x

, x

) ∈ R × R such that V (x

, x

) = f

) +

) serves as a Lyapunov function for (3.299) and (3.300).

Problem 3.52. Prove i) of Theorem 3.19 using Krasovskii’s theorem

(Theorem 3.6).

Problem 3.53. Consider the nonlinear dynamical sys tem (3.1) with

f(x) = A(x)x, where A : R

→ R

n×n

is a continuous matrix-valued function.

Show that if th ere exist positive deﬁnite m atrices P ∈ R

n×n

and R ∈ R

n×n

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STABILITY THEORY FOR NONLINEAR SYSTEMS 197

such that

0 = A

(x)P + P A(x) + R, x ∈ R

, x 6= 0, (3.301)

then the zero solution x(t) ≡ 0 to (3.1) is globally asymptotically stable.

Furth ermore, prove or refute that if, for each x ∈ R

, all of the eigenvalues

of A(x) lie in the open left half complex plane, then the zero solution x(t) ≡ 0

to (3.1) is globally asymptotically s table.

Problem 3.54. Consider the scalar nonlinear system

˙x(t) = x

(t) −x(t), x(0) = 0, t ≥ 0. (3.302)

Using Zubov’s method construct a Lyapunov function for (3.302) and

provide a characterization for the domain of attraction of the zero solution

x(t) ≡ 0 to (3.302).

Problem 3.55. Consider the nonlinear dynamical system representing

a rigid spacecraft given by (3.11)–(3.13) in Example 3.1. Use the energy-

Casimir method to show that the equilibrium solution x(t) ≡ x

, w here

= [x

, 0, 0]

, to (3.11)–(3.13) is Lyapunov stable.

Problem 3.56. Consider the controlled rigid s pacecraft given by

(3.215) –(3.217) in Problem 3.15 with u

(t) ≡ 0 and u

(t) ≡ 0. Show

that with u

(t) = −k

(t)x

(t), where k > c and c

△

= (I

− I

)/(I

the closed-loop system can be written as

˙z

(t) = a

(t)z

(t), z

(0) = z

, t ≥ 0, (3.303)

˙z

(t) = a

(t)z

(t), z

(0) = z

, (3.304)

˙z

(t) = cz

(t)z

(t), z

(0) = z

, (3.305)

where z

= I

, z

= I

, z

= (c/(c − k))I

, a

△

= (c − k)/c < 0,

△

= (I

−I

)/(I

), and b

△

= (I

−I

)/(I

). Furthermore, us e the en ergy-

Casimir method to show that the equilibrium solution z(t) ≡ z

, where

= [0, z

, 0]

and z

= I

, to (3.303)–(3.305) is Lyapunov stable.

(Hint: Sh ow that C

(z) =

(

+ a

) and C

(z) =

+ z

+ a

)

are Casimir functions for (3.303)–(3.305).)

Problem 3.57. In the spread of epidemics in large populations the

basic susceptible-infected-removed (SIR) model has been p roposed in the

literature. In particular, the model of a SIR epidemic is given by

˙x

(t) = −

(t)x

(t) −µx

(t) + u, x

(0) = x

, t ≥ 0, (3.306)

˙x

(t) =

(t)x

(t) −(γ + µ)x

(t), x

(0) = x

, (3.307)

˙x

(t) = γx

(t) −µx

(t), x

(0) = x

, (3.308)

where x

denotes the number of su sceptibles, x

denotes the number of

infectives, x

denotes the number of immun es, µ > 0 is the death rate

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198 CHAPTER 3

co eﬃcient, N is a constant denoting the total size of the population, that is,

(t) + x

(t) = N, u is the rate of recruitment of new members into

the susceptible pool and is assumed to be a constant that just makes up for

the deaths, that is, u = µN, γ > 0 is a rate constant for recovery, and λ > 0

is the mean constant rate per person for contacts that transmit the disease.

For α ≤ 1 and α > 1, where α

△

= λ/(γ+µ), compute all equilibria of (3.306)–

(3.308). Using L yapunov’s linearization metho d, analyze the stability of

these equilibria. Repeat the above analysis for the case corresponding to a

zero death rate, that is, µ = 0. (Hint: Note that s ince x

(t)+x

(t) =

N, (3.308) is superﬂuous and need not be considered in the analysis.)

