On the Relation between Discounted and Average Optimal Value Functions

Lars Grune

Fachbereich Mathematik Johann Wolfgang Goethe{Universitat

Postfach 11 19 32

60054 Frankfurt a.M., Germany E-Mail: gruene@math.uni-frankfurt.de

Abstract: We investigate the relation between discounted and average deterministic optimal con- trol problems for nonlinear control systems. In particular we are interested in the corresponding optimal value functions. Using the concepts of Viability, Chain Controllability and Controllability a global convergence result for vanishing discount rate is obtained. Basic ingredients for the analysis are an Abelian type theorem, controllability properties of the system and the Morse decomposition of the corresponding control ow.

Keywords:

Deterministic nonlinear optimal control, discounted optimal control, average optimal control, Morse decomposition, Viable Sets, Chain control sets, Control sets

AMS Classication:

49J10, 93C15

1 Introduction

In this paper we investigate the relation between average and discounted deterministic nonlinear optimal control problems for discount rate tending to zero. Whereas the re- lation between discounted and average integrals has already been explored more than a century ago leading to the Abelian and Tauberian theorems (see e.g. [25, Chapter 10]), corresponding results in nonlinear optimal control theory are much more recent. For sto- chastic optimal control problems in the Markovian setup the corresponding convergence result is almost classical, see e.g. the survey [2], or [28], where also estimates about the rate of convergence are given. The usual assumptions made in the Markovian case, how- ever, exclude the deterministic case. In the deterministic setup, which we will consider in this paper, Colonius [8] in 1989 published a convergence result for vanishing discount rate on invariant control sets, a similar result for arbitrary control sets has been obtained in 1993 by Wirth [27]. These results have in common that assumptions on the optimal trajectories are made which are dicult to check and in general not satised even for sim- ple one-dimensional systems. For invariant control sets this restriction could be removed

This paper has been written while the author was a member of the Graduiertenkolleg "Nonlinear Problems in Analysis, Geometry und Physics" (GRK 283) at the Universitat Augsburg, Germany, nanced by the DFG and the State of Bavaria. Research partially supported by DFG-Grant Co 124/12-1

1

in [18, Theorem 2.11]. Arisawa [3], [4] treats a similar problem (under the name ergodic problem) but from a somewhat dierent point of view: Maximal subsets of convergence of the discounted functional are characterized by introducing a controllability concept in connection with attractivity properties, where again invariance plays a crucial role. This problem goes back to Lions [24], who studied the convergence properties of solutions of Hamilton-Jacobi equations. The name ergodic problem is motivated by the fact that for an uncontrolled system (i.e. an ordinary dierential equation) the convergence property is equivalent to the ergodicity of that system, see [3, Appendix 1].

The main purpose of this paper is to develop results without assuming invariance and without making assumptions on optimal trajectories but by assuming certain qualitative properties of the system. We obtain a global convergence result by merging estimates from three basic concepts | Viability (Section 5) allowing us to state results on extremal values of the value functions, Chain controllability (Section 6) enabling us to give estimates for all possible trajectories of the system and Controllability (Section 7), which is used in order to characterize the behaviour of certain optimal trajectories | into one global picture in Section 8. This kind of approach was inspired by the analysis of the Lyapunov spectrum of bilinear control systems as carried out in [13]. By this procedure we are also able to characterize the subsets of uniform convergence. Furthermore we present a penalizing strategy for the restriction to certain regions of the state space in Section 9. The assumptions we impose can be interpreted as robustness conditions, cp. Remark 8.5, and are generically satised for families of systems under an inner pair condition, cp. Remark 8.6.At the very heart of our analysis two tools are used: In Section 3 we thoroughly investigate the relation between discounted and average functionals using a similar technique as in [18]

and [20]. This can be interpreted as a stronger version of the Abelian theorem, allowing also results on uniform convergence. In Section 4 we investigate the control ow associated to our control system (cf. [11]). Here the concept of attractivity (which is also used in [3]) ts into the general framework of dynamical systems from which we adopt the concept of Morse decompositions.

Apart from the main theorem which is presented in Section 8 we have also formulated the partial results in the Sections 5{7 in a self contained way since they provide useful estimates in themselves. Throughout this paper we assume that the state space is a compact manifold

M

; in Section 9, however, we give some hints about how to overcome this restriction.

