Infinite time regular synthesis

INFINITE TIME REGULAR SYNTHESIS

B. PICCOLI

Abstract. In this paper we provide a new suciency theorem for regular syntheses. The concept of regular synthesis is discussed in 12], where a suciency theorem for nite time syntheses is proved.

There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarily bounded. This research is motivated by the fact that one of the main tools toward the construction of optimal syntheses is the proof of a strong suciency theorem.

The regularity assumptions of the main theorem in 12] are veri ed by every piecewise smooth feedback control generating extremal tra- jectories that reach the target in nite time with a nite number of switchings (indeed even by more complicate syntheses like the Fuller one presenting trajectories with an in nite number of switchings). In the case of this paper the situation is even more complicate, since we admit both trajectories with nite and in nite time. It is important to notice that, in spite of its complexity, this situation is encountered in many simple cases like linear-quadratic problems (see the example of the last section and 13]).

We use weak dierentiability assumptions on the synthesis and weak continuity assumptions on the associated value function. However, in this paper we need the value function to be continuous at the origin (see Remark 3.3 for more details). The general case of synthesis gen- erated by general piecewise smooth feedback deserves a further careful investigation.

1. Introduction

Given an optimization problem for a control system with xed initial data, there are various ways of giving a solution. One can nd an open loop con- trol or try to solve all the problems obtained by varying the initial data.

Beside the classical concept of feedback, one can construct a trajectory for every initial data in such a way that the resulting collection has nice reg- ularity properties. This gives raise to what is called a regular synthesis.

Firstly introduced by Boltianskii, the concept of regular synthesis has been developed and used by many authors in connection with theoretical prob- lems as well as for special classes of systems 2{8], 10{16], 18]. A synthesis is a mathematical object with more \structure" with respect to a feedback, and there are examples showing that a feedback alone is not sucient to describe the solution to an optimization problem, see 12].

In this paper we provide a new suciency theorem for regular synthesis.

The concept of regular synthesis is discussed in 12], where it is proved a

SISSA-ISAS, via Beirut 2-4, 34014 Trieste, Italy. Email: piccoli@sis sa. it.

Received by the journal August 21, 1997. Revised June 17, 1988. Accepted for publi- cation October 8, 1998.

c Societe de Mathematiques Appliquees et Industrielles. Typeset by TEX.

suciency theorem for syntheses formed by nite time trajectories. We refer to 12] and the references therein for synthesis theory.

There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarily bounded, see 13].

One of the main tools toward the construction of optimal syntheses is the proof of a strong suciency theorem. If an optimization problem is su- ciently regular then every optimal trajectory must satisfy the Pontryagin Maximum Principle (in this case we say that the trajectory is extremal).

Hence a rst necessary condition for optimality is extremality. However, in general extremality is not a sucient condition for a single trajectory (see 7], 8], 15]). Moreover, no regularity property can ensure the optimality of an extremal trajectory(see 15]). On the other hand, the suciency theo- rems guarantees that every extremal synthesis that is suciently regular is indeed optimal.

Usually one builds a candidate optimal synthesis using regularity prop- erties of extremal trajectories. Thus extremality is a condition granted by construction. The regularity assumptions of the main theorem in 12] are veried by every piecewise smooth feedback control generating extremal tra- jectories that reach the target in nite time with a nite number of switch- ings. Moreover, the same theorem applies also to the system considered by Fuller, that presents trajectories with an innite number of switchings (however it is not clear under which conditions it is possible to apply the theorem to a generic piecewise smooth feedback). In the case of this paper the situation is even more complicate since we admit trajectories dened on unbounded domains. We need weak dierentiability assumptions on the synthesis and weak continuity assumptions on the value function associated to the synthesis. However, to treat innite time trajectories, we have to assume the continuity of the value function at the origin, while only weak upper semicontinuity was assumed in 12] (see Remark 1 for more details).

Moreover, we give two dierent denitions of weak dierentiability, see Re- mark 3.3 of Section 3. The rst relies on strong integrability properties (of the Jacobians of the dynamics and the cost) along the trajectories, the other, veried by the linear-quadratic example of Section 4, requires fast in time convergence of trajectories to the target.

