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

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STABILITY THEORY FOR NONLINEAR SYSTEMS 155

proposition is needed.

Proposition 3.2. Let f, g : R

→ R

be continuously diﬀerentiable

functions such that f (0) = 0. Then for every x ∈ R

there exists α ∈ [0, 1]

such that

(x)f(x) = g

(x)

∂f

∂x

(αx)x. (3.71)

Proof. L et p ∈ R

and note th at it follows from the m ean value

theorem (Theorem 2.16) that for every x ∈ R

there exists α ∈ [0, 1] such

that

(p)f(x) = g

(p)[f(x) −f (0)]

= g

(p)



∂f

∂x

(αx)x



Hence, there exists α ∈ [0, 1] such that

(p)f(p) = g

(p)



∂f

∂x

(αp)p



. (3.72)

Now, since p is arbitrary, the result follows.

Theorem 3.6 (Krasovskii’s Theorem). Let x(t) ≡ 0 be an equilibrium

point for the nonlinear dynamical system

˙x(t) = f(x(t)), x(0) = x

, t ≥ 0, (3.73)

where f : D → R

is continuously diﬀerentiable and D is an open set

with 0 ∈ D. Assume there exist positive deﬁnite matrices P ∈ R

n×n

and

R ∈ R

n×n

such that



∂f

∂x

(x)



P + P



∂f

∂x

(x)



≤ −R, x ∈ D, x 6= 0. (3.74)

Then, the zero solution x(t) ≡ 0 to (3.73) is a unique asymptotically stable

equilibrium with Lyapunov function V (x) = f

(x)P f(x). If, in addition,

D = R

, then the zero solution x(t) ≡ 0 to (3.73) is a unique globally

asymptotically stable equilibrium.

Proof. Suppose, ad absurdum, that there exists x

∈ D such that

6= 0 and f (x

) = 0. I n this case, it follows from Proposition 3.2 that for

every x

∈ D there exists α ∈ [0, 1] such that x

P f (x

) = x

∂f

∂x

(αx

Hence,

(



∂f

∂x

(αx

)



P + P



∂f

∂x

(αx

)



)

= 0, (3.75)

which contradicts (3.74). Hence, there does n ot exist x

∈ D, x

6= 0, such

that f(x

) = 0. Next, note that V (x) = f

(x)P f(x) ≥ λ

min

(P )kf(x)k

≥ 0,

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156 CHAPTER 3

x ∈ D, which implies that V (x) = 0 if and only if f (x) = 0 or, equivalently,

V (x) = 0 if and only if x = 0. Hence, V (x) = f

(x)P f(x) > 0, x ∈ D,

x 6= 0. Next, computing the derivative of V (x) along th e trajectories of

(3.73) and using (3.74) yields

V (x) = V

′

(x)f(x)

= 2f

(x)P

∂f(x)

∂x

f(x)

= f

(x)

(



∂f

∂x

(x)



P + P



∂f

∂x

(x)



)

f(x)

≤ −f

(x)Rf(x)

≤ −λ

min

(R)kf(x)k

≤ 0, x ∈ D. (3.76)

Now, since f (x) = 0 if and only if x = 0, it follows that

V (x) < 0,

x ∈ D, x 6= 0, which proves that the zero solution x(t) ≡ 0 to (3.73) is

a unique asymptotically stable equilibrium with Lyapunov function V (x) =

(x)P f(x).

Finally, in th e case w here D = R

we need only show that V (x) → ∞

as kxk → ∞. Note that it follows from Proposition 3.2 that for every x ∈ R

and some α ∈ (0, 1),

P f (x) = x

∂f

∂x

(αx)x

(



∂f

∂x

(αx)



P + P



∂f

∂x

(αx)



)

≤ −

min

(R)kxk

, x ∈ R

, (3.77)

which implies

P f (x)|

kxk

≥

min

(R), x ∈ R

, x 6= 0. (3.78)

Hence, since |x

P f (x)| ≤ λ

max

(P )kxkkf(x)k, x ∈ R

, it follows from (3.78)

that kf(x)k ≥

min

(R)

2λ

max

(P )

kxk, which implies that V (x) → ∞ as kxk → ∞.