Problem 3.58. Consider the model of two populations which interact

in a predator-prey relationship given by the Lotka-Volterra equations

˙x

(t) = αx

(t) −βx

(t)x

(t), x

(0) = x

, t ≥ 0, (3.309)

˙x

(t) = −γx

(t) + δx

(t)x

(t), x

(0) = x

, (3.310)

where α, β, γ, δ > 0, x

denotes the number of prey, and x

denotes the

number of predators. Compute all equilibria of (3.309) and (3.310). Analyze

the stability of these equilibria. (Hint: In the case wh ere Lyapunov’s

linearization method provides no stability information u se the Lyapunov

function candidate V (x) = E(x)−E(x

), where E(x) = δx

+βx

−γ ln x

−

α ln x

. Make sure you j ustify that V (x), x ∈ R

, is a valid Lyapunov

function candidate.)

Problem 3.59. Consider the nonlinear dynamical system (3.1). Show

that if the zero solution x(t) ≡ 0 to (3.1) is asymptotically stable and

f : D → R

is continuously diﬀerentiable, then V : D → R satisfying

the conditions of Theorem 3.9 additionally satisﬁes kV

′

(x)k ≤ ν(kxk),

x ∈ B

(0), where ν : [0, δ] → [0, ∞) is a class K function.

Problem 3.60. Consider the nonlinear dynamical system (3.1). Show

that if the zero solution x(t) ≡ 0 to (3.1) is exponentially stable and f : D →

is continuously diﬀerentiable, then V : D → R satisfying the conditions

of Theorem 3.10 additionally satisﬁes kV

′

(x)k ≤ νkxk

p−1

, x ∈ B

(0), where

ν > 0. Can p = 2 without loss of generality? Justify your answer.

Problem 3.61. Let f = [f

, . . . , f

]

: D → R

, where D is an

open subset of R

that contains R

. Then f is essentially nonnegative if

(x) ≥ 0, for all i = 1, . . . , n, and x ∈ R

such that x

= 0, where x

denotes the ith component of x. Consider the nonlinear dynamical system

(3.1). Show that the following statements hold:

i) Suppose R

⊂ D. Then R

is an invariant set with respect to (3.1) if

and only if f : D → R

is essentially nonn egative.

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STABILITY THEORY FOR NONLINEAR SYSTEMS 199

ii) Suppose f(0) = 0 and f : D → R

is essentially nonnegative and

continuously diﬀerentiable on R

. Then A

△

∂f

∂x



x=0

is essentially

nonnegative (see Problem 3.7).

iii) If f(x) = Ax, where A ∈ R

n×n

, then f is essentially nonnegative if

and only if A is essentially nonnegative.

Problem 3.62. Let f : [0, ∞) → R. Show that if f (·) ∈ L

∩ L

∞

and

f(·) is bounded, then lim

t→∞

f(t) = 0.

Problem 3.63. Consider the nonlinear gradient dynamical system giv-

en by

˙x(t) = −



∂V

∂x

(x(t))



, x(0) = x

, t ≥ 0, (3.311)

where x ∈ D ⊆ R

and V : D → R is a two-times continuously diﬀerentiable

function. Show that:

V (x) ≤ 0, x ∈ D, and

V (x) = 0 if and only if x is an equilibrium point

of (3.311).

ii) Let D

△

= {x ∈ R

: V (x) ≤ c} be compact for every c > 0 and let

∈ D

. Then there exists a uniqu e solution to (3.311) that is deﬁned

for all t ≥ 0.

iii) Let

∂V

∂x

= 0 for a ﬁnite numb er of points p

, . . . , p

. Then for every

solution x(t), t ≥ 0, of (3.311), lim

t→∞

x(t) = p

, i ∈ {1, . . . , q}.

iv) If x

∈ D is an isolated minimizer of V (·), then x

is an asymptotically

stable equilibrium point of (3.311).

Problem 3.64. Consider the nonlinear dynamical system

˙x

(t) = f

(t), x

(t)), x

(0) = x

, t ≥ 0, (3.312)

˙x

(t) = f

(t), x

(t)), x

(0) = x

, (3.313)

where f

(·, ·) and f

(·, ·) are continuously diﬀerentiable functions. Show

that a necessary condition for (3.312) and (3.313) to be a gradient system

(see Problem 3.63) is

∂f

∂x

∂f

∂x

. Is this condition also suﬃcient? For

, x

) = x

+ 2x

and f

, x

) = x

+ x

−x

show that (3.312) and

(3.313) is a gradient system and ﬁnd its potential function V (x

, x

Problem 3.65. Consider the nonlinear functional dynamical system

˙x(t) = f (x(t), x(t − τ

)), x(θ) = φ(θ), −τ

≤ θ ≤ 0, (3.314)

