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

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 85

Proof. Since kxk

= x

x it follows that

(t)x(t) = 2x

(t) ˙x(t) = 2x

(t)f(x(t)), t ∈ [t

, t

]. (2.175)

Now, since f (0) = 0 and f : D → R

is uniformly Lipschitz continuous

on D it follows that kf(x)k

≤ Lkxk

. Hence, using the Cauchy-Schwarz

inequality it follows that



(t)x(t)



≤ 2kx(t)k

kf(x(t))k

≤ 2Lkx(t)k

, t ∈ [t

, t

]. (2.176)

Now, deﬁning q(t)

△

= x

(t)x(t) and q

△

= q(0) = x

, it follows from (2.176)

that

−2Lq(t) ≤ ˙q(t) ≤ 2Lq(t), t ∈ [t

, t

], (2.177)

which implies

−

2Lds ≤

≤

2Lds. (2.178)

Hence,

−2L|t−t

≤ q(t) ≤ q

2L|t−t

, t ∈ [t

, t

]. (2.179)

The result is now immediate by noting that q(t) = kx(t)k

and q

= kx

2.8 Extendability of Solutions

In Section 2.7 we showed that und er the conditions of Lipschitz continuity on

D, the nonlinear dynamical system (2.136) is guaranteed to have a unique

solution over the closed interval [t

, τ]. In this section, we consider the

question of whether it is possible to extend the solution x : [t

, τ] → R

of (2.136) to a larger interval. This question can be intuitively answered

by noting that if (2.136) h as a unique solution over [t

, τ], then by re-

applying Theorem 2.25 w ith τ serving as the initial time and x(τ) as the

initial condition to (2.136), it follows th at there exists a unique solution to

(2.136) over the interval [τ, τ

]. By iteratively repeating this process and

concatenating all solutions, it follows that there does not exist a largest

closed bou nded interval over which (2.136) has a unique solution. Hence,

the solution w ill exist over the semiopen interval [t

, τ

max

). This implies

that th e solution x(t) to (2.136) must approach the boundary of D, tend

to inﬁnity, or both. For negative time, an identical argument for the left-

hand limit can be used to show that the maximal interval of existence for

a s olution to (2.136) is (τ

min

, τ

max

) ⊆ R. Before summarizing the above

discussion we consider a simp le example.

Example 2.35. Consider the scalar non linear dynamical system

˙x(t) = 1 + x

(t), x(0) = 0, t ∈ R. (2.180)

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86 CHAPTER 2

This system has a unique solution given by x(t) = tan t. Since x(t) → ±∞ as

t → ±π/2, the m aximal interval of existence of (2.180) is (−π/2, π/2) ⊂ R.

△

The following theorem shows that if a solution to (2.136) cannot be

extended, it must leave any given compact set of the state space.

Theorem 2.29. Consider the nonlinear dynamical system (2.136).

Assume f : D → R

is Lipschitz continuous on D. Furthermore, let

x(t) den ote the solution of (2.136) on the maximal interval of existence

= (τ

min

, τ

max

) ⊂ R with τ

max

< ∞. T hen given any compact set

⊂ D, there exist t

∈ (τ

min

, t

) and t

∈ (t

, τ

max

) such that x(t

) 6∈ D

and x(t

) 6∈ D

Proof. Suppose, ad absurdum, that there does not exist t ∈ I

such

that x(t) 6∈ D

. Then, x(t) ∈ D

, t ∈ (τ

min

, τ

max

). Since f : D → R

continuous on D

, it follows from Theorem 2.13 that there exists α > 0 such

that kf(x)k ≤ α for all x ∈ D

. Next, let σ ∈ (t

, τ

max

) and note that for

min

< t

< t < τ

max

kx(t) −x(t

)k =



f(x(s))ds



≤

kf(x(s))kds

≤ α(t −t

), (2.181)

which shows that x(·) is uniform ly continuous on I

Now, since x(·) is uniformly continuous on I

, it follows that

x(τ

max

) = x(σ) + lim

t→τ

max

f(x(s))ds

= x(σ) +

max

f(x(s))ds. (2.182)

Thus ,

x(t) = x(σ) +

f(x(s))ds, t ∈ [σ, τ

max

], (2.183)

which implies that ˙x(τ

max

) = f(x(τ

max

)), and hence, x(·) is (left) diﬀer-

entiable at τ

max

. Hence, x : [σ, τ

max

] → R

is a solution to (2.136).