The applications of our results are immediate, since discounted optimal control problems enjoy a number of features that averaged ones do not have in general: The correspond- ing optimal value functions are Hoelder continuous and can be characterized as viscosity solution of Hamilton-Jacobi-Bellman equations (cf. [23]), the problems admit a numerical solution (cf. [6], [19]) and the construction of optimal controls in open loop and feedback form (cf. e.g. [5], [7] and [17]).

Nevertheless it is often desirable to solve average optimal control problems, because they can be formulated in order to determine asymptotic properties of a given control system.

One example is the exponential behaviour of bi- and semilinear systems measured by Lyapunov exponents (cf. [10], [12]). The approximation by a discounted optimal control problem enables us to obtain stabilizing optimal controls of feedback type (see [17] and [20]) and to compute the whole Lyapunov spectrum numerically (cf. [18]). In particular for

the analysis of the complete asymptotic behaviour of a system a global convergence result is needed; the result of the present paper in fact closes the gap in the convergence analysis in [18].

2 Problem statement

We consider nonlinear control systems of the type

x

_(

t

) =

f

⁰(

x

(

t

)) +^Xm

i⁼¹

u

i(

t

)

f

i(

x

(

t

)) (2.1) on some compact smooth manifold

M

where the vector elds

f

i,

i

= 0

;:::m

are assumed to be Lipschitz and the control function

u

() satises

u

()^2U :=^f

u

:^R!

U

^j

u

() measurable^g

where

U

^Rmis compact and convex. For a given initial value

x

^{0 2}

M

at time

t

= 0 and a given control function

u

()^2U we denote the trajectories of (2.1) by

'

(

t;x

⁰

;u

())

In order to dene the optimal control problems we assume that a cost function

g

:

M

^Rm!R

; g

(

x;u

) :=

g

⁰(

x

) +^Xm

i⁼¹

u

i

g

i(

x

) (2.2) which is Lipschitz continuous and bounded, i.e. ^j

g

(

x;u

)^j

M

g for some constant

M

g, is given.

Using this cost function we dene the averaged functionals along a trajectory by

J

⁰(

x

⁰

;u

()) := limsup_t

!1

1

t

Z t

0

g

(

'

(

s;x

⁰

;u

())

;u

(

s

))

ds

(2.3) and

J

⁰(

x

⁰

;u

()) := liminf_t

!1

1

t

Z t

0

g

(

'

(

s;x

⁰

;u

())

;u

(

s

))

ds

(2.4) and for a positive discount rate

>

0 we dene the discounted functional

J

(

x

⁰

;u

()) :=

^{Z 1}

0

e

^s

g

(

'

(

s;x

⁰

;u

())

;u

(

s

))

ds

(2.5) (The scaling of the integral by the discount rate

is introduced in order to obtain a more consistent notation in what follows.)

The optimization problem now is to minimize these functionals for any initial value with respect to the control function

u

() ^{2 U}. More precisely we consider the optimal value functions

v

⁰(

x

⁰) := inf_u

()2U

J

⁰(

x

⁰

;u

()) and

v

⁰(

x

⁰) := inf_u

()2U

J

⁰(

x

⁰

;u

()) (2.6)

and

v

(

x

⁰) := inf_u

()2U

J

(

x

⁰

;u

()) (2.7)

Note that the corresponding maximization problem is obtained by simply replacing

g

by

g

.

Both criteria are dened over an innite time horizon. Here the averaged functionals indeed only measure asymptotic properties, i.e. everything that happens up to some bounded time

t

⁰ does not contribute to the integral. In contrast to this for the discounted functional the boundedness of

g

implies that essentially only the behaviour on a nite horizon is measured:

For any

>

0 and any

" >

0 there exists

t >

0 such that

j

J

(

x

⁰

;u

())

^{Z t}

0

e

^s

g

(

'

(

s;x

⁰

;u

())

;u

(

s

))

ds

^j

"

for all

x

^{0 2}

M

and all

u

()^2U. However, for decreasing

^!0 and xed

" >

0 this time increases. Hence the question, whether

v

approximates

v

⁰ and

v

⁰ for small

>

0 arises naturally. It is this question that we want to investigate in this article.