The result now is immediate by repeating the steps of the ﬁrst part of the

proof.

Example 3.9. Consider the nonlinear d y namical system

˙x

(t) = −3x

(t) + x

(t), x

(0) = x

, t ≥ 0, (3.79)

˙x

(t) = x

(t) −x

(t), x

(0) = x

. (3.80)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 157

Note that (0, 0) is the only equilibrium point of (3.79) and (3.80). Next,

computing

∂f

∂x

∂f

(x)

∂x

∂f

(x)

∂x

∂f

(x)

∂x

∂f

(x)

∂x



−3 1

1 −1 − 3x



, (3.81)

it can easily be shown that (3.74) holds with P = I

and R = I

so that

all the conditions of Theorem 3.6 are satisﬁed. Hence, the zero solution

(t), x

(t)) ≡ (0, 0) to (3.79) and (3.80) is globally asymptotically stable

with Lyapunov function V (x) = f

(x)P f(x) = f

(x)f(x) = (−3x

)

− x

)

. △

Next, we present Zubov’s m ethod for constructing Lyapunov functions

for nonlinear systems. Unlike the variable gradient method and Krasovskii’s

method, Zubov’s method add itionally characterizes a domain of attraction

for a given nonlinear system.

Theorem 3.7 (Zubov’s Theorem). Consider the nonlinear dynamical

system (3.1) with f (0) = 0. Let D ⊂ R

be bounded and assume there exist

a continuous ly diﬀerentiable function V : D → R and a continuous function

h : R

→ R such that V (0) = 0, h(0) = 0, and

0 < V (x) < 1, x ∈ D, x 6= 0, (3.82)

V (x) → 1 as x → ∂D, (3.83)

h(x) > 0, x ∈ R

, x 6= 0, (3.84)

′

(x)f(x) = −h(x)[1 −V (x)]. (3.85)

Then, the zero solution x(t) ≡ 0 to (3.1) is asymptotically stable with

domain of attraction D.

Proof. It follows from (3.82), (3.84), and (3.85) that in a neighborhood

(0) of the origin V (x) > 0 and

V (x) < 0, x ∈ B

(0). Hence, the origin

is locally asymptotically stable. Now, to show that D is the domain of

attraction we need to show that x(0) ∈ D implies x(t) → 0 as t → ∞ and

x(0) 6∈ D implies x(t) 6→ 0 as t → ∞. Let x(0) ∈ D. Then, by (3.82),

V (x(0)) < 1. Next, let β > 0 be such that V (x(0)) ≤ β < 1 and deﬁne

△

= {x ∈ D : V (x) ≤ β}. Note that D

⊂ D and D

is bounded since

lim

x→∂D

V (x) = 1 and β < 1. Furthermore, since

V (x) < 0, x ∈ D

, it

follows that D

is a positively invariant set. Now, using (3.82) it follows

that

V (x) = 0, x ∈ D, implies that h(x) = 0, x ∈ D, which further implies

x = 0. Hence, it follows from Theorem 3.3 that x(t) → 0 as t → ∞.

Next, let x(0) 6∈ D and assume, ad absurdum, that x(t) → 0 as t → ∞.

In this case, x(t) → D for some t ≥ 0. Hence, there exist ﬁnite times t

and t

such th at x(t

) ∈ ∂D and x(t) ∈ D for all t ∈ (t

, t

]. Next, deﬁne

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158 CHAPTER 3

W (x)

△

= 1 − V (x) and note that

W (x) = h(x)W (x) or, equivalently,

W (x(t))

h(x(s))ds, (3.86)

where W

△

= W (x(t

)). Integrating (3.86) and rearranging terms yields

1 −V (x(t

)) = [1 −V (x(t))]e

−

h(x(s))ds

. (3.87)

Now, taking t = t

, letting t

→ t

, and using (3.83) it follows that

lim

→t

[1−V (x(t

))] = 0 and lim

→t

[1−V (x(t

))]e

−

h(x(s))ds

> 0, which

is a contradiction. Hence, for x(0) 6∈ D, x(t) 6→ 0 as t → ∞.