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200 CHAPTER 3

where x ∈ R

, f : R

× R

→ R

is Lipschitz continuous and satisﬁes

f(0, 0) = 0, and φ ∈ C([−τ

, 0], R

), where C([−τ

, 0], R

) is a Banach

space of continuous functions mapp ing the interval [−τ

, 0] into R

with

topology of uniform convergence and designated norm given by |||φ||| =

sup

−τ

≤θ≤0

kφ(θ)k. Let φ : [−τ

, 0] → R

be a continuous vector-

valued function specifying the initial state of the system. Furtherm ore, let

∈ C([−τ

, 0], R

) deﬁned by x

(θ)

△

= x(t + θ), θ ∈ [−τ

, 0], denote the

(inﬁnite-dimensional) state of (3.314) at time t corresponding to the piece

of trajectories x between t − τ

and t or, equivalently, the element x

the space of continuous functions d eﬁ ned on the interval [−τ

, 0] and taking

values in R

. The zero solution x

≡ 0 of (3.314) is Lyapunov stable if, for

all ε > 0, there exists δ > 0 such that kφ(θ)k < δ, θ ∈ [−τ

, 0], implies

kx(t, φ(θ))k < ε for t ≥ 0, where x(t, φ(θ)) denotes the solution to (3.314).

The zero solution x

≡ 0 of (3.314) is asymptotically stable if it is Lyapunov

stable and there exists δ > 0 such that kφ(θ)k < δ, θ ∈ [−τ

, 0], implies

that lim

t→∞

kx(t, φ(θ))k = 0. Show th at if there exists a continuously

diﬀerentiable Lyapunov functional V : C([−τ

, 0], R

) → R such that

α(kψ(0)k) ≤ V (ψ), (3.315)

V (ψ) ≤ 0, (3.316)

where ψ ∈ C([−τ

, 0], R

), α(·) is a class K

∞

function, and

V (ψ)

△

= lim

h→0

[V (x

(ψ)) − V (ψ)], then th e zero solution x

≡ 0 to (3.314) is Lyapunov

stable. If, in addition, there exists a class K

∞

function γ(·) such that

V (ψ) ≤ −γ(kψ(0)k), (3.317)

show that the zero solution x

≡ 0 to (3.314) is asymptotically stable.

Problem 3.66. A mapping T : D → R

is a diﬀeomorphism on

D ⊆ R

if it is invertible on D, that is, there exists a function T

−1

(x)

such that T

−1

(T (x)) = x, x ∈ D, and T (x) and T

−1

(x) are continuously

diﬀerentiable. Consider the nonlinear dynamical system (3.1) and let

T : R

→ R

be a diﬀeomorphism in the neighborhood of the origin such

that z = T (x) and T (0) = 0. Show that the zero solution z(t) ≡ 0

of the transformed nonlinear system ˙z(t) =

f(z(t)), z(0) = z

, where

f(z) =

∂T (x)

∂x

f(x)



x=T

−1

(z)

, is Lyapunov stable (respectively, asymptotically

stable) if and only if the zero solution x(t) ≡ 0 to (3.1) is Lyapunov stable

(respectively, asymptotically stable).

Problem 3.67. Consider the nonlinear dynamical system

˙x(t) = f (x(t)), x(0) = x

, t ≥ 0, (3.318)

where f : D ⊆ R

→ R is continuous ly diﬀerentiable w ith 0 ∈ D. Ass ume

there exist continuous model validity functions w

: R

→ R, i = 1, . . . , r,

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STABILITY THEORY FOR NONLINEAR SYSTEMS 201

such that (3.318) can be r ep resented by

˙x(t) =

i=1

(x(t))A

x(t), (3.319)

where w

(x) > 0, i = 1, . . . , r, and A

∂f

∂x



x=x

. Show that if there exists

a single positive-deﬁnite matrix P ∈ R

n×n

satisfying

P + P A

< 0, i = 1, . . . , r, (3.320)

then the zero solution x(t) ≡ 0 to (3.318) is globally asymptotically stable.

Problem 3.68. Show that Theorem 3.7 also holds in the case where

(3.85) is replaced by

′

(x)f(x)

[1 − V (x)][1 + kf(x)k

]

1/2

< 0, x ∈ D. (3.321)

Problem 3.69 (Convergence Lemma). Let V : [0, ∞) → R be a

continuously diﬀerentiable function such that

V (t) + αV (t) ≤ 0, t ≥ 0, (3.322)

where α ∈ R. Show that V (t) ≤ V (0)e

−αt

, t ≥ 0.