3 Discounted and averaged functionals

In this section we will investigate the relation between discounted and averaged integrals, functionals (along trajectories) and value functions. We start with a lemma giving an estimate for these integrals which can be interpreted as a stronger version of the classical Abelian theorem that can be found e.g. in [25, Theorem 10.2].

Lemma 3.1

Let

q

:^R!Rbe a measurable function satisfying^j

q

(

s

)^j

< M

q for almost all

s

^2R. Assume there exists a time

T >

0 such that

1

t

Z t

0

q

(

)

d <

for all

t

T

Then for any

" >

0 and all 0

< <

⁽_Mq ^"

+j⁺"^j)T the following inequality holds:

^{Z 1}

0

e

q

(

)

d

+

"

A proof of this lemma can be found in [20, Appendix], which uses essentially the same techniques as the proof of [18, Theorems 2.1 and 2.2] combined with a careful evaluation of the constants.

Note that the converse inequalities are easily obtained by replacing

g

by

g

.

In order to carry over these results to our functionals and value functions we introduce some further denitions.

Denition 3.2

For the control system (2.1) and the cost function (2.2) we dene

J

^0t(

x;u

()) := 1

t

Z t

0

g

(

'

(

s;x;u

())

;u

(

s

))

ds J

^0t(

x;u

()) := sup

t

J

⁰(

x;u

())

J

^0t(

x;u

()) := inf

t

J

⁰(

x;u

())

v

^t0(

x

) := inf_u

()2U

J

^0t(

x;u

())

v

^t0(

x

) := inf_u

()2U

J

^0t(

x;u

())

The following lemma shows the relation to the averaged functionals and value functions from Section 2.

Lemma 3.3

tlim^!1

J

^0t(

x;u

()) =

J

⁰(

x;u

())

;

_tlim

!1

J

^0t(

x;u

()) =

J

⁰(

x;u

())

and _tlim

!1

v

^0t(

x

) =

v

⁰(

x

)

;

_tlim

!1

v

^t0(

x

) =:

v

⁰¹(

x

)

v

⁰(

x

)

Proof:

The rst two equalities are immediately clear from the denitions. We prove the third assertion, the fourth follows by similar arguments.

Recalling the denition of

v

^0t(

x

) and

v

⁰(

x

) using the notation of Denition 3.2 this equality states

tlim^!1_uinf

()2U

sup

t

J

⁰(

x;u

()) = inf_u

()2U

limsup_t

!1

J

^0t(

x;u

()) We prove the equality by proving both inequalities.

\" Fix

" >

0 and

u

"()^2U such that limsup_t

!1

J

^0t(

x;u

"())

<

_uinf

()2U

limsup_t

!1

J

^0t(

x;u

())+

"

Then there exists

t

"0 such that

J

⁰(

x;u

"())

<

_uinf

()2U

limsup_t

!1

J

^0t(

x;u

())+ 2

"

for all

t

". Since

" >

0 was arbitrary this implies \".

\" Fix

" >

0 and

t

"

>

0 such that

tlim^!1_uinf

()2U

sup

t

J

⁰(

x;u

())+

" >

_uinf

()2U

supt^"

J

⁰(

x;u

()) Then there exists a control function

u

"()^2U such that

tlim^!1_uinf

()2U

sup

t

J

⁰(

x;u

())+ 2

" >

_uinf

()2U

supt^"

J

⁰(

x;u

())+

" >

sup

t^"

J

⁰(

x;u

"())

This implies

tlim^!1_uinf

()2U

sup

t

J

⁰(

x;u

())+ 2

" >

limsup_t

!1

J

^0t(

x;u

"()) and since

" >

0 was arbitrary \" follows.

These \nite time" averaged functionals and value functions can now be used to give uniform bounds for

J

and

v

for small discount rate

>

0.

Lemma 3.4

For all

t >

0, all

" >

0 and

⁰ = ²_M^"gt the estimate

J

(

x;u

())²[

J

^0t(

x;u

())

";J

^0t(

x;u

())+

"

]

holds for all

⁰. In particular if the limit limt^!1

J

^0t(

x;u

()) exists the equality

lim^!0

J

(

x;u

()) =

J

^0t(

x;u

()) =

J

^0t(

x;u

()) is implied

Proof:

Follows immediately from Lemma 3.1 by observing that it is sucient to consider

j

+

"

^j

M

g.