Note that in the case where D is unbou nded or D = R

, (3.83) becomes

V (x) → 1 as kxk → ∞. In latter case, the conditions of Theorem 3.7, with

(3.83) replaced by V (x) → 1 as kxk → ∞, guarantee that the zero solution

x(t) ≡ 0 to (3.1) is globally asymptotically stable.

Example 3.10. Consider the second-order nonlinear dynamical system

adopted from [178] given by

˙x

(t) = −f

(t)) + f

(t)), x

(0) = x

, t ≥ 0, (3.88)

˙x

(t) = −f

(t)), x

(0) = x

, (3.89)

where f

(0) = 0, σf

(σ) > 0, σ ∈ (−a

, b

), for i = 1, 2, 3, a

= a

, b

= b

and

(s)ds → ∞ as y → −a

or y → b

, for i = 2, 3. Next, let h(x) =

) and let V (x) be of the form V (x) = 1 − V

), where

: R → R, V

(0) = 1, i = 1, 2. Now, it follows from (3.85) that

′

) + f

)]V

)

+[V

′

) −V

′

)] = 0, (3.90)

which can be satisﬁed by s etting V

′

) = −f

) and V

′

) =

−f

). Hence, (3.85) holds with

V (x) = 1 − e

−[

(s)ds+

(s)ds]

. (3.91)

Note that (3.91) satisﬁes V (0) = 0, (3.82), and (3.83) for all x ∈ D =

{x ∈ R

: −a

< x

< b

}, i = 1, 2. Furthermore, it can be easily shown that

V (x) = −f

)[1 − V (x)] ≤ 0, which p roves Lyapunov stability of

(3.88) and (3.89). To show asymptotic stability note that

V (x) = 0 imp lies

) = 0, which further implies x

= 0. Furthermore, x

(t) ≡ 0

implies f

(t)) ≡ 0, which further implies x

(t) ≡ 0. Hence, with D

= D,

it follows from Theorem 3.3 that the zero solution (x

(t), x

(t)) ≡ (0, 0) to

(3.88) and (3.89) is asymptotically stable with domain of attraction D. △

Finally, we present the energy-Casimir method for constructing Lya-

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STABILITY THEORY FOR NONLINEAR SYSTEMS 159

punov functions for nonlinear dynamical systems. This method exploits the

existence of dynamical invariants, or integrals of motion, called Casimir

functions of the nonlinear dynamical system (3.1). In particular, a function

C : D → R is an integral of motion of (3.1) if it is conserved along the ﬂow of

(3.1), that is, C

′

(x)f(x) = 0. For the statement of our n ext result let r ≥ 2

and let C

: D → R, i = 1, . . . , r, be two-times continuously diﬀerentiable

Casimir functions. Furthermore, deﬁne

E(x)

△

i=1

(x), (3.92)

for µ

∈ R, i = 1, . . . , r.

Theorem 3.8 (Energy-Casimir Theorem). Consider the nonlinear

dynamical system (3.1) where f : D → R

is Lipschitz continuous on D. Let

∈ D be an equilibrium point of (3.1) and let C

: D → R, i = 1, . . . , r, be

Casimir functions of (3.1). Assum e that the vectors C

′

), i = 2, . . . , r, are

linearly independ ent, and suppose there exists µ = [µ

, µ

, . . . , µ

]

∈ R

such that µ

6= 0, E

′

) = 0, and x

′′

)x > 0, x ∈ M, where

△

= {x ∈ D : C

′

)x = 0, i = 2, . . . , r}. Then, there exists α ≥ 0

such that

′′

) + α

i=2



∂C

∂x

)





∂C

∂x

)



> 0. (3.93)

Furth ermore, the equilibrium solution x(t) ≡ x

of (3.1) is Lyapunov stable

with Lyapunov fun ction

V (x) = E(x) − E(x

) +

i=2

(x) − C

)]

. (3.94)

Proof. Note that

V (x) = V

′

(x)f(x)

= E

′

(x)f(x) + α

i=2

(x) − C

)]C

′

(x)f(x)

i=1

′

(x)f(x) + α

i=2

(x) − C

)]C

′

(x)f(x)

= 0, x ∈ D. (3.95)

Now, it need only be shown that V (x

) = 0 and V (x) > 0, x ∈ D, x 6= x

Clearly, V (x

) = 0. Furthermore,

′

(x) = E

′

(x) + α

i=2

(x) − C

)]C

′

(x), (3.96)

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160 CHAPTER 3

and h en ce, V

′

) = 0.

Next, note that

′′

(x) = E

′′

(x) + α

i=2



′

(x)]

′

(x) + [C

(x) − C

)]C

′′

(x)



, (3.97)

and h en ce,

′′

) = E

′′

) + α

i=2



∂C

∂x

)





∂C

∂x

)



. (3.98)

Now, to show the existence of α satisfying (3.93), n ote that since C

′

)x =

0, i = 2, . . . , r, x ∈ M, it follows that there exists an invertible matrix

S ∈ R

n×n

such that [0, I

r−1

]Sx = 0, x ∈ M, and [I

n−(r−1)

, 0]Sx = 0,

x ∈ M

[288]. Hence, for x

∈ D,

′′

) = S





S (3.99)

and

i=2



∂C

∂x

)





∂C

∂x

)



= S



0 0

0 N



S, (3.100)

where E

∈ R

(n−r+1)×(n−r+1)

, E

∈ R

(n−r+1)×(r−1)

, E

∈ R

(r−1)×(r−1)

, and

N ∈ R

(r−1)×(r−1)

are symmetric, and E

and N are positive deﬁnite. Next,

substituting (3.99) and (3.100) into (3.98) yields

′′

) = S



+ αN



△

= S

QS. (3.101)

Now, choosing α ≥ 0 such that Q > 0, (3.93) is satisﬁed, and hence,

′′

) > 0. Since V (·) is two-times continuously diﬀerentiable, it follows

that (see Problem 3.2) V (x), x ∈ D, is positive deﬁnite in the neighborhood

of x

It is clear from Theorem 3.8 that the existence of energy-Casimir

functions for (3.1) can be used to construct Lyapunov functions for (3.1).

In particular, suppose we can construct a function H : D → R su ch that

H(x) = 0 along the trajectories of the nonlinear dynamical system (3.1). If

, . . . , C

are Casimir functions for (3.1), then

[H + E(C

, . . . , C

)](x(t)) ≡ 0 (3.102)

for every function E : R

→ R. Hence, even if H is not positive deﬁnite at

the equilibrium x

∈ D, the function V (x) = H(x) + E(C

(x), . . . , C

(x))

can be made positive deﬁnite at x

∈ D by properly choosing E so that

V (x) is a Lyapunov function for (3.1).

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STABILITY THEORY FOR NONLINEAR SYSTEMS 161

Example 3.11. This example is adopted from [450] and considers the

nonlinear d ynamical system representing a rigid spacecraft given by (3.11)–

(3.13) in Example 3.1. To show that the equilibrium solution x(t) ≡ x

where x

= [0, 0, x

]

, to (3.11)–(3.13) is Lyapunov stable note that

(x) =

+ I

), (3.103)

(x) =

+ I

), (3.104)

are Casimir functions for (3.11)–(3.13). Now, letting E(x) = µ

(x) +

(x) it follows that E

′

) = 0 and x

′′

)x > 0, x ∈ M, x 6= 0, are

satisﬁed with µ

= −I

and µ

= 1. Next, using

V (x

, x

) = E(x

, x

)−E(0, 0, x

, x

)−C

(0, 0, x

)]

(3.105)

it follows that Q in (3.101) is given by

Q =





− I

) 0 0

0 I

− I

) 0

0 0 αI





. (3.106)

Note that Q > 0 for every α > 0. Hence, it follows from Theorem 3.8 that

the equilibr ium solution x(t) ≡ x

to (3.11)–(3.13) is Lyapunov stable with

Lyapunov function (3.105). △

3.5 Converse Lyapunov Theorems

In the previous sections the existence of a Lyapunov function is assumed

while stability properties of a nonlinear dynamical s ys tem are deduced. Th is

raises the question of whether or not there always exists a Lyapunov function

for a Lyapunov stable and an asymptotically stable nonlinear dynamical

system. A number of results concerning the existence of continuously

diﬀerentiable Lyapunov functions for Lyapunov stable and asymptotically

stable time-varying nonlinear systems known as converse Lyapunov theorems

address this problem [178, 298, 306, 307, 445, 474]. However, unlike converse

theorems f or time-varying systems where the existence of a continuously

diﬀerentiable Lyapunov function for a Lyapunov s table sys tem is ensured,

in the time-invariant case Lyapunov stability does not in general imply

the existence of a continuously diﬀerentiable or even a continuous time-

independent Lyapunov function (see Example 4.15). For further discussion

on this often overlooked fact the interested reader is r eferred to [134, 178].

However, the existence of a lower semicontinuous Lyapu nov function for

a Lyapun ov stable system is guaranteed (see Section 4.8). As sh own

below, however, there always exists a continuously diﬀerentiable time-

independ ent Lyapunov function for asymptotically stable, time-invariant

nonlinear dynamical systems. I n order to state and prove the converse

Lyapunov th eorems we need s everal deﬁn itions and one key lemma.

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162 CHAPTER 3

Deﬁnition 3.3. A continuous function γ : [0, a) → [0, ∞), wh ere a ∈

(0, ∞], is of class K if it is strictly increasing an d γ(0) = 0. A continuous

function γ : [0, ∞) → [0, ∞) is of class K

∞

if it is strictly increasing, γ(0) =

0, and γ(s) → ∞ as s → ∞. A continuous function γ : [0, ∞) → [0, ∞)

is of class L if it is strictly decreasing and γ(s) → 0 as s → ∞. Finally, a

continuous function γ : [0, a) × [0, ∞) → [0, ∞) is of class KL if, for each

ﬁxed s, γ(r, s) is of class K with respect to r and, for each ﬁxed r, γ(r, s) is

of class L with respect to s.

The following key lemma due to Massera [306] is needed for the main

result of this section.

Lemma 3.1 (Massera’s Lemma). Let σ : [0, ∞) → [0, ∞) be a class

L function and let λ be a given positive scalar. Then there exists an

inﬁnitely diﬀerentiable function γ : [0, ∞) → [0, ∞) such that γ(·), γ

′

(·) ∈ K,

∞

γ[σ(t)]dt < ∞, and

∞

′

[σ(t)]e

λt

dt < ∞.

Proof. Since σ(·) is strictly decreasing there exists an unbounded

sequence {t

}

∞

n=1

such that

σ(t

) ≤

n + 1

, n = 1, 2, . . . .

Now, deﬁne η : R

→ R

η(t)

△

(





, 0 < t ≤ t

(n+1)t

n+1

−nt

−t

n(n+1)(t

n+1

−t

)

, t

≤ t ≤ t

n+1

, n = 1, 2, . . . ,

where p >

2(t

−t

)

. Note that η(·) is strictly decreasing, σ(t) < η(t), t ∈

, ∞), and lim

t→0,t>0

η(t) = ∞. Since η(·) is an inﬁnitely diﬀerentiable

function except at a countable number of values of t, it can be approximated

by an inﬁnitely diﬀerentiable function on [0, ∞). Next, it can be shown that

−1

(·) is a strictly decreasing f unction such that lim

s→0,s>0

−1

(s) = ∞ and

lim

s→∞

−1

(s) = 0. Now, let µ : R

→ R

be such that µ(0) = 0 and

µ(s) = e

−(λ+1)η

−1

(s)

, s > 0. Note that µ(·) is inﬁnitely diﬀerentiable on

(0, ∞) and µ(·) is strictly increasing, which implies that µ(·) is a class K

function.

Next, deﬁne γ : [0, ∞) → [0, ∞) by

γ(r)

△

µ(s)ds.

Note that γ(·) and γ

′

(·) = µ(·) are class K functions. Fu rthermore, note

that

∞

′

[σ(t)]e

λt

dt =

µ[σ(t)]e

λt

dt +

∞

µ[σ(t)]e

λt

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STABILITY THEORY FOR NONLINEAR SYSTEMS 163

µ[σ(t)]e

λt

dt +

∞

µ[η(t)]e

λt

µ[σ(t)]e

λt

dt +

∞

−t

< ∞.

Next, s ince for all t ≥ t

and s ∈ (0, η(t)], t ≤ η

−1

(s), it follows that

η(t)

µ(s)ds =

η(t)

−(λ+1)η

−1

(s)

≤

η(t)

−(λ+1)t

= e

−(λ+1)t

η(t)

≤ e

−(λ+1)t

Hence,

∞

γ[σ(t)]dt =

∞

σ(t)

µ(s)dsdt

∞

η(t)

µ(s)dsdt

≤

∞

−(λ+1)t

< ∞.

Now, the result follows from the fact that

∞

γ[σ(t)]dt < ∞ if and only if

∞

γ[σ(t)]dt < ∞.

Theorem 3.9. Assume that the zero solution x(t) ≡ 0 to (3.1) is

asymptotically stable, f : D → R

is continuously diﬀerentiable, and let

δ > 0 be s uch that B

(0) ⊂ D is contained in the domain of attraction of

(3.1). Then there exists a continuously diﬀerentiable function V : B

(0) → R

such that V (0) = 0, V (x) > 0, x ∈ B

(0), x 6= 0, and V

′

(x)f(x) < 0,

x ∈ B

(0), x 6= 0.

Proof. Sin ce the zero solution x(t) ≡ 0 to (3.1) is, by assumption,

asymptotically stable it follows that (see Problem 3.76) there exist class

K and L functions α(·) and β(·), respectively, such that if kx

k < δ then

kx(t)k ≤ α(kx

k)β(t), t ≥ 0, where k · k is the Euclidean norm. Next,

let x ∈ B

(0) and let s(t, x) denote the solution to (3.1) with the initial

condition x(0) = x so that for all kxk < δ, ks(t, x)k ≤ α(kxk)β(t), t ≥ 0.

NonlinearBook10pt November 20, 2007

164 CHAPTER 3

Furth ermore, note that s(t, x) = x +

f(s(τ, x))dτ , which implies th at

∂s(t, x)

∂x

= I +

′

(s(τ, x))

∂s(τ, x)

∂x

dτ. (3.107)

Hence,



∂s(t, x)

∂x



≤ 1 +



∂s(τ, x)

∂x



dτ, (3.108)

where λ

△

= max

x∈B

(0)

′

(x)k. Fur th ermore, it follows f rom Lemma 2.2

that k

∂s(t,x)

∂x

k ≤ e

λt

. Now, it follows from Lemma 3.1 that there exists an

inﬁnitely diﬀerentiable function γ(·) such that γ(·) and γ

′

(·) are class K

functions,

∞

γ(α(δ)β(t))dt < ∞, and

∞

′

(α(δ)β(t))e

λt

dt < ∞. Since

ks(t, x)k ≤ α(kxk)β(t) ≤ α(δ)β(t), t ≥ 0, it follows that the function

V (x) =

∞

γ(ks(t, x)k)dt (3.109)

is well deﬁned on B

(0). Furthermore, note that V (0) = 0.

Next, s ince f(·) in (3.1) is uniformly Lipschitz continuous on B

(0) it

follows from Proposition 2.30 that

V (x) =

∞

γ(ks(t, x)k)dt ≥

∞

γ(kxke

−Lt

)dt,

where L denotes the Lipschitz constant of f(·) on B

(0), which implies that

V (x) > 0, x ∈ D

, x 6= 0. Now, note that since ks(t, x)k =

(t, x)s(t, x),

′

(x) =

∞

′

(ks(t, x)k)

(t, x)

ks(t, x)k

∂s(t, x)

∂x

dt, (3.110)

and h en ce,

′

(x)k ≤

∞

′

(ks(t, x)k)



∂s(t, x)

∂x



≤

∞

′

(α(δ)β(t))e

λt

< ∞, x ∈ B

(0), (3.111)

which proves that V

′

(·) is bounded. Now, (3.110) implies that V

′

(·) is

continuous, and hence, V (·) is continuously diﬀerentiable.

Finally, note that since f(·) is Lipschitz continuous on D the trajectory

s(t, x), t ≥ 0, is unique, and hence,

V (s(t, x)) =

∞

γ(ks(τ, s(t, x))k)dτ

∞

γ(ks(τ + t, x)k)dτ