Problem 3.70. Consider the nonlinear dynamical system given in

Example 3.7. Show that if kx(0)k

< 1, then kx(t)k

< αe

−2t

, t ≥ 0,

α > 0. Alternatively, show that if kx(0)k

> 1, then kx(t)k tends to inﬁnity

in ﬁnite time. (Hint: Use the convergence lemma of Problem 3.69.)

Problem 3.71. Let γ

: [0, a) → [0, ∞) and γ

: [0, a) → [0, ∞) be

class K functions, γ

: [0, ∞) → [0, ∞) and γ

: [0, ∞) → [0, ∞) be class K

∞

functions, and β : [0, a) × [0, ∞) → [0, ∞) be a class KL function. Show

that the following statements hold:

i) γ

(σ

+ σ

) ≤ γ

(2σ

) + γ

(2σ

) for all σ

, σ

∈ [0, a).

ii) γ

−1

: [0, γ

(a)) → [0, ∞) and γ

−1

(·) ∈ K.

iii) γ

−1

: [0, ∞) → [0, ∞) and γ

−1

(·) ∈ K

∞

iv) γ

◦ γ

∈ K.

v) γ

◦ γ

∈ K

∞

vi) γ

(β(γ

(·), ·)) ∈ KL .

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202 CHAPTER 3

Problem 3.72. Let V : D ⊆ R

→ R be a continuous function such

that V (0) = 0, V (x) > 0, x ∈ D, x 6= 0. Show that there exist class K

functions α(·) and β(·) deﬁned on [0, ε] such that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ B

(0), (3.323)

where B

(0)

△

= {x ∈ D : kxk < ε}. Furthermore, show that if D = R

and

V (x) → ∞ as kxk → ∞, then α(·) and β(·) can be chosen to be class K

∞

and (3.323) holds for all x ∈ R

Problem 3.73. Show that if the zero solution x(t) ≡ 0 of the nonlinear

dynamical system (3.1) is asymptotically stable with Lyapunov function

V (x), x ∈ D, then there exist class K functions α(·), β(·), and γ(·) deﬁned

on [0, ε] such that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ B

(0), (3.324)

′

(x)f(x) ≤ −γ(kxk) < 0, x ∈ B

(0), x 6= 0. (3.325)

Problem 3.74. Suppose the zero solution x(t) ≡ 0 to (3.1) is

asymptotically stable. Show that the domain of attraction D

of (3.1) is

an open, connected invariant set.

Problem 3.75. Show that the zero solution x(t) ≡ 0 to (3.1) is

Lyapunov stable if and only if there exist δ > 0 and a class K function

α(·) such that if kx

k < δ, then kx(t)k ≤ α(kx

k), t ≥ 0.

Problem 3.76. Show that the zero solution x(t) ≡ 0 to (3.1) is

asymptotically stable (respectively, globally asymp totically stable) if and

only if there exist δ > 0 and class K and L functions α(·) and β(·),

respectively, such that if kx

k < δ (respectively, x

∈ R

), then kx(t)k ≤

α(kx

k)β(t), t ≥ 0.

Problem 3.77. Show that the zero solution x(t) ≡ 0 to (3.1) with

f(x) = Ax, where A ∈ R

n×n

, is asymptotically stable if and only if it is

exponentially stable.

Problem 3.78. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (3.326)

˙x

(t) = x

(t) + x

(t) −x

(t)x

(t), x

(0) = x

. (3.327)

Show that the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.326) and (3.327) is

unstable.

Problem 3.79. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (3.328)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 203

˙x

(t) = −x

(t) −x

(t), x

(0) = x

. (3.329)

Show that the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.328) and (3.329) is

unstable. (Hint: Consider the function V (x

, x

) = (2x

− x

)

− x

Problem 3.80. Consider the nonlinear dynamical system

˙x

(t) = αx

(t), x

(0) = x

, t ≥ 0, (3.330)

˙x

(t) = −βx

(t) + µ[γ − x

(t)]

(t), x

(0) = x

, (3.331)

where α, β, γ, and µ are positive constants. Use Theorem 3.12 to show that

the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.330) and (3.331) is unstable.