Corollary 3.5

For all

t >

0, all

" >

0 and

⁰ = ²_M^"g_t the estimate

v

(

x

)²[

v

^t0(

x

)

";v

^t0(

x

) +

"

]

holds for all

⁰. In particular if

v

⁰(

x

) and

v

⁰¹(

x

) agree the equality

lim^!0

v

(

x

) =

v

⁰(

x

) =

v

⁰(

x

) =

v

¹⁰ (

x

) is implied.

Proof:

Follows immediately from the preceding lemma.

Hence the goal of this paper will be to give estimates for

v

^t0(

x

) and

v

^t0(

x

) and characterize the situations in which the limits coincide. In particular we are interested in uniform estimates in

t

on certain subsets of

M

which then imply uniform estimates for

v

for small

>

0. A special case for these subsets will be those where

v

⁰(

x

) =

v

⁰(

x

) =

v

¹⁰ (

x

)

const

. Keeping Corollary 3.5 in mind we will not | except for the main Theorem 8.4 | explicitly formulate the implications of the estimates in the following sections on

v

.

Sometimes it will be useful to restrict the state space to some subset

B

M

. We will denote the corresponding value functions as follows.

Denition 3.6

For a subset

B

M

we dene

v

⁰(

x;B

) := inf^f

J

⁰(

x;u

())^j

u

()^2U

; '

(

t;x;u

())²

B

for all

t

0^g for those points

x

²

B

for which at least one trajectory exists that stays inside

B

. In the same way we dene

v

⁰(

x;B

),

v

⁰¹(

x;B

),

v

^0t(

x;B

),

v

^0t(

x;B

) and

v

(

x;B

).

We end this section with two lemmas showing some useful properties of averaged functionals which will be used in the next sections.

Lemma 3.7

Let

q

:^R!Rbe a measurable function,

t >

0 and

t

¹²(0

;t

). Let

t

² =

t t

¹. Then(i) the following equality holds

1

t

t

Z

0

q

(

)

d

=

t

¹

t t

¹¹

t¹

Z

0

q

(

)

d

+

t

²

t t

¹²

t²

Z

0

q

(

+

t

¹)

d

(ii) if^j

q

^jis bounded by some constant

M

q the following estimates hold

j

1

t

t

Z

0

q

(

)

d t

¹¹

t¹

Z

0

q

(

)

d

^j2

M

q

t

²

t

^{and j1}

t

t

Z

0

q

(

)

d t

¹²

t²

Z

0

q

(

+

t

¹)

d

^j2

M

q

t

¹

t Proof:

(i) follows by a simple calculation, (ii) from (i) using the property

j

1

s

s

Z

0

q

(

)

d

^j

M

q

which holds for all

s >

0.

Lemma 3.8

Let

q

:^R!Rbe a measurable function bounded by some constant

M

q. Let

t >

0 be arbitrary and

:= 1

t

Z t

0

q

(

)

d

Then for any

" >

0 there exists a time

t

^(2M2_M^qq^")t such that 1

s

Z s

0

q

(

t

+

)

d

+

"

for all

s

²(0

;t t

]. Here

t t

²_M^"tq

!1 as

t

^!1.

Proof:

Let

:= sup

s²⁽⁰;t^]1

s

Z s

0

q

(

)

d

and x

" >

0. If

+

"

the assertion follows with

t

= 0.

Otherwise let

t

:= sup

s

²(0

;t

]

1

s

Z s

0

q

(

)

d

+

"

By the continuity in

s

of this averaged integral the equality

t

1

Z t

0

q

(

)

d

=

+

"

is implied. By Lemma 3.7(ii) it follows from ¹_t^R0t

q

(

)

d

=

that

t t

=

t

^{2 2}_M^"tq and hence

t

^(2M2_M^qq")t. We claim that

t

satises the desired property:

Dening ~

q

(

s

) :=

q

(

s

)

"

it follows from the denition of

t

that

t

1

Z t

0

q

~(

)

d

= 0 and 1

s

Z s

0

q

~(

)

d <

0 for all

s

²(

t

;t

]. Hence also

Z t

0

q

~(

)

d

= 0 and ^{Z s}

0

q

~(

)

d <

0

holds implying ^Z _s

t

q

~(

)

d <

0 for all

s

²(

t

;t

] which yields the assertion.

