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

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 115

iii) If C = R

and f is convex, then f

′′

(x) ≥ 0, x ∈ C.

iv) f(x) = x

P x is convex if and only if P ≥ 0.

v) f (x) = x

P x is strictly convex if and only if P > 0.

Problem 2.33. Let f : R

→ R

be continuously diﬀerentiable and let

α > 0. Show that if

′

(x) − f

′

(y)]

(x −y) ≥ αkx − yk

, x, y ∈ R

, (2.250)

then f is strictly convex. Furthermore, show that if f is two-times

continuously diﬀerentiable, then (2.250) is equivalent to f

′′

(x) ≥ αI

, for

every x ∈ R

Problem 2.34. Let C ⊂ R

be a convex set and let f : C → R be a

convex function. Show that if x ∈ C is a lo cal m inimum of f , then x ∈ C is

also a global minimum of f. I f, in addition, f is strictly convex, show that

there exists at most one global minimum of f.

Problem 2.35. Show that the function f : R → R given by

f(x) =



sin(

), x 6= 0,

0, x = 0,

is continuous at x = 0.

Problem 2.36. Show that the function f : R

→ R given by

f(x

, x

) =



/(x

+ x

), (x

, x

) 6= (0, 0),

0, (x

, x

) = (0, 0),

is discontinuous at (x

, x

) = (0, 0).

Problem 2.37. Show that the function f : R → R given by f (x) =

sin(

), x 6= 0, and f (0) = α, α ∈ R, is discontinuous at x = 0.

Problem 2.38. Show that the function f : R

→ R given by

f(x

, x

) =



+ x

)

/(x

+ x

), (x

, x

) 6= (0, 0),

0, (x

, x

) = (0, 0),

is continuous at (x

, x

) = (0, 0).

Problem 2.39. Let f : R

→ R

be given by f (x) = Ax, where

x ∈ R

and A ∈ R

n×m

. Show that f is continuous on R

. Furthermore,

for m = n and det A 6= 0, show that f is a d iﬀeomorphism .

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116 CHAPTER 2

Problem 2.40. Let f : R → R be given by

f(x) =



x sin(

), x 6= 0,

0, x = 0.

Show that f is not diﬀerentiable at x = 0.

Problem 2.41. Let f : [a, b] → R, where [a, b] ⊂ R, be s uch that

f is diﬀerentiable on (a, b) and f

′

(x) > 0, x ∈ (a, b). Show that f

−1

[f(a), f(b)] → R is diﬀerentiable on (f(a), f (b)).

Problem 2.42. Let f : (0, ∞) → R be given by

f(x) =



1/x, 0 < x ≤ 1,

1/2, x > 1.

Show that f is left continuous, piecewise continuous, left piecewise continu-

ous, and upper s emicontinuous on (0, ∞).

Problem 2.43. Let f : R → R be given by

f(x) =



sin(1/x), x < 0,

0, x ≥ 0.

Show that f is right continuous on R. Is f piecewise continuous on R?

Problem 2.44. Let f : R → R be given by f(x) = |x|. Show that f is

continuous on R and diﬀerentiable at all x ∈ R, x 6= 0. Furthermore, show

that f is piecewise continuously diﬀerentiable on R. Does f

′

(0) exist?

Problem 2.45. Let f : R → R and g : R → R be such that f and g

are Lipschitz continuous on R. Show that f + g, f g, and g ◦f are Lipschitz

continuous on R.

Problem 2. 46. Let k·k : R

→ R be a vector norm on R

. Show that

k· k is continuous on R

. Is k · k uniformly continuous on R

Problem 2.47. Let X ⊆ R

, Y ⊆ R

, and Z ⊆ R

, and let f : X → Y

and g : Y → Z be continuous functions. Show that g ◦ f : X → Z is

continuous on X.

Problem 2.48. Let D ⊂ R

and let k·k : D → R and k·k

′

: D → R b e

vector norms on D. Show that if D is open (respectively, closed, bounded,

compact) under k·k, then D is open (respectively, closed, bounded, compact)

under k · k

′

Problem 2.49. A function f : R

→ R

is called homogeneous

(respectively, positively homogeneous) of degree r if f (λx) = λ

f(x) for

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 117

all λ ∈ R (respectively, λ > 0). Show that if 0 < r < 1, then f is not

Lipschitz continuous at x = 0.