(Hint: Use Bendixson’s theorem to show that no limit cycles exist in the

simply connected region −γ

1/4

< x

< γ

1/4

Problem 3.81. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t), x

(0) = x

, t ≥ 0, (3.332)

˙x

(t) = x

(t) −x

(t) + x

(t)x

(t), x

(0) = x

. (3.333)

Use Chetaev’s instability theorem to show that the zero solution (x

(t),

(t)) ≡ (0, 0) to (3.332) and (3.333) is unstable. (Hint: Consider the

function V (x

, x

) = x

and the subregion Q = {(x

, x

) ∈ R

: x

0, x

> 0, and x

+ x

< 1}.)

Problem 3.82. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (3.334)

˙x

(t) = −x

(t) + x

(t)x

(t) −x

(t), x

(0) = x

. (3.335)

Using Chetaev’s in stability theorem show that the zero solution (x

(t), x

(t))

≡ (0, 0) to (3.334) and (3.335) is unstable. (Hint: Consider th e function

V (x

, x

) =

−

Problem 3.83. Consider the nonlinear controlled dynamical system

˙x

(t) = x

(t)x

(t), x

(0) = x

, t ≥ 0, (3.336)

˙x

(t) = x

(t) + u(t), x

(0) = x

. (3.337)

Show that there does n ot exist a linear feedback control law (that is, u(t) =

−k

(t) − k

(t)) that renders the zero s olution (x

(t), x

(t)) ≡ (0, 0) to

(3.336) and (3.337) locally asymptotically stable. (Hint: First try using

Lyapunov’s indirect method. If this fails to yield any information, use

Chetaev’s instability theorem with function V (x

, x

) =

− x

).)

Problem 3.84. Consider the nonlinear dynamical system

˙x

(t) = 2x

(t) + x

(t), x

(0) = x

, t ≥ 0, (3.338)

˙x

(t) = x

(t) −x

(t) + x

(t)x

(t), x

(0) = x

. (3.339)

NonlinearBook10pt November 20, 2007

204 CHAPTER 3

Use Chetaev’s instability theorem to show that the zero solution (x

(t),

(t)) ≡ (0, 0) to (3.338) and (3.339) is unstable. (Hint: C onsider the

function V (x

, x

) = x

and the subregion Q = {(x

, x

) ∈ R

: x

0, x

> 0, and x

+ x

< 1}.)

Problem 3.85. Use Chetaev’s instability theorem to show that the

rigid spacecraft dynamical system spinning about its intermediate axis

considered in Problem 3.14 is unstable. (Hint: Translate the system

co ordinates so that the equilibrium (0, x

, 0), x

6= 0, becomes the

origin. Then use the function V (x

, x

) = x

and the subregion

Q = {(x

, x

− x

, x

) ∈ B

ε/2

(0) : x

> 0, x

> 0}.)

Problem 3.86. Prove Theorem 3.12 using Chetaev’s instability theo-

rem. (Hint: Consider the subregion Q = {x ∈ B

(0) : V (x) > 0}, where

(0) ⊂ D.)

Problem 3.87. Prove Theorem 3.13 using Chetaev’s instability theo-

rem.

3.9 Notes and References

The original work on Lyapunov stability theory is due to the Russian

mathematician Aleksandr Mikhailovich Lyapunov [293]; it was trans lated

into French in 1907 [294] and English in 1992 [295]. Lyapunov stability

theory is extensively developed in the classical textbooks by Hahn [178],

Krasovskii [245], LaSalle and Lefschetz [260], and Yoshizawa [474]. A more

modern textbook treatment is given by Vidyasagar [445] and Khalil [235].

See also the papers by Kalman and Bertram [228] and L aSalle [258]. The

invariant set stability theorems are due to Barbashin and Krasovskii [23,245]

and LaSalle [258]. These theorems were ﬁrst proved by Barbashin and

Krasovskii [23] for autonomous systems, and by Krasovskii [245] for periodic

systems. The invariant set theorem for autonomous systems was later

rediscovered by LaSalle [258]. In the western literature the invariant set

theorem is often incorrectly referred to as LaSalle’s theorem. Th e systematic

construction of Lyapunov functions was originally studied by Kr asovskii

[245] and Zubov [481]. The earliest result on stability using integrals of

motion is due to Routh [370] with a Lyapunov function foundation given by

Rouche, Habets, and Laloy [368]. A tutorial exposition of these results is

given by Wan, Coppola, and Bernstein [450]. Th e treatment here is adopted

from Wan, Coppola, and Bernstein [450]. Converse Lyapunov theorems

are due to Persidskii [350], Malkin [298], and Massera [306, 307]. For an

excellent textbook treatment see Vidyasagar [445]. The Lyapunov instability

theorems can be found in Chetaev [92] and Vidyasagar [445].