4 The control ow,

^(";T)

-chains and their values

As already pointed out in the introduction, the concept of attractivity forms one of the basic tools for the analysis of our problem, since this enables us to formulate results for all possible trajectories with initial values in some specied set. Instead of using the control system (2.1) itself we will develop these results in terms of the corresponding control ow. Although this requires some denitions it will turn out that this procedure admits an elegant and straightforward approach to the desired results, since techniques from dynamical systems theory can be applied directly. We will start by dening the control ow , see [11] for details.

By endowing the space ^U of measurable control functions with the weak-topology we obtain a compact metric space. On this space for

t

^2Rwe dene the right shift by

:^{RU !U}

;

(

t;u

()) =

u

(+

t

)

This generates a continuous ow on ^U. Using this shift we dene the control ow

:^RU

M

^!U

M;

(

t;u

()

;x

) = ((

t;u

())

;'

(

t;x;u

())) (4.1) In fact, this generates a continuous ow on the product space ^U

M

. For convenience of notation we abbreviate

p

= (

u

()

;x

) for the elements of this product space. For a set

BU

M

we denote by

M^B:=^f

x

²

M

^jthere exists

u

()^2U with (

u

()

;x

)^{2B g} the natural projection onto

M

.

Note that the functionals

J

and

J

^0t depend continuously on

p

due to the fact that

f

and

g

are ane in

u

.

One of the main tools in our analysis is the concept of attractors for ows on metric spaces.

In order to dene these objects we have to dene omega limit sets and invariance. We refer to [1] for more information about ows and dynamical systems.

Denition 4.1

For a subset ^BX the

!

-limit set is dened by

!

(^B) :=^f

p

^2Xjthere exist points

p

k^2B and times

t

k ^!1 with lim_k

!1

(

t

k

;p

k) =

p

^g The

!

-limit set is dened analogously for the time reversed system.

A subset ^{B X} is called forward invariant if (

t;p

) ^2B for all

p

^{2 B} and all

t

0. It is called backward invariant if (

t;p

)^2B for all

p

^{2 B} and all

t

0 and invariant if it is forward and backward invariant.

Now we can introduce the concept of attractors.

Denition 4.2

Let be a continuous ow on a compact metric space ^X. A compact invariant set

A

^X is called an attractor if it admits a neighborhood^N such that

!

(^N) =

A

.

For an attractor

A

the set

A

:=^f

p

^2Xj

!

(

p

)⁶

A

^gis called the complementary repeller.

The domain of attraction of an attractor

A

is the set

A

(

A

) :=^{X n}

A

. Note that a repeller is an attractor for the time reversed ow.

The following Lemma on uniform attraction will be used in this section.

Lemma 4.3

Let

A

be an attractor and ^{K X n}

A

be a compact set. Let ^N be some open neighborhood of

A

. Then there exists a time

T

such that

(

t;p

)^2N for all

t

T

and all

p

^2K

Proof:

By the denition of an attractor

!

( ~^N) =

A

holds for some open neighborhood ~^N of

A

. Hence there exists

T

¹

>

0 such that (

t;p

)^2N for all

p

^2N~ and all

t

T

¹.

From the assumptions on ^K it follows that for any point

p

^{2 K} there exists a time

t

p

>

0 such that (

t

p

;p

) ^{2 N}~. The continuity implies that (

t

p

; p

~) ^{2 N}~ for all ~

p

in some neighborhood of

p

. Since^Kis compact we obtain that these times

t

p are bounded by some

T

², hence the assertion follows with

T

=

T

¹+

T

².

We will now somewhat generalize the nite time average functionals by introducing (

"; T

)- chains and their averaged values. The basic idea is to allow small jumps between nite time trajectory pieces and dene inmal chain values by letting these jumps tend to 0 and the time length of the trajectory pieces tend to innity.

Denition 4.4

For

p; q

^2X and

";T >

0 an (

"; T

)-chain

is given by a number

n

^2N together with points in^X

p

⁰ =

p;p

¹

;:::;p

n=

q

and times

t

⁰

;:::;t

n ¹

T

such that

d

((

t

i

;p

i)

;p

i⁺¹)

< "

for

i

= 0

;:::;n

1. The total time of a chain is given by

T

(

) :=^Pn_i⁼⁰¹

t

i.