Now, using Theorem 2.25 it follows that the solution x(t) to (2.136) can

be extended to the interval (τ

min

, τ

∗

), τ

∗

> τ

max

, which contradicts the

claim that I

= (τ

min

, τ

max

) is the maximal interval of existence of (2.136).

Hence, there exists t

∈ (t

, τ

max

) such that x(t

) 6∈ D

. The existence of

∈ (τ

min

, t

) such that x(t

) 6∈ D

follows in a similar manner.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 87

As mentioned earlier, in most physical dyn amical systems and espe-

cially feedback control systems we are more interested in the fu tu re behavior

of the system as opposed to the past. Hence, we give a version of Theorem

2.29 wherein t ∈ [0, τ

max

). This is sometimes referred to in the literature as

the right maximal interval of existence.

Theorem 2.30. Consider the nonlinear dynamical system (2.136).

Assume f : D → R

is Lipschitz continuous on D. Furthermore, let

x(t) denote th e solution to (2.136) on the maximal interval of existence

= [0, τ

max

) with τ

max

< ∞. Then given any compact set D

⊂ D, there

exists t ∈ I

such that x(t) 6∈ D

Proof. The proof is virtu ally identical to the proof of Theorem 2.29

and, hence, is omitted.

The next result shows that if τ

max

< ∞ and if lim

t→τ

max

x(t) exists,

then x(t) → ∂D as t → τ

max

Corollary 2.4. C onsider the nonlinear dynamical system (2.136).

Assume f : D → R

is Lipschitz continuous on D. Furthermore, let x(t)

denote the solution to (2.136) on the maximal interval of existence [0, τ

max

)

with τ

max

< ∞. If lim

t→τ

max

x(t) exists, then lim

t→τ

max

x(t) ∈ ∂D.

Proof. Assume lim

t→τ

max

x(t) exists and is given by x

∗

. Then,

s(t, x

)

△



x(t), t ∈ [0, τ

max

∗

, t = τ

max

(2.184)

is continuous on [0, τ

max

]. Next, let

△

= {y ∈ R

: y = s(t, x

) for some t ∈ [0, τ

max

]}. (2.185)

Since D

is the image of the compact set [0, τ

max

] under the continuous map

s(τ, x

), D

is a compact su bset of D. Now, suppose, ad absurdum, that

∗

∈ D. Then D

⊂ D and by Theorem 2.30 there must exist t ∈ (0, τ

max

)

such that x(t) 6∈ D

, which is a contradiction. Hence, x

∗

6∈ D. However,

since x(t) ∈ D for all t ∈ [0, τ

max

), it follows that x

∗

= lim

t→τ

max

x(t) ∈ D,

and h en ce, x

∗

∈ D \ D = ∂D.

Example 2.36. Consider the scalar non linear dynamical system

˙x(t) = −[2x(t)]

−1

, x(0) = 1, t ≥ 0. (2.186)

This system has a solution given by x(t) =

√

1 −t on the maximal interval of

existence I

= [0, 1). Note that on D = (0, ∞), the function f(x) = −1/2x

is continuously diﬀerentiable. Now, noting that ∂D = {0}, it follows that

lim

t→1

x(t) = 0 ∈ ∂D. △

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88 CHAPTER 2

An immediate and very important corollary to Theorem 2.30 is the

contrapositive statement of the theorem, which states that if a solution to

(2.136) lies entirely in a compact set, then τ

max

= ∞, and hen ce, there exists

a unique solution to (2.136) for all t ≥ 0.

Corollary 2.5. Consider the nonlinear dynamical system (2.136).

Assume f : D → R

is Lipschitz continuous on D. Furthermore, let D

⊂ D

be compact and suppose for x

∈ D

, the solution x : [0, τ] → D lies entirely

in D

. Then there exists a unique solution x : [0, ∞) → D to (2.136) f or all

t ≥ 0.

Proof. Let [0, τ) be the maximal interval of existence of (2.136) with

the s olution x(t), t ∈ [0, τ), lying entirely in D

. Then, by assumption

{y ∈ D : y = x(t) for some t ∈ [0, τ)} ⊂ D

or, equivalently, x([0, τ)) ⊂ D

Hence, it follows from Theorem 2.30 that τ = ∞.

2.9 Global Existence and Uniqueness of Solutions

In this section, we give suﬃcient conditions for global existence and

uniqueness of solutions over all time. First, however, we present a theorem

that strengthens the results of Theorem 2.26 on uniform convergence with

respect to initial conditions.