We state a result valid for a point target problem. The same result can be used for Bolza problems without nal condition when the form of the Lagrangian naturally forces the trajectories to tend asymptotically to a given point. This is the case of some examples, see 13] and Section 4.

We remark that in our problem we admit both trajectories with compact domain and trajectories with unbounded domains at the same time. As remarked in 12], our denition of regular synthesis is quite mild compared to the other denitions given in previous papers 2], 5], 6]. Moreover, these conditions can be checked easily in some interesting examples. The main result is stated for presynthesis even if we do not know interesting examples in which the concept of presynthesis plays a crucial role.

Finally, let us mention that another well known approach ensuring su- ciency theorems is the theory of viscosity solutions for the Hamilton{Jacobi{

Bellmann equations, see 1],9].

In Section 2 we give basic denitions, in Section 3 we state and prove the main result and in section 4 we give examples.

2. Basic definitions Consider a control system:

x

_ =

f

(

xu

)

x

²

u

²

U

(2

:

1) and the minimization problem:

minimize

Z

L

(

(

t

)

(

t

))

dt

(2

:

2) where (

) is a trajectory{control pair satisfying some admissibility condi- tions that will be stated later.

Our basic assumption is

(H). is an open subset of^Rn,

U

is a set,

f

:

U

^!Rnand

L

:

U

^!R are maps such that

f

(

u

) and

L

(

u

) are of class

C

¹ for every xed

u

²

U

. We use the same notation of 12]. In particular, we use ~

f

to denote the

R

n+1valued map (

xu

)^!(

f

(

xu

)

L

(

xu

)).

A control is a map

:

I

^!

U

, whose domain is a subinterval

I

of ^R(not necessarily bounded).

If

:

A

^!

B

is a map, we use Dom(

) to indicate the domain of

, i.e.

the set

A

. In particular, if

:

I

^!

U

is a control, then Dom(

) =

I

.

A trajectory

for a control

is an absolutely continuous map

: Dom(

)^! that satises _

(

t

) =

f

(

(

t

)

(

t

)) for almost every

t

²Dom(

).

If Dom(

) =

ab

] (so that Dom(

) =

ab

] as well), and

x

=

(

a

),

y

=

(

b

), we say that

(or the pair (

)) goes from

x

to

y

, and that

steers

x

to

y

, and we write

^; =

x

,

⁺ =

y

. In the same way if Dom(

) =

a

+¹) then we simply write

^; =

x

. Moreover, if lim^t!+¹

(

t

) =

y

then we write

⁺=

y

.

A control

:

I

^!

U

is admissible if the map

I

³(

xt

) ^!

f

~(

xt

) =

f

~(

x

(

t

))^2Rn+1satises the following

C

¹ Caratheodory conditions:

(A) ~

f

is measurable as function of (

tx

)

(B) for every compact

K

and every compact interval

J

I

, there exists an integrable function

'

^KJ such that for every

x

²

K

and

t

²

J

:

k

f

~(

xt

)^k+^k

D

^x

f

^~(

xt

)^k

'

^KJ(

t

)

:

(2

:

3) If

is an admissible control,

a trajectory corresponding to

, such that the integral in (2.2) is dened (possibly equal to +¹), then we say that (

) is an admissible pair for ~

f

. We use Adm( ~

f

) to denote the set of all admissible pairs for ~

f

. For an admissible pair (

), we use

J

(

) to denote the cost of (

), i.e. the value of the integral of (2.2) over Dom(

).

Given

^2Rn(the space of row

n

vectors),

0 ^2R,

x

² and

u

²

U

, we dene:

H(

x

0

u

) =^h

f

(

xu

)ⁱ+

0

L

(

xu

) (2

:

4)

H

(

x

0) = inf^fH(

x

0

u

) :

u

²

U

^g

:

(2

:

5) The functions ^H: ^RnR

U

^!Rand

H

: ^RnR!Rf;1g are known as the Hamiltonian and the minimized Hamiltonian of ~

f

.