We say that

lies in^BX if (

t;p

i)^2B for all

t

²[0

;t

i] and all

i

= 0

;:::;n

1.

The averaged value of a chain is given by

J

⁰(

) := 1

T

(

)

n^X1

i⁼⁰

t

i

J

^0ti(

p

i) with

J

^0t(

p

) :=

J

^0t(

x;u

()) from Denition 3.2 for

p

= (

u

()

;x

).

For a subset ^BX we dene the inmal chain value over ^B for

"

^!0 and

T

^!1 by

(^B) := inf

(

^2R

there exist

"

k ^!0

; T

k^!1 and (

"

k

; T

k)-chains

k in ^B such that limk^!1

J

⁰(

k)^!

)

Note that this setup and hence all results in this section can be generalized to arbitrary ows on compact metric spaces and arbitrary average functionals, provided they can be written in a suitable integral form.

The following equality is an immediate consequence of the previous denitions.

Proposition 4.5

For the inmal chain value over some forward invariant subset ^{B X} the following equality holds

(^B) = inf

(

^2R

there exist

t

k ^!1 and points

p

k ^2K

such that limk^!1

J

^0tk(

p

k)^!

)

i.e. the jumps in the chains do not change the minimal value over ^K.

Proof:

\": This follows from the fact that each trajectory is a (trivial) chain.

\": Let

be an arbitrary (

";T

)-chain in ^B. Then by the denition of

J

⁰(

) there exists a time

t

i

T

and a point

p

i ^{2 B} in the chain such that

J

^0ti(

p

i)

J

⁰(

). Hence by the denition of

(^B) there exist sequences of times

t

k ^{! 1} as

k

^{! 1} and points

p

k ^{2 B}

such that limsup_k^!1

J

^0tk(

p

k)

(^B) which implies the assertion.

For certain points we can even establish a stronger relation between

,

J

^0t,

J

⁰ and

J

⁰.

Proposition 4.6

For the inmal chain value over some compact forward invariant subset

KX there exists a point

p

^2Ksuch that

J

^0t(

p

)

(^K) for all

t

0 and lim_t

!1

J

^0t(

p

) =

(^K) In particular for this point the limit exists and

J

⁰(

p

) =

J

⁰(

p

) =

(^K).

Proof:

By Proposition 4.5 we nd a sequence of points

p

k ^{2 K} and times

t

k ^{! 1} as

k

^!1 such that

J

^0tk(

p

k)

<

(^K) +

"

k where

"

k ^!0 for

k

^!1. Dening ~

"

k := ^p1_tk

!0 for

k

^!1 we apply Lemma 3.8 to

q

(

s

) :=

g

((

p

k

;s

)) for each

k

^2Nand obtain times

t

_k such that

J

^0s((

t

_k

;p

k))

(^K) +

"

k + ~

"

k

for all

s

² (0

;t

k

t

_k] where

t

k

t

_k ²_M^tkg. Dening points ~

p

k := (

t

_k

;p

k) and times

~

t

k :=

t

k

t

_k ^!1as

k

^!1 we obtain

J

^0s(~

p

k)

(^K) +

"

k + ~

"

k

for all

s

²(0

;

^~

t

k].

Since ^B is compact we may assume that the points ~

p

k converge to some

p

^2B. Now x arbitrary

t >

0 and

" >

0 and consider

J

^0t(

p

). Since

J

^0t is continuous we nd

k

^{0 2N}such that ^j

J

^0t(

p

)

J

^0t(

p

k)^j

< "

for all

k

k

⁰. Hence

J

^0t(

p

)

<

(^K)+

"

k+ ~

"

k+

"

follows for all

k

k

⁰. Since

" >

0 was arbitrary and

"

k + ~

"

k ^! 0 for

k

^{! 1} we can conclude

J

^0t(

p

)

(^K) which implies the rst assertion since

t >

0 was arbitrary.

This immediately implies limsup_t^!1

J

^0t(

p

)

(^K). Now assume liminft^!1

J

^0t(

p

)

<

(^K). This implies the existence of a sequence

t

k such that limk^!1

J

^0tk(

p

)

<

(^K) which contradicts Proposition 4.5.