Theorem 2.31. Consider the nonlinear dynamical system (2.136).

Assume that f : D → R

is Lipschitz continuous on D. Fu rthermore, let

x(t) d en ote a solution to (2.136) with initial condition x(t

) = x

deﬁned on

the closed interval [t

, t

]. Then there exists a neighborhood N ⊂ D of x

and a constant L > 0 such that for all y(t

) = y

∈ N, there exists a unique

solution y(t) of (2.136) deﬁned over the closed interval [t

, t

] satisfying

kx(t) − y(t)k ≤ kx

− y

L|t−t

, (2.187)

for all t ∈ [t

, t

]. Moreover,

lim

→x

s(t, y

) = s(t, x

), (2.188)

uniformly for all t ∈ [t

, t

], where s(t, x

) denotes the solution to (2.136)

with initial condition x

Proof. Since [t

, t

] is compact and s(·, x

) is continuous on [t

, t

], it

follows that {x ∈ R

: x = s(t, x

), t ∈ [t

, t

]} is a compact subset of D.

Now, since D is an open set, there exists ε > 0 such that the compact set

= {x ∈ R

: kx − s(t, x

)k ≤ ε, t ∈ [t

, t

]} ⊂ D.

Next, since by assumption f : D → R

is Lipschitz continuous on D, it

follows from Proposition 2.23 that there exists L > 0 such that kf(y) −

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 89

f(x)k ≤ Lky − xk, x, y ∈ D

. Let δ > 0 be such that δ ≤ ε and δ ≤

εe

−L(t

−t

)

. Now, let y

∈ N

) and let s(t, y

) be the solution to (2.136)

over the maximal interval of existence (τ

min

, τ

max

). Suppose, ad absurdum,

that τ

max

≤ t

. Then it follows that s(t, y

) ∈ D

for all t ∈ (τ

min

, τ

max

). To

see this, suppose, ad absurdum, that there exists τ ∈ (τ

min

, τ

max

) such that

s(t, x

) ∈ D

for some t ∈ (τ

min

, τ] and s(τ, y

) ∈ ∂D

. In this case,

ks(t, y

) − s(t, x

)k ≤ ky

− x

k +

kf(y(σ)) − f(x(σ))kdσ

≤ ky

− x

k + L

ks(σ, y

) − s(σ, x

)kdσ, (2.189)

for all t ∈ (τ

min

, τ]. Now, since τ < τ

max

≤ t

, it follows from Gronwall’s

lemma that

ks(τ, y

) −s(τ, x

)k ≤ ky

− x

L|τ−t

< δe

L(τ−t

)

≤ ε. (2.190)

Hence, s(τ, y

) ∈

◦

, which leads to a contradiction. Thus, s(t, y

) ∈ D

for all t ∈ (τ

min

, τ

max

). However, by Theorem 2.30, (τ

min

, τ

max

) cannot

be the maximal interval of existence of s(t, y

), leading to a contradiction.

Thus , t

< τ

max

. Similarly, it can be shown that τ

min

< t

, and hence,

, t

] ⊂ (τ

min

, τ

max

). Hence, for all y

∈ N

), y(t) is the un ique solution

to (2.136) deﬁned over the closed interval [t

, t

Next, to show that s(t, y

) ∈ D

for all t ∈ [t

, t

], suppose, ad

absurdum, that there exists τ ∈ [t

, t

) such that s(t, y

) ∈ D

for all

t ∈ [t

, τ) and s(τ, y

) ∈ ∂D

. Repeating the identical s teps as above, it

can be sh own that this leads to a contradiction, and h en ce, s(t, y

) ∈ D

for all t ∈ [t

, t

]. Now, a reapplication of Gronwall’s lemma over the closed

interval [t

, t

] yields

ks(t, y

) − s(t, x

)k ≤ ky

− x

L|t−t

, t ∈ [t

, t

]. (2.191)

Finally, (2.188) is now immediate.

The main d iﬀerence between Theorem 2.26 and Theorem 2.31 is that

in Theorem 2.26 we assumed that both solutions to (2.136) are deﬁ ned on

the same closed interval. Alternatively, Theorem 2.31 shows that solutions

to (2.136) that start nearby will be deﬁned on the same interval and

remain nearby over a ﬁnite time. An extension of Theorem 2.31 involving

continuous dependence on initial conditions and system parameters can

be easily addressed as in Theorem 2.31. Hence, the following result is

immediate.