Problem 2.50. Let f : R

→ R. Then f is H¨older continuous with

exponent α > 0 at x ∈ R

if there exist a constant k > 0 (called the H¨older

constant) and an open neighborhood D of x such that

|f(x) − f(y)| ≤ kkx −yk

, x ∈ D. (2.251)

f is simply said to be H¨older continuous at x if f is H¨older continuous at

x with some exponent α > 0. Show that H¨older continuity at x imp lies

continuity at x. In addition, show that if α > 1, then H¨older continuity at

x implies diﬀerentiability at x.

Problem 2.51. Let f : R

→ R be two-times continuously diﬀeren-

tiable on B

(x) ⊂ R

. Show that:

i) For all y ∈ R

such that x + y ∈ B

(x),

f(x + y) = f(x) + y

′

(x) +





′′

(x + σy)dσ





(2.252)

ii) For all y ∈ R

such that x + y ∈ B

(x), there exists α ∈ [0, 1] such

that

f(x + y) = f(x) + y

′

(x) +

′′

(x + αy)y. (2.253)

iii) For all y ∈ R

such that x + y ∈ B

(x),

f(x + y) = f(x) + y

′

(x) +

′′

(x)y + O(kyk

). (2.254)

Problem 2.52. Let D ⊂ R

be bounded, let ε > 0 be such that kx −

yk < ε for all x, y ∈ D, and let z ∈ D. Show that D ⊆ B

(z).

Problem 2.53. Let C ⊆ R

be a convex set and let f : C → R be a

convex function. Show that the [0, β]-sublevel set of f is convex.

Problem 2.54. Let D ⊆ R

be a connected set and let f : D → R

be continuous. Show that the image of D under f is connected.

Problem 2.55. Let C ⊆ R

be a convex set and let f : C → R be a

convex function. Show that f is continuous on

◦

Problem 2.56. Show that every [α, β]-sublevel set of a continuous

function is closed.

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118 CHAPTER 2

Problem 2.57. Show that every hyperplane is a subspace.

Problem 2.58. Let D ⊆ R

and let f : D → R

. Show th at the

following statements are equivalent:

i) f is continuous.

ii) For all closed Q ⊆ R

, the inverse image of Q under f is closed relative

to D.

Problem 2.59. Let D ⊆ R

, let f : D → R

be continuous on Q ⊆ D,

and assume that Q is bounded and ∂Q ⊆ D. Show that f is uniformly

continuous on Q.

Problem 2.60. Let f : (α, β) → R, where α, β ∈ R. Show th at the

following statements hold:

i) I f f is s tr ictly convex, then f is continuous.

ii) If f is strictly increasing or strictly decreasing, then f is injective.

iii) If f is injective and strictly convex, then f is strictly increasing or

strictly decreasing.

Problem 2.61. Let f : [0, ∞) → [0, ∞). Assume that f is

diﬀerentiable, f ∈ L([0, ∞)), and assume that there exists α ∈ [0, ∞) such

that |f

′

(x)| ≤ α for all x ∈ [0, ∞). Show that lim

x→∞

f(x) = 0.

Problem 2.62. Let C ⊆ R

be a convex set and f : C → R

continuously diﬀerentiable. Furthermore, suppose 0 ∈ C and f (0) = 0.

Show that

f(x) =

∂f

∂x

(σx)xdσ, x ∈ C. (2.255)

Problem 2.63. Show that if f : R

→ R

is Lipschitz continuous on

(0) = {x ∈ R

: kxk < ε} and f(0) = 0, then there exists L > 0 such

that kf (x)k ≤ Lkxk for all x ∈ B

(0).

Problem 2.64. Let k · k

and k · k

, where p, q ∈ [1, ∞], denote two

vector norms on R

. Show that f : R

→ R

is Lipschitz continuous under

k· k

if and only if f (·) is Lipschitz continuous under k ·k

Problem 2.65. Show that if f : D → R

is continuously diﬀerentiable

at x

∈ D, then there exists δ > 0 and L > 0 such that kf(x) − f (x

)k <

Lkx − x

k for all x ∈ B

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 119

Problem 2.66. Let (C[0, T ], ||| · |||

2,2

) denote the space of continuous

functions f : [0, T ] → R with norm

|||f |||

2,2



|f(t)|



1/2

Show that (C[0, T ], ||| · |||

2,2

) is not a Banach space. (Hint: Consider the

sequence of functions {f

}

∞

n=1

⊂ C[0, T ], where f

(t) = t/(|t| + 1/n).)