The Propositions 4.5 and 4.6 show in particular that when considering inma over compact subsets of ^X the chain values, the nite time average values and the averaged values are equivalent. Note, however, that for a single point these equalities will in general not hold.

The main advantage of the concept of chains and their values is that we can formulate the following result on continuous dependence for arbitrary times

t >

0.

Proposition 4.7

Let ^{K X} be a compact forward invariant set for the control ow . Then for any

>

0 there exists a neighborhood ^N(^K) and a time

T >

0 such that

J

^0t(

p

)

>

(^K)

for all

t

T

and all

p

^2N(^K) with (

s;p

)^2N(^K) for all

s

t

.

Proof:

Fix

>

0. Assume that for arbitrary neighborhoods ^N(^K) and arbitrary times

T >

0 there exist points

p

T;^{N 2 N}(^K) such that

J

^0t(

p

)

<

(^K)

for some

t

T

and (

s;p

)^2N(^K) for all

s

t

Now choose an arbitrary

" >

0 and a

²(0

;"

) such that for all times 0

s

2

T

and all points

p; q

^2X with

d

(

p;q

)

<

the inequalities

j

J

^0s(

p

)

J

^0s(

q

)^j

< "

and

d

((

s;p

)

;

(

s;q

))

< "

(4.2) hold. The trajectories (

s;p

T;^N) can now be partitioned into pieces with times

n ²

[

T;

2

T

]. By choosing ^N(^K) suciently small by the choice of

for every point

p

m :=

(^P_mn⁼⁰

n

;p

T;^N) there exists a point

q

m in^Ksuch that (4.2) is satised. Hence this yields an (

";

2

T

)-chain

in ^K satisfying

J

⁰(

)

(^K)

+

"

. Since

"

and

T

were arbitrary a contradiction to the denition of

(^K) follows.

The following corollary shows how this result can be extended to attractors and their domain of attraction.

Corollary 4.8

Let

A

^X be an attractor for . Let^Kbe a compact subset of the domain of attraction of

A

. Then for any

>

0 there exists a time

T >

0 such that

J

^0t(

p

)

>

(

A

)

for all

p

^2Kand all

t

T

.

Proof:

For any

>

0 we nd a neighborhood ^N(

A

) of

A

and a time

T

⁰ such that the assertion of Proposition 4.7 holds with

=

2. By Lemma 4.3 there exists a time

T

¹ such that (

s;p

)^2N(

A

) for all

s

T

¹ and all

p

^2K. Now the assertion follows by Lemma 3.7 by choosing

T

suciently large compared to

T

¹.

In the next step we will investigate the nite time average value on nested attractors. We start with two attractors.

Lemma 4.9

Let

A

⁰

A

¹be attractors of and

A

⁰

A

¹be the complementary repellers.

Then for any

>

0 there exists a

T >

0 such that

J

^0t(

p

)min^f

(

A

⁰)

;

(

A

^1\

A

⁰)^g

for all

p

²

A

¹ and all

t

T

.

Proof:

Fix

>

0. Then we nd an open neighborhood ^N(

A

^1\

A

⁰) and a a time

T

¹

>

0 such that the assertion of Proposition 4.7 holds for

=

2. For^K:=

A

^1nN(

A

^1\

A

⁰) there exists a time

T

²

>

0 such that also the assertion of Corollary 4.8 also holds for

=

2.

We claim that

T

:= max^f

T

¹+

T

²

; M

^{2 gmaxf}

T

¹

;T

^2gg

;

where

M

g is the bound on

g

, satises the assertion: Pick an arbitrary point

p

²

A

¹. For

p

^2Kthe assertion follows from Corollary 4.8. For

p

²

A

^1\

A

⁰ the assertion follows from Proposition 4.7. For all other

p

dene

t

⁰ := min^f

t

0^j(

t;p

)^{2K g}. In order to estimate

J

^0t(

p

) for

t

T

we distinguish three cases:

(i)

t

⁰

< T

¹: Lemma 3.7(ii) implies

J

^0t(

p

)

J

^{0t t0}((

t

⁰

;p

)

| {z }

2K

) 2

M

q

t

⁰

t

(

A

⁰)

(ii)

t

⁰

T

¹

;t

t

⁰+

T

²: Here Lemma 3.7(ii) implies

J

^0t(

p

)