We say that the pair (

)²Adm( ~

f

) is extremal if there exist an abso- lutely continuous map

: Dom(

) ^{! Rn}, called the adjoint vector, and a constant

0 0, not both zero, such that the adjoint equation

_ =^;

@

^H

@x

⁽

(

t

)

0

(

t

)

(

t

)) (2

:

6) and the minimization condition:

H

(

(

t

)

(

t

)

0) =^H(

(

t

)

(

t

)

0

(

t

)) = 0 (2

:

7) hold for almost every

t

²Dom(

). For a discussion of the validity of PMP under our assumptions see 18].

3. The main result

We consider the problem (2.1), (2.2) with a point target that, without loss of generality, will be assumed to be the origin. We consider only trajectories

whose domain is bounded below, that is either Dom(

) =

ab

] or Dom(

) =

a

+¹) for some

a

^{2 R}. Hence a trajectory has to start at some time

a

from a point

x

and either reach the origin in nite time or converge to it as

t

tends to +¹. We call Adm⁰( ~

f

) the set of such trajectories (0 indicates the fact that they end at the origin).

If ;Adm⁰( ~

f

) is an arbitrary set of admissible pairs, we dene a function

V

; : ^!Rf1g, called the value function of ; , by letting

V

;(

x

) be, for

x

², the inmum of the costs

J

(

), taken over the set of all (

)²; such that

goes from

x

to 0. (In particular, if there is no (

)²;going from

x

to 0 then

V

;(

x

) = +¹.) If ;= Adm⁰( ~

f

), then the function

V

; is the value function of our problem, and in that case we will write

V

^f~ rather than

V

_Adm⁰_{( ~}^f₎. A pair (

)²Adm⁰( ~

f

) is optimal if

V

^f~(

^;) =

J

(

).

A presynthesis for our problem on a set

S

is a set ; = ^f(

^x

^x) :

x

²

S

^g of admissible pairs (

^x

^x) ²Adm⁰( ~

f

) such that

^x; =

x

. The set

S

is called the domain of ;. If the domain

S

of ;consists of all points that can be steered to the origin by an admissible pair, then we say that ; is total. Given a presynthesis ;we will always renormalize the time along every (

^x

^x) in such a way that the domain of

^x (and of

^x) either is of the form 0

T

^x] for some non negative number

T

^x or it is 0

+¹).

A presynthesis is memoryless if whenever

x

²

S

and

t

²Dom(

^x) it follows that

y

=

^x(

t

) belongs to

S

and

^yis the renormalization of the restriction of

^xto the interval

tT

^x] or

t

+¹). A synthesis is a memoryless presynthesis.

If each pair of a presynthesis ;is optimal (resp. extremal) then we say that ;is optimal (resp. extremal). In particular, a presynthesis ;is a total optimal presynthesis if and only if

V

;

V

^f~.

Our goal is to prove that, under suitable regularity assumptions on ; and

V

;, a total extremal presynthesis is optimal. We rst describe these regularity assumptions in detail.

Given a locally Lipschitz vector eld

X

on , we say that

V

has the NDJ (\no downward jumps") property along

X

if the following holds:

(NDJ).For every

integral curve of

X

,

t

²Dom(

), if

a

= inf Dom(

) then for every

t

²Dom(

)^nf

a

^g we have

limsup

h!0+

V

(

(

t

))^;

V

(

(

t

^;

h

))0

:

We say that

V

satises the weak continuity conditions for the control problem (2.1), (2.2) if

V

is lower semicontinuous, continuous at the origin, and has the NDJ property along the vector eld

x

^!

f

(

xu

) for every

u

²

U

. We now dene the \weak dierentiability conditions" for ;. In this def- inition it is understood that ~

f

(

y

^x(

t

)) = 0 for every

t =

² Dom(

^x). A presynthesis ;=^f(

^x

^x) :

x

²

S

^g is (

fL

){dierentiable at a point !

x

² if the following assumption is satised:

(A) there exist an open set

W

, containing ^f

^x(

t

) :

t

²Dom(

^x)^g, a neigh- borhood

N

of !

x

(in ) such that

N

Dom(;), with the property that (A1) for every compact

K

, there exist an integrable function

'

^K : 0

+¹] ^{! R}such that ^j

'

^JK(

t

)^j

'

^K(

t

) for every compact

J

I

,

t

² Dom(

'

^JK(

t

)) (here

'

^JK are dened as in (2.3) for the control

^x), and, for suciently small

" >

0, integrable functions

^": 0

+¹]^!Rsuch that lim^"!0^R+¹

0

^"(

t

)

dt

= 0, and the inequalities

k

f

~(

y

^x(

t

))^;

f

~(

y

^x(

t

))^k

^"(

t

)

k

D

^y

f

^~(

y

^x(

t

))^;

D

^y

f

^~(

y

^x(

t

))^k

^"(

t

)

hold for every

y

²

W

,

x

²

N

such that ^k

x

^;

x

!^k

"

,

t

²0

+¹).

(A2) the map

v

^!

^v, where

^v is the integrable ^Rn+1valued function on 0

+¹) given by

^v(

t

) = ~

f

(

^x(

t

)

^x+^v(

t

))^;

f

~(

^x(

t

)

^x(

t

))

is vaguely dierentiable at

v

= 0 or it is weak-dierentiable at

v

= 0, regarded as a map into the space of ^Rn+1{valued Borel measures. That is: for every continuous function

: 0

+¹) ^{! R}, that vanishes at +¹ (lim^t!+¹

(

t

) = 0), the map ^Rn3

v

^{! R}₀⁺¹

^v(

t

)

(

t

)

dt

^2Rn+1is dier- entiable at

v

= 0.

Moreover the map^Rn3

v

^!R₀⁺¹

^00v(

t

)

dt

^2R, where

^00v is the (

n

+1)-th component of

^v, is dierentiable at

v

= 0.

A set

A

is (

n

^;1) dimensional rectiable if there exist

A

1,

A

2, such that

A

=

A

1

A

2,

A

1 is a nite or countable union of connected embedded

C

¹ submanifolds of positive codimension and

A

2 veries ^Hn;1(

A

2) = 0 where^Hn;1 is the

n

^;1 dimensional Hausdor measure.

We can now state the main theorem (cf. 12], 18]):

Theorem 3.1. Let,

U

,

f

,

L

be such that0² and assumption (H) hold.

Let; be a total extremal presynthesis. Assume that

(i) the associated value function

V

;satises the weak continuity conditions, (ii)

V

;(0) = 0,

(iii) ; is (

fL

){dierentiable at all points in the complement of a (

n

^;1) dimensional rectiable set

A

.

Then; is optimal.

Proof. We rst prove the relation between the dierential of the value func- tion

V

; and the adjoint covector elds along the extremal trajectories of the presynthesis. This is of interest in itself so we state it as a separate

theorem. ^tu

Theorem 3.2. Let ,

U

,

f

,

L

be such that 0 ² and assumption (H) hold. Let ; be an extremal presynthesis. If ; is (

fL

){dierentiable at !

x

and if

denote the adjoint covector associated to the pair(

^x

^x)²; then

(0) =

D

^y

V

;(!

x

).

Proof of Theorem 3.2. Fix a point !

x

of (

fL

){dierentiability for ; . Let

W

,

N

,

^", be as given by Condition (A). Pick

>

0 such that the compact set

W

⁰ = ^f

x

:

d

(

x

^f

^x(

t

) :

t

²Dom(

^x)^g)

^g

satises

W

⁰

W

. ^tu

Let

X

be the Banach space

C

0(0

+¹)

^R) of continuous real-valued func- tions on 0

+¹), that vanishes at +¹, endowed with the sup norm. We recall that usually

X

is dened as the completion for the sup norm of the space of continuous functions with compact support.

With

^v : 0

+¹)^{! Rn+1}dened as in (A2) for

v

^2Rn, ^jj

v

^jj small, let

^1v

:::

^nv+1be the components of

^v, so each

^jv is an integrable real-valued function on 0

+¹). Then for each

²

X

and each

j

^2f1

:::n

+ 1^g the function

v

^{! R}₀⁺¹

(

t

)

^jv(

t

)

dt

is dierentiable at

v

= 0, so there exists, for each

j

, a unique

-dependent vector

w

^j(

)^2Rnsuch that

Z +¹

0

(

t

)

^jv(

t

)

dt

=^h

w

^j(

)

v

ⁱ+

o

(^jj

v

^jj) as

v

^!0

:

(3

:

1) It is clear that

w

^j(

) depends linearly on

, so each component

w

^ji(

) (

i

= 1

:::n

) of

w

^j(

) depends linearly on

as well. For every

v

^{2 Rn} suciently small,

^jv

dt

is a nite Borel measure, hence an element of

X

(the dual of

X

). Let us indicate with by^kkX the norm of

X

. For every xed

²

X

, from the dierentiability at

v

= 0, it follows that there exists

C

1

>

0 such that for every

v

⁶= 0 suciently small

1

k

v

^k

Z

^jv

dt C

1

:

Hence the Uniform Boundedness Theorem implies that there exists a con- stant

C

2 such that

k

^jvkL1

k

v

^{k =}

^jv

k

v

^k

X

C

2

(3

:

2) for

v

small and

j

^{2 f}1

:::n

+ 1^g. (Otherwise there would exist a

j

and a sequence ^f

v

^`g such that

v

^{` !} 0,

v

^{` 6}= 0, and ^R₀^+1jj

^jv`(

t

)^jj

dt > K

^`jj

v

^`jj, with

K

^{` !</sup> +<sup>1</sup>. By passing to a subsequence, we may assume that <sup>jjv`jjv`} converges to a unit vector

v

. Then by (3.1) the continuous linear functionals

^` :

X

^!Rgiven by

^`(

) = 1

jj

v

^`jj

Z +¹

0

(

t

)

^jv`(

t

)

dt

converge pointwise on

X

to the map

^{! h}

w

^j(

)

v

ⁱ. So the sequence

f

^`(

)^g1`₌₁ is bounded for each

, and then there exists |by Uniform Boundedness| a

C >

0 such that^jj

^`jjX

C

for all

`

. But

jj

^`jjX = 1

jj

v

^`jj

Z +¹

0 ^jj

^jv`(

t

)^jj

dt > K

^`

and we have derived a contradiction.) From (3.2), it follows in particular that

jh

w

^j(

)

v

^ij

C

^jj

^jjjj

v

^jj for

²

X v

^2Rn

so each

w

^ji is a bounded linear functional on

X

and^jj

w

^ijjjX

C

. Therefore each

w

^ji is given by a (signed) Borel measure on 0

+¹), that will also be denoted by

w

^ji. Let ^

w

^ji denote the indenite integral of

w

^ji, i.e.

w

^^ji(

t

) =

w

^ij(0

t

]) for

t

²(0

+¹)

w

^^ij(0) = 0

:

Let

BV

(0

¹) denote the space of all functions

w

: 0

+¹) ^{! R} of bounded variation such that

w

(0) = 0 and

w

is right-continuous at every

t

² (0

+¹). Let^M(0

¹) denote the space of all monotonically nondecreasing functions

w

: 0

+¹) ^{! R}such that

w

(0) = 0 and

w

is right-continuous at every

t

² (0

+¹). Then there is a canonical correspondence between members of

BV

(0

¹) and signed Borel measures on 0

+¹), that assigns to a function

w

²

BV

(0

¹) the unique Borel measure

^w such that

^w(0

t

]) =

w

(

t

) for 0

< t <

+¹. (The measure of a single point set ^f

t

^g is then

^w(

t

+)^;

^w(

t

^;), which is equal to

^w(

t

)^;

^w(

t

^;) if

t >

0 and to

^w(

t

+) if

t

= 0. In particular,

^w(^f

t

^g) = 0 i

^w is continuous at

t

.) From now on we identify a function

w

²

BV

(0

¹) with its corresponding measure

^w, and write ^R

dw

for ^R

d

^w. We emphasize that, if

w

is not continuous, then for an integral such as^Rst

dw

to be well dened we have to be careful to specify whether the integral is to be interpreted as^R

^st]

dw

or^R

(^st]

dw

or

R

^st)

dw

or^R

(^st)

dw

, so we will never write^Rst

dw

unless

w

is continuous.

Clearly, each ^

w

^ij belongs to

BV

(0

¹), and (3.1) can be rewritten as

Z +¹

0

(

t

)

^jv(

t

)

dt

=^Xn

i=1

v

^iZ

d w

^^ij+

o

(^jj

v

^jj)

(3

:

3)

where

v

= (

v

¹

:::v

ⁿ).

We now let ^

^jv(

t

) = ^R₀^t

^jv(

s

)

ds

, so each ^

^jv(

t

) is an absolutely continuous function 0

+¹)^!Rand a member of

BV

(0

¹), that satises

j

^^jv(

t

)^j

C

2^jj

v

^jj for small

v t

²0

+¹)

j

^2f1

:::n

+ 1^g

:

We let

Y

be the space

L

¹(0

+¹)

^R)of all real-valued bounded measurable functions on 0

+¹), endowed with the weak topology of

Y

regarded as the dual of

L

¹(0

+¹)

^R), so a net ^f

y

^d:

d

²

D

^g of members of

Y

|where

D

is a directed set| converges to a

y

²

Y

i ^R

(

t

)

y

^d(

t

)

dt

^{! R}

(

t

)

y

(

t

)

dt

for all

²

L

¹(0

+¹)

^R). We remark that the topology of

Y

is metrizable on subsets of

Y

that are bounded in the

L

¹ norm. Therefore, as long as we are dealing with bounded sets the topology is entirely characterized by the convergence of sequences.

We show that

^^jv =^Xn

i=1

v

ⁱ

w

^^ij+

o

(^jj

v

^jj)

(3

:

4) in the sense that

lim

v !0 1

jj

v

^jj

^^{jv ;Xn}

i=1

v

ⁱ

w

^^ji

= 0 in

Y

for

j

= 1

:::n

+ 1

:

(3

:

5) To see this, x

j

, and write

Q

^j(

v

)(

t

) = 1

jj

v

^jj

^^jv(

t

)^;Xn

i=1

v

ⁱ

w

^^ij(

t

)

:

(3

:

6) Let

^jv(

t

) =

^jv+(

t

)^;

^j;v (

t

), where

^jv+(

t

) = max(

^jv(

t

)

0), and dene

^^jv+(

t

) =

Z

t

0

^jv+(

s

)

ds

^^j;v (

t

) =

Z

t

0

^j;v (

s

)

ds

so each ^

^jv+, ^

^j;v is a monotonically nondecreasing continuous function on 0

+¹) with the property that ^

^jv+(0)=^

^j;v (0)=0, and lim^t!+¹

^^jv+(0) +

^^j;v (0)

C

^jj

v

^jj. If^f

v

^kg1k₌₁is a sequence of nonzero vectors in^Rnconverging to 0, then it follows from Helly's Theorem that there exists a subsequence

f

v

^k(^`)^gand functions

!

^j+,

!

^j; belonging to^M(0

¹), such that 1

jj

v

^k(^`)^jj

^^jvk(`)+ (

t

)^!

!

^j+(

t

) and 1

jj

v

^k(^`)^jj

^^j;vk(`)(

t

)^!

!

^j;(

t

) (3

:

7) for all

t

²0

+¹) that are points of continuity of

!

^j+and

!

^j;. By passing to a subsequence, if necessary, we may assume that there is a vector

v

such that^jj

v

^jj= 1 for which

lim

`!1

v

^k(^`)

jj

v

^k(^`)^jj =

v:

(3

:

8)

It then follows that 1

jj

v

^k(^`)^jj

Z +¹

0

(

t

)

^jvk(`)+ (

t

)

dt

^!Z

d!

^j+

and 1

jj

v

^k(^`)^jj

Z +¹

0

(

t

)

^j;vk(`)(

t

)

dt

^!Z

d!

^j;

for all

²

X

. Indeed, if

(

t

) = ^Pnk₌₁

a

^k

^tktk+1](

t

), for some

a

^{k 2R}and some points

t

^{k 2R}of continuity for

!