Problem 2.67. Let X 6= Ø be a s et. Show that F = {f : X → X}

is a semigroup with identity u nder operation of composition g ◦ f , where

f, g ∈ F.

Problem 2.68. Let Q = {x : x = p/q, p, q ∈ Z, q 6= 0}. Show that Q

is an Abelian group with respect to addition (+) on R, and Q \ {0} is an

Abelian group with respect to multiplication (·) on R.

Problem 2.69. Show that every subspace of R

is closed.

Problem 2.70. Let x, y ∈ R

and S ⊂ R

. Then x and y are

orthogonal if x

y = 0. Furthermore, the orthogonal complement of S is

deﬁned as S

⊥

△

= {x ∈ R

: x

y = 0, y ∈ S}. Show that if S

and S

are

subspaces of R

with S

⊆ S

, then S

⊥

⊆ S

⊥

Problem 2.71. Let S be a collection of bounded, measurable sub s ets

of R with measure µ : S → [0, ∞). Show that the following statements hold:

i) µ(Ø) = 0.

ii) µ([a, b)) = µ([a, b]) = µ((a, b)) = µ((a, b]) = b − a.

iii) If S

, S

∈ S, then

µ(S

∪ S

) = µ(S

) + µ(S

) − µ(S

∩ S

iv) If {S

}

∞

i=1

⊆ R is such that S

i+1

⊆ S

and S

∈ S, then

∞

i=1

= µ( lim

i→∞

) = lim

i→∞

µ(S

v) If {S

}

∞

i=1

⊆ R is such that S

∩ S

= Ø and S

∈ S for i, j = 1, 2, . . . ,

i 6= j, then

∞

[

i=1

∞

i=1

µ(S

Problem 2.72. Let S = {a}, where a ∈ R. Show that µ(S) = 0.

(Hint: Use the properties in Prob lem 2.71.)

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120 CHAPTER 2

Problem 2.73. Let S =

be a ﬁnite or countable union of sets in

where µ(S

) = 0 for each i. Show that µ(S) = 0.

Problem 2.74. Show that the following statements hold:

i) I f S

⊂ S

, then µ

∗

) ≤ µ

∗

ii) If S =

∞

i=1

, then µ

∗

(S) ≤

∞

i=1

∗

iii) A countable union of measurable sets is measurable.

iv) Closed sets are measurable.

v) The complement of a measurable set is measurable.

vi) A countable intersection of measurable sets is measurable.

vii) Let {S

}

∞

i=1

⊆ R

be such that S

∩S

= Ø f or i, j = 1, 2, . . . , j, i 6= j,

and S =

∞

i=1

. Then µ(S) =

∞

i=1

µ(S

Problem 2.75. Give a function that is Riemann integrable but not

Lebesgue integrable. (Hint: Consider improper integrals.)

Problem 2.76. Let D ⊆ R

be measurable, and let f : D → R. Show

that the following statements are equivalent:

i) f is measurable.

ii) {x ∈ D : f(x) > α} is measurab le for every α ∈ R.

iii) {x ∈ D : f (x) ≥ α} is measurable for every α ∈ R.

iv) {x ∈ D : f (x) < α} is measurable for every α ∈ R.

v) {x ∈ D : f(x) ≤ α} is measurab le for every α ∈ R.

Problem 2.77. Let D ⊆ R

be measurable, and let I be a dense subset

of R. Show that if {x ∈ D : f(x) > α} is measurable for all α ∈ I, then

f : D → R is measurable.

Problem 2.78. Let f : D → R and g : D → R be measurable, where

D ⊆ R

is measurable. Show that the following statements hold:

i) (f + g) : D → R is measurable.

ii) f g : D → R is measurable.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 121

iii) If g(x) 6= 0 for almost all x ∈ D, then (f /g) : D\{x ∈ D : g(x) = 0} →

R is measurable.

iv) f

, where p ∈ Z

, is measurable.

Problem 2. 79. Let {f

}

∞

n=1

be a sequence of measurable functions on

D ⊆ R

. Show that th e following statements hold:

i) s up

(x) is measurable.

ii) inf

(x) is measurable.

iii) lim sup

n→∞

(x) is measurable.

iv) lim inf

n→∞

(x) is measurable.

v) If lim

n→∞

(x) = f(x), then f (x) is measurable.

Problem 2.80. Prove or refute that the composition of two m easurable

functions is measurable.

Problem 2.81 (Bounded Convergence Theorem). Let {f

}

∞

n=1

be a

sequence of measurable functions on D ⊂ R

, where D is measurable with

µ(D) < ∞, such that lim

n→∞

= f almost everywhere on D. Assume

there exists α ≥ 0 such that |f

(x)| ≤ α for all x ∈ D and n ∈ Z

. Show

that f ∈ L(D) and

lim

n→∞

(x)dx =

f(x)dx.

Problem 2. 82 (Monotone Convergence Theorem). Let {f

}

∞

n=1

be a

sequence of measurable fu nctions on D ⊂ R

such that f

n+1

(x) ≥ f

(x) for

all x ∈ D an d n ∈ Z

. Assume that lim

n→∞

(x) = f almost everywhere

on D. Show that f ∈ L(D) and

lim

n→∞

(x)dx =

f(x)dx.

Problem 2.83 (Dominated Convergence Theorem). Let g ∈ L(D)

and let {f

}

∞

n=1

be a sequence of measurable functions on D such that

(x)| ≤ g(x) for all x ∈ D. Assume that lim

n→∞

= f almost everywhere

on D. Show that f ∈ L(D) and

lim

n→∞

(x)dx =

f(x)dx.

Problem 2. 84. Let X be a normed linear space with norm |||·||| : X →

R. Show th at |||x||| − |||y||| ≤ |||x − y|||, x, y ∈ X.

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122 CHAPTER 2

Problem 2.85. Show that every Cauchy sequence in a normed linear

space is bounded.

Problem 2. 86. Let X be a normed linear space with norm |||·||| : X →

R. Show th at ||| ·||| is uniformly continuous.

Problem 2.87. An inner product space is a linear vector space X with

associated ﬁeld F and mapping < ·, · >: X ×X → F such that the following

axioms hold:

i) < x, x >≥ 0, x ∈ X.

ii) < x, x >= 0 if and only if x = 0.

iii) < x, αy >= α < x, y >, x, y ∈ X and α ∈ F.

iv) < x, y + z >=< x, y > + < x, z >, x, y, z ∈ X.

v) < x, y >= < y, x >, x, y ∈ X.

For ||| · ||| : X → R deﬁned by |||x||| =< x, x >

1/2

show that ||| · ||| is a norm

on X, and hen ce, X is a normed linear space. (Hint: First show that

| < x, y > | ≤ |||x||||||y|||, where x and y belong to the inner product space.)

Problem 2.88. Let X be an inner product space (see Problem 2.87)

with in ner produ ct < ·, · >: X × X → R. Show that for each y ∈ X, the

function mapp ing x into < x, y > is uniformly continuous.

Problem 2.89. A complete inner product space (see Problem 2.87)

with norm deﬁned by the inner product is called a Hilbert space. Discus s

the Hilbert space R

. What are the inner product and corresponding n orm?

Problem 2.90. Let f : [0, ∞) → R

n×m

. Which matrix norm (as a

spatial norm) makes L

a Hilbert space (see Problem 2.89)?

Problem 2.91. Let X = R and deﬁne T : R → R by T (x) = x −

tan

−1

(x) + π/2. Sh ow th at T (·) satisﬁes (2.128) but T has no ﬁ xed point in

Problem 2.92. Let X be a Banach space with norm ||| · ||| : X → R,

let S be a subset of X, and let T : S → X. Suppose there exists a constant

ρ ∈ [0, 1) such th at

|||T (x) − T (y)||| ≤ ρ|||x −y|||, x, y ∈ S, (2.256)

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 123

and suppose there exists x

∈ S such that

B =



x ∈ X : |||x −x

||| ≤

|||T (x) − T (x

)|||

1 −ρ



⊂ S. (2.257)

Show that there exists unique x

∗

∈ S such that T (x

∗

) = x

∗

. Furthermore,

show that for each x

∈ S, the sequence {x

}

∞

n=0

⊂ S deﬁned by x

n+1

T (x

) converges to x

∗

. Finally, show that

|||x

∗

− x

||| ≤

1 − ρ

|||T (x

) −x

|||, n ≥ 0. (2.258)

Problem 2.93 (Schauder Fixed Point Theorem). Let C ⊆ R

be a

nonempty, convex, and closed set, let f : C → C be continuous, and assume

f(C) is bounded. Show that there exists x ∈ C such th at f (x) = x.