Theorem 2.32. Consider the nonlinear dynamical system (2.162).

Assume f : D×R

→ R

is such that for every λ ∈ R

, f(·, λ) is Lipschitz

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90 CHAPTER 2

continuous on D and for every x ∈ D, f(x, ·) is Lipschitz continuous on R

Furth ermore, let x(t) denote a solution to (2.162) with system parameter λ

and initial condition x(t

) = x

deﬁned over the closed interval [t

, t

]. Then

there exist neighborhoods N

) ⊂ D and N

(µ) ⊂ R

, and a constant

L > 0 such that for all y(t

) = y

∈ N

) and λ ∈ N

(µ) there exists a

unique solution y(t) of (2.162) deﬁned over the closed interval [t

, t

] such

that

kx(t) − y(t)k ≤ [kx

− y

k + kλ − µk]e

L|t−t

, (2.192)

for all t ∈ [t

, t

]. Moreover,

lim

,µ)→(x

,λ)

s(t, y

, µ) = s(t, x

, λ), (2.193)

uniformly for all t ∈ [t

, t

], where s(t, x

, λ) denotes the solution to (2.162)

with initial condition x

and p arameters λ.

Next, we turn our attention to the question of global existence and

uniqueness of solutions to (2.136). For th is result the following lemma is

needed.

Lemma 2.3. Let y : R → R

be a continuously diﬀerentiable function

on (α, β). If

k˙y(t)k ≤ q(t), y(t

) = y

, t ∈ (α, β), (2.194)

then

ky(t)k ≤ ky(t

)k +

q(s)ds, t ∈ (α, β). (2.195)

Proof. Note that for t ∈ (α, β),



˙y(s)ds



≤

k˙y(s)kds ≤

q(s)ds, (2.196)

which implies

ky(t) − y(t

)k ≤

q(s)ds, t ∈ (α, β).

Hence,

ky(t)k = ky(t) − y(t

) + y(t

)k ≤ ky(t

)k +

q(s)ds,

for all t ∈ (α, β).

Theorem 2.33. Consider the nonlinear dynamical system (2.136).

Assume that f : R

→ R

is globally Lipschitz continuous with Lipschitz

constant L. Then, for all x

∈ R

, (2.136) has a unique solution x :

(−∞, ∞) → R

over all t ∈ R.

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 91

Proof. Let x(t) be the solution to (2.136) on the maximal interval of

existence I

= (τ

min

, τ

max

). Then it follows from the triangle inequality for

vector norms and the globally Lipschitz condition on f that

k˙x(t)k = kf (x(t))k

= kf(x(t)) −f (x

) + f (x

≤ kf(x(t)) −f (x

)k + kf(x

≤ Lkx(t) −x

k + α, t ∈ (τ

min

, τ

max

), (2.197)

where α

△

= kf (x

)k. Now, suppose, ad absurdum, that τ

max

< ∞. In

this case, it follows from Lemma 2.3, with y(t) = x(t) − x

and q(t) =

Lkx(t) − x

k + α, that

kx(t) − x

k ≤

[Lkx(s) − x

k + α]ds

≤ α(τ

max

−t

) + L

kx(s) − x

kds, (2.198)

for all t ∈ (t

, τ

max

). Next, using Gronwall’s lemma (2.198) implies

kx(t) − x

k ≤ α(τ

max

−t

L(t−t

)

, t ∈ [t

, τ

max

), (2.199)

which further implies that the solution to (2.136) through the point x

time t = t

is contained in the compact set

= {x ∈ R

: kx −x

k ≤ α(τ

max

− t

L(τ

max

−t

)

} ⊂ R

Hence, it follows from Theorem 2.30 that the su pposition τ

max

< ∞ leads

to a contradiction. Thus, τ

max

= ∞. Similarly, it can be shown that τ

min

−∞. Hence, for all x

∈ R

, (2.136) h as a unique solution x(t) over all

t ∈ R, where uniquen ess follows from Th eorem 2.25.

Example 2.37. Consider the linear dyn amical system

˙x(t) = Ax(t), x(t

) = x

, t ∈ [t

, t

], (2.200)

where x(t) ∈ R

, t ∈ [t

, t

], and A ∈ R

n×n

. Since A is constant, kAk ≤ α,

where k · k : R

n×n

→ R is the equi-induced matrix norm induced by the

vector norm k· k. Now, with f (x) = Ax it follows that

kf(x) −f (y)k = kA(x −y)k ≤ kAkkx − yk ≤ αkx − yk, (2.201)

for all x, y ∈ R

, and hen ce, (2.136) is globally Lipschitz on R

. Hence, it

follows from Theorem 2.33 that linear systems have unique solutions over

all t ∈ R. △

The following corollary to Theorem 2.33 is immediate.