J

^0t0(

p

) 2

M

q

t t

⁰

t

(

A

^1\

A

⁰)

(iii)

t

⁰

T

¹

;t

t

⁰+

T

²: In this case Lemma 3.7(i) implies

J

^0t(

p

)

t

⁰

t J

^0t0(

p

) +

t t

⁰

t J

^{0t t0((}

t

⁰

;p

)

| {z }

2K

)

t

⁰

t

⁽

A

^1\

A

⁰) +

t t

⁰

t

⁽

A

⁰)

2 Hence in all three cases the assertion follows.

The main goal of this section is to give uniform estimates for

J

^0t and

J

^0t on a Morse decomposition of^X corresponding to the ow . In order to obtain such a result we need the following denitions.

Denition 4.10

Let be a continuous ow on a compact metric space^X. Let ^;=

A

⁰

A

¹

:::

A

d =^X be a sequence of attractors and let^X =

A

⁰

A

¹

:::

A

_d =^; be the complementary repellers. Then for any

i

= 1

;:::;d

the set

Mi=

A

d i^+1\

A

_{d i}

is called a Morse set and the collection ^Mi,

i

= 1

;:::;d

is called a Morse decomposition.

For any Morse set ^Mi we dene the corresponding domain of attraction by

A

(^Mi) :=^f

p

^2Xj

!

(

p

)^Mi^g

Lemma 4.11

For any sequence of Morse sets ^Mj¹

;:::;

^Mj², 1

j

¹

j

²

d

we have

[

i⁼j¹;:::;j²

A

(^Mi) =

A

_{d j}²ⁿ

A

_{d j}¹⁺¹

Proof:

\" Let

p

²

A

(^Mi) for some

i

^2f

j

¹

;:::;j

^2g. This implies

!

(

p

)

A

d i⁺¹, hence

p

⁶²

A

_{d i}⁺¹

A

_{d j}¹⁺¹.

On the other hand

!

(

p

) ^2Mi implies

!

(

p

)

A

_{d i}. Since

p

⁶²

A

_{d i} implies

!

(

p

)

A

d i

and

A

d i^\

A

_{d i}=^; it follows that

p

²

A

_{d i}

A

_{d j}².

\" Let

p

²

A

_{d j}^{2 n}

A

_{d j}¹⁺¹. Then by the inclusion of the repellers there exists

i

²

f

j

¹

;:::;j

^2g such that

p

²

A

_{d i}ⁿ

A

_{d i}⁺¹ By the invariance

p

²

A

_{d i} implies

!

(

p

)

A

_{d i}. By the denition of the repeller

p

⁶²

A

_{d i}⁺¹implies

!

(

p

)

A

d i⁺¹, hence

!

(

p

)^Mi. We will now discuss the order of the Morse sets. Note that the attractor sequence induces a total order of the Morse sets. However, it is possible that dierent attractor sequences generate the same Morse sets but with a dierent order. Hence we dene a stronger order relation for the Morse sets.

Denition 4.12

Consider a Morse decomposition ^M1

;:::;

^Md for the ow . Then for two Morse sets ^Mi ⁶= ^Mj we dene ^Mi

<

^Mj if there exist points

p

¹

;:::;p

k and Morse sets ^Mi⁰ = ^Mi

;

^Mi¹

;:::;

^Mi^d = ^Mj such that

!

(

p

l) ^{2 M}l^{i 1} and

!

(

p

l) ^{2 M}lⁱ for all

l

= 1

;:::;d

.

Remark 4.13

Note that^Mi

<

^Mj implies

i < j

.

However, even a stronger relation to the attractor sequence can be established.

Proposition 4.14

Let ^M1

;:::

^Md be a Morse decomposition of the ow generated by an attractor sequence

A

⁰

A

¹

:::

A

d. Then for any Morse set ^Mi there exists an attractor sequence ~

A

⁰

A

^~1

:::

A

^~d generating a Morse decomposition ~^M1

;:::

^M~d

satisfying

~

Ml⁽j⁾=^Mj

for all

j

^2f1

;:::;d

^gand a bijective function

l

:^f1

;:::;d

^g!f1

;:::;d

^gand

~

Ml⁽i⁾

<

^M~j

for all

j > l

(

i

).