Problem 2.94 (Brouwer Fixed Point Theorem). Let C ⊆ R

be a

nonempty, compact, and convex set, and let f : C → C be continuous. Show

that there exists x ∈ C such that f (x) = x.

Problem 2.95. Let X be a Banach space with norm |||·|||

′

: X → R, let

T : X → X, let I : X → X be the identity operator, and let |||T ||| < 1, where

|||T |||

△

= sup

x∈X, x6=0

|||T x|||

′

|||x|||

′

Show that the range of I −T is X and (I −T )

−1

exists and is bounded, and

satisﬁes

|||(I − T )

−1

||| ≤

1 − |||T |||

Furth ermore, show that

∞

k=0

= (I − T )

−1

Problem 2.96. Consider the nonlinear dynamical system given by

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (2.259)

˙x

(t) = −sgn[x

(t) + x

(t)], x

(0) = x

, (2.260)

where sgn(x)

△

= x/|x|, x 6= 0, and sgn(0)

△

= 0. Show that the dynamical

system (2.259) and (2.260) has a unique solution for all initial conditions.

Furth ermore, show that the solution reaches the manifold {(x

, x

) ∈ R×R :

+ x

= 0} in ﬁnite time and slides on the manifold towards the origin for

large values of time.

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124 CHAPTER 2

Problem 2.97. Consider the scalar nonlinear dynamical s y s tem

˙x(t) = f (x(t)), x(0) = 0, t ≥ 0, (2.261)

where

f(x)

△



−1 − x

, x ≥ 0,

1 + x

, x < 0.

(2.262)

Show that there does not exist a continuously diﬀerentiable function x(·)

satisfying (2.261).

Problem 2.98. Consider the nonlinear dynamical system (2.136) and

let s(t, x), t ≥ 0, denote the solution to (2.136) with initial condition x(0) =

. Show that if f : D → R

is Lipschitz continuous on D, then for every

ε, T > 0, and x

∈ D, there exists δ(ε, T, x

) > 0 such that if kx

− yk <

δ(ε, T, x

), y ∈ D, then ks(t, x

) −s(t, y)k < ε f or all t ∈ [0, T ].

Problem 2.99. Let x : (α, β) → R

be a continuously diﬀerentiable

function on (α, β). Show that

kx(t)k

≤ k˙x(t)k

, t ∈ (α, β).

Problem 2.100. Consider the nonlinear dynamical system (2.136)

with f : D → R

continuously diﬀerentiable on D. Let x

(t) and x

(t)

be two solutions of (2.136) over intervals I

and I

, respectively. Show that

∈ I

∩I

and if I ⊂ I

∩I

is any open interval containing t

, show that

(t) = x

(t) for all t ∈ I.

Problem 2.101. Consider the nonlinear dynamical s ys tem (2.136).

Picard’s method of successive approximations is based on the fact that x(t) is

a solution to (2.136) if and only if x(t) is a solution to (2.137). In particular,

the successive approximations of the solution to (2.137) are given by the

sequence of functions

(t) = x

, (2.263)

n+1

(t) = x

f(y

(s))ds, (2.264)

for n = 0, 1, . . .. Use Picard’s method of successive approximations to show

that su ccessive approximations for the linear dynamical system

˙x(t) = Ax(t), x(0) = x

, t ≥ 0, (2.265)

where x(t) ∈ R

, t ≥ 0, and A ∈ R

n×n

, converge to x(t) = e

, t ≥ 0.

Problem 2.102. Prove Theorem 2.24 using Picard’s method of su c-

cessive approximations (see Problem 2.101).

Problem 2.103. Consider the nonlinear dynamical s ys tem (2.136).

Show that if lim

t→τ

max

x(t) = x

∗

exists and x

∗

∈ D, then τ

max

= ∞.