Corollary 2.6. C onsider the nonlinear dynamical system (2.136).

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92 CHAPTER 2

Assume that f : R

→ R

is continuously diﬀerentiable and its derivative

is uniformly bounded on R

. Then, for all x

∈ R

, (2.136) has a unique

solution x : (−∞, ∞) → R

over all t ∈ R.

Proof. It follows from Proposition 2.26 that if f : R

→ R

continuously diﬀerentiable on R

and its derivative is uniformly bounded

on R

, then f (·) is globally Lipschitz continuous. Existence and uniqueness

of solutions over R now follows from Theorem 2.33.

2.10 Flows and Dynamical Systems

As shown in Section 2.6, a dynamical system d eﬁ nes a continuous ﬂow in the

state space. This ﬂow is a one-parameter family of homeomorphisms and

is one possible mathematical expression of the behavior of the dynamical

system. In this section, we provide some additional properties of ﬂows

deﬁned by diﬀerential equations. In particular, we consider the non linear

dynamical system

˙x(t) = f (x(t)), x(0) = x

, t ∈ I

, (2.202)

where x(t) ∈ D, t ∈ I

, D is an open subset of R

, f : D → R

is Lipschitz

continuous on D, and I

= (τ

min

, τ

max

) ⊆ R is the maximal interval of

existence for the solution x(·) of (2.202). As in Section 2.6, for t ∈ I

, the set

of mappings s : I

×D → D deﬁned by s

) = s(t, x

) will denote the ﬂow

of (2.202). Furthermore, we deﬁne the set S

△

= {(t, x

) ∈ R × D : t ∈ I

}

so that s : S → D.

Theorem 2.34. Let s : S → D be the ﬂ ow generated by (2.202) and

let x

∈ D. If t ∈ I

and τ ∈ I

)

, then τ + t ∈ I

and

t+τ

) = s

)). (2.203)

Proof. Let t ∈ I

and let τ ∈ I

)

be such that τ > 0. Furthermore,

let I

= (τ

min

, τ

max

) and deﬁne the solution curves s

: (τ

min

, τ + t] → D

(σ) =



s(σ, x

), τ

min

< σ ≤ t,

s(σ − t, s

)), t ≤ σ ≤ τ + t.

(2.204)

Clearly, s

(σ) is the solution curve of (2.202) over th e interval (τ

min

, τ + t]

with initial condition s

(0) = x

. Hence, τ + t ∈ I

. Furthermore, since

f : D → R

is Lipschitz continuous on D it follows from the uniqueness

property of solutions that

t+τ

) = s

(t + τ ) = s(τ, s

)) = s

)), (2.205)

which proves (2.203) for the case τ > 0. Next, in the case where τ = 0,

(2.203) is immediate. Finally, let τ < 0 and deﬁne the solution curve s

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 93

[τ + t, τ

max

) → D by

(σ) =



s(σ, x

), t ≤ σ < τ

max

s(σ − t, s

)), τ + t ≤ σ ≤ t.

(2.206)

Note that s

(σ) is the solution curve of (2.206) over the interval [τ +t, τ

max

)

with initial condition s

(0) = x

. Now, using an identical argument as

above shows (2.203).

Theorem 2.34 shows that s

τ+t

) is deﬁned for τ + t ∈ I

if and only

if s

)) is deﬁned for t ∈ I

and τ ∈ I

)

Theorem 2.35. Let s : S → D be the ﬂ ow generated by (2.202) and

let x

∈ D. Then S is an open subset of R ×D and s : S → D is continuous

on S.

Proof. To show that S is open, let (t

, x

) ∈ S and t

≥ 0. In this

case, the solution x(t) = s(t, x

) to (2.202) is deﬁned over [0, t

], and hence,

by Theorem 2.25, can be extended on [0, t

+ τ ] for some τ > 0. Hence,

repeating this argument for the left-side interval, s(t, x

) is deﬁned over

− τ, t

+ τ ] for each x

∈ D.