^j, the convergence easily follows from (3.7). By approximation, we obtain the conclusion for every

. Therefore, if we let

!

^j =

!

^j+;

!

^j;, we see that

!

^{j 2}

BV

(0

¹) and

1

jj

v

^k(^`)^jj

Z +¹

0

(

t

)

^jvk(`)(

t

)

dt

^!Z

d!

^j for all

²

X

.

On the other hand, (3.3) and (3.8) imply that 1

jj

v

^k(^`)^jj

Z +¹

0

(

t

)

^jvk(`)(

t

)

dt

^!Xn

i=1

v

^iZ

d w

^^ji for all

²

X

. So

!

^{j Pni}₌₁

v

ⁱ

w

^^ji, and then (3.7) implies that

1

jj

v

^k(^`)^jj

^^jvk(`)(

t

)^!Xn

i=1

v

ⁱ

w

^^ji(

t

) for all

t

²0

+¹) except possibly on a countable set. But

1

jj

v

^k(^`)^jj

n

X

i=1

v

^ki₍^`₎

w

^^ji(

t

) ^!Xn

i=1

v

ⁱ

w

^^ji(

t

) for all

t

. So

Q

^j(

v

^k(^`))(

t

) = 1

jj

v

^k(^`)^jj

^^jvk(`)(

t

)^;Xn

i=1

v

^ki₍^`₎

w

^^ji(

t

)

!0 (3

:

9) for almost all

t

. Therefore, since

Q

^j(

v

^k(^`))(

t

) is bounded by a xed constant, independent of

t

and

`

, the Dominated Convergence Theorem implies that

Q

^j(

v

^k(^`))^!0 in

Y

. So we have shown that every sequence ^f

v

^kgconverging to 0 in ^Rn has a subsequence ^f

v

^k(^`)^g for which

Q

^j(

v

^k(^`)) ^! 0 in

Y

. This implies that (3.6) holds.

We can rewrite (3.6) in vector form by introducing the column-vector- valued functions ^

^v :

ab

] ^{! Rn+1} given by ^

^v(

t

) = ^R₀^t

^v(

s

)

ds

, and the matrix-valued function ^

W

: 0

+¹) ^{! Rn+1n} (with

n

+ 1 rows and

n

columns) whose entry in column

i

, row

j

, is ^

w

^ji. Then (3.6) clearly implies that the scalar function

t

^{! 1}

jj

v

^jj

^^v(

t

)^;

W

^(

t

)

:v

converges to 0 in

Y

. It will also be convenient to split

^vinto a

n

-dimensional part

^0v corresponding to the rst

n

components and a scalar part

^00v corre- sponding to the

n

+ 1-th component. With the obvious denition of ^

^0v, ^

^00v,

W

^⁰, ^

W

⁰⁰, we can conclude that the scalar functions

t

^{! 1}

jj

v

^jj

^^0v(

t

)^;

W

^⁰(

t

)

:v

and

t

^{! 1}

jj

v

^jj

^^00v(

t

)^;

W

^⁰⁰(

t

)

:v

converge to 0 in

Y

.

Now consider

x

= !

x

+

"v

,

v

^2Rn,^k

v

^k= 1 and

"

^2Rsmall.

Let

'

be such that ^k

f

(

x

^x(

t

))^;

f

(

y

^x(

t

)^k

'

(

t

)^k

x

^;

y

^k for every

xy

²

W

⁰ and

t

²

ab

]. There exists !

"

such that:

Z +¹

0

^"(

t

)

dt <

₂

e

^k'kL1

for every

"

²0

"

!]. From now on we consider only those

x

corresponding to

" "

!. Fix now such an

x

. There exists

T

(

x

) such that:

k

^x(

T

(

x

))^k

<

₄

e

^k'kL1

^k

^x(

T

(

x

))^k

<

₄

e

^k'kL1

(3

:

10

a

) and for

t

T

(

x

)

k

^x(

t

)^;

^x(

t

)^{k k}

x

^;

x

!^k

:

(3

:

10

b

) Notice that we may have that the trajectory

^x (or

^x) reaches the origin in nite time. But in this case we consider it as prolonged on 0

+¹) by

^x(

t

) = 0 for

t

sup(Dom(

^x)). This is compatible with the denition

f

^(

^x(

t

)

^x(

t

)) = 0 for

t =

²

Dom

(

^x).

Dene

^x(

t

) =

^x(^;

t

),

^x(

t

) =

^x(^;

t

) so that

^x: ^;

T

(

x

)

0]^!Rnis

^x run backward in time and it satises the equation:

_^x(

t

) =^;

f

(

^x(

x

)

^x(

t

))

:

Now

k

^x(

t

)^;

^x(

t

)^{k k}

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)^k

f

(

^x(

s

)

^x(

s

))^;

f

(

^x(

s

)

^x(

s

))^k

ds

k

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)^k

f

(

^x(

s

)

^x(

s

))^;

f

(

^x(

s

)

^x(

s

))^k

ds

+

Z

t

;T(^x)^k

f

(

^x(

s

)

^x(

s

))^;

f

(

^x(

s

)

^x(

s

))^k

ds

k

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)

^"(^;

s

)

ds

+

Z

t

0

'

(^;

s

)^k

^x(

s

)^;

^x(

s

)^k

ds

as long as

^x(

s

)

^x(

s

)²

W

⁰ for every

s

^2;

T

(

x

)

0]. Then

k

^x(

t

)^;

^x(

t

)^k

e

^k'kL1

k

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k+

Z

t

;T(^x)

^"(^;

s

)

ds

!

e

^{k'kL1 k}

^x(

T

(

x

))^k+^k

^x(

T

(

x

))^k+

Z +¹

0

^"(

s

)

ds

4 +

4 +

2 =

:

(3

:

11) Assume by contradiction that

^x(

t

) is not in

W

⁰for some

t

^2;

T

(

x

)

0] and let !

t

the rst time in which this happens. Using the estimate (3.11) at time

!

t

we obtain

k

^x(!

t

)^;

^x(!

t

)^k

<

(3

:

12) that gives us the contradiction. Hence we can conclude that for

"

² 0

"

!],

^x+^"v(

t

) ²

W

⁰ for

t

^2;

T

(

x

)

0] and from now on we consider only those

x

= !

x

+

"v

for which

" "

!.

For simplicity, denote by

'

the integrable function

'

^W0 of (A1). From (A1)it follows

sup

y 2W 0

k

D

^y

f

^~(

y

^x(

t

))^k sup

y 2W 0

k

D

^y

f

^~(

y

^x(

t

))^;

D

^y

f

^~(

y

^x(

t

))^k + sup

y 2W 0

k

D

^y

f

^~(

y

^x(

t

))^k

^"(

t

) +

'

(

t

)

:

(3

:

13) Notice that from (3.12) we know that the segment joining

^x(

t

) to

^x(

t

) is inside

W

⁰ thus we can apply the mean value inequality and use (3.13).

Hence

k

^x(

t

)^;

^x(

t

)^{k k}

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)^k

f

(

^x(

s

)

^x(

s

))^;

f

(

^x(

s

)

^x(

s

))^k

ds

+

Z

t

;T(^x)^k

f

(

^x(

s

)

^x(

s

))^;

f

(

^x(

s

)

^x(

s

))^k

ds

k

^x(^;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)(

^"(

s

) +

'

(

s

))^k

^x(

s

)^;

^x(

s

)^k

ds

+

Z

t

;T(^x)^k

^"v(^;

s

)^k

ds

and applying Gronwall Lemma, from (A2)

k

^x(

t

)^;

^x(

t

)^k

e

^k'kL1+k"kL1k

^x;

T

(

x

))^;

^x(^;

T

(

x

))^k +

Z

t

;T(^x)^k

^"v(^;

s

)^k

ds e

^k'kL1+k"kL1(^k

x

^;

x

!^k+

C

^k

"v

^k)

=

e

^k'kL1+k"kL1(1 +

C

)^k

x

^;

x

!^k

:

(3

:

14)