Now, it follows from Theorem 2.31 that there exists δ > 0 and L > 0

such that for all y

∈ B

), s(t, y

) is deﬁned on [t

− τ, t

+ τ] × D

)

and satisﬁes

ks(t, y

) −s(t, x

)k ≤ ky

− x

L|t−t

, t ∈ [t

− τ, t

− τ ]. (2.207)

Thus , (t

− τ, t

+ τ ) ×D

) ⊂ S, and hence, S is open in R ×D.

To show that s : S → D is continuous at (t

, x

) note that since f :

D → R

is Lipschitz continuous on D and D

= s([t

−τ, t

−τ]×B

)) ⊂ D

is a compact set, it follows from Pr oposition 2.23 that f : D → R

uniformly Lipschitz continuous on D

. Next, let α = m ax{kf(x)k : x ∈ D

}

and let δ > 0 be such that δ < τ and B

) ⊂ D

. Furthermore, let y be

such that y ∈ B

). Now, su ppose |t

∗

− t

| < δ and ky − x

k < δ. Then,

ks(t

∗

y) −s(t

, x

)k = ks(t

∗

, y) − s(t

∗

, x

) + s(t

∗

, x

) −s(t

, x

≤ ks(t

∗

, y) − s(t

∗

, x

)k + ks(t

∗

, x

) −s(t

, x

)k.

(2.208)

Since s(·, x

) is continuous in t, it follows that ks(t

∗

, x

) −s(t

, x

)k → 0 as

δ → 0. Alternatively, it follows from (2.207) that ks(t

∗

, y)−s(t

∗

, x

)k < δe

Lδ

and hence, ks(t

∗

, y) − s(t

∗

, x

)k → 0 as δ → 0. Hence, s(·, ·) is continuous

on S. An analogous proof holds for the case where t

< 0.

Finally, we show that s

: D → D is a one-to-one m ap and has a

NonlinearBook10pt November 20, 2007

94 CHAPTER 2

continuous inverse.

Theorem 2.36. Let s

: D → D be the ﬂow generated by (2.202) and

let x

∈ D. If (t, x

) ∈ S then th ere exists a neighborh ood N of x

such

that {t} × N ⊂ S. Furthermore, U = s

(N) ⊂ D is open, {−t} × U ⊂ S,

and

−t

(x)) = x, x ∈ N, (2.209)

−t

(y)) = y, y ∈ U. (2.210)

Proof. Let (t, x

) ∈ S. Using similar arguments as in the proof of

Theorem 2.35 it follows that there exists a neighborhood N of x

and τ > 0

such that (t − τ, t + τ) × N ⊂ S, and hence, {t} × N ⊂ S. Next, to show

that s

has a continuous inverse, let y = s

(x) for x ∈ N and t ∈ I

. Since

φ(τ) = s(τ + t, y) is a solution of (2.202) over the interval [−t, 0] with initial

condition φ(−t) = y, it follows that −t ∈ I

. Hence, s

−t

is deﬁned on

U = s

(N). Now, Theorem 2.35 yields s

−t

(x)) = s

(x) = x for all x ∈ N

and s

−t

(y)) = s

(y) = y for all y ∈ U. Finally, to show that U is open

let s

−t

: U

max

→ D, where U

max

⊃ U is such that R ×U

max

= S. Since S is

open, U

max

is open. Furthermore, since, by Theorem 2.35, s

is continuous

it follows that s

−t

: U

max

→ D is also continuous. Now, since the inverse

image of the open set N under the continuous map s

−t

is open, and since

this inverse image is U , that is, s

−t

(U) = N, it follows that U is open.

2.11 Time-Varying Nonlinear Dynamical Systems

In Section 2.6 we deﬁned a dynamical system as a pr ecise mathematical

object satisfying a set of axioms. A key implicit assu mption in Deﬁnition

2.47 was that the solution curve of the dyn amical system G remained

unchanged under translation of time. In many dynamical systems this

assumption does not hold, giving rise to time-varying or nonautonomous

diﬀerential equations. Even though this allows for a more general class of

dynamical systems, the fundamental theory developed in Sections 2.7–2.10

does not change signiﬁcantly for the class of time-varying systems. In this

section, we outline the salient changes for time-varying dynamical systems

as compared to time-invariant dynamical systems. Since these results very

closely parallel the results on time-invariant systems, we present the most

important theorems needed for later developments. Furthermore, we leave

the proofs of these theorems as exercises for the reader. For the following

deﬁnition let D be an open subset of R

Deﬁnition 2.48. A time-varying dynamical system on D is the triple

(D, R, s), where s : R × R ×D → D is such that the following axioms hold: