A Model of Technological Growth under Emission Constraints

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Interim Report IR-06-021

A Model of Technological Growth Under Emission Constraints

Elena Rovenskaya

Approved by

Arkady Kryazhimskiy (kryazhim@iiasa.ac.at) Leader, Dynamic Systems

May 2006

Abstract

We suggest and analyze a model of global technological growth under a prescribed con- straint on the annual emission of greenhouse gases (GHG). The model assumes that in- dustrial GHG emission is positively related to the world's production output driven by the development of the \production" technology stock. \Cleaning" technology is developed in parallel to keep the annual GHG emission within a \safety" zone. The ratio between annual investment in \cleaning" technology and annual investment in \production" tech- nology acts as a time-variable control parameter in the model. Under a set of natural assumptions we nd an optimal control which maximizes an integral utility characterizing the rate of economic growth over a given time period. In substantial terms, the optimal control strategy suggests that \production" technology is developed at a maximum rate until a critical point is reached, at which the annual emission hits the prescribed upper bound. In the subsequent period investment in \production" and \cleaning" technology is planned so that the annual emission \tracks" the prescribed upper bound. One should note that the proposed control strategy optimal with respect to the chosen utility, is the most risky one since it assumes a minimum distance to the boundary of the \safety" zone.

Contents

1 Introduction 1

2 Model 2

3 Equivalent problem formulation 3

4 Discrete approximation 4

5 Solution of

(

)

-problem 8

A Model of Technological Growth Under Emission Constraints

Elena Rovenskaya

1 Introduction

The issue of the reduction of emission of greenhouse gases (GHG) is to a considerable extent associated with climate change. Some works (see, e.g., 6], 1]) relate the increase in GHG emission driven by global economic growth to the raise of the average temperature in the world. Even a small temperature increase is believed to produce a strong negative impact on the environment. Accordingly, it is believed that a reduction of GHG emission should reduce this negative impact. The reduction of industrial GHG emission implies investment in development of \cleaning" mechanisms or/and new technology producing less GHG emissions. However, an excessive investment in development of \cleaning" mechanisms may lead to a recession in technological growth, which may in turn restrict possibilities of investment in \cleaning" in the future. As a result, the atmospheric GHG concentration may, in a time perspective, become high in spite of today's eorts to decreasing it in the same time humanity may face a shortage of resources for developing both \cleaning"

technology and \production" technology.

A principal goal of environmental economics is therefore to nd an optimal balance in development of \production" technology and \cleaning" technology so that the eective- ness of economic growth is maximized under the constraint that a \safe" level of negative environmental eects is not exceeded. While planning an optimal policy, it should be taken into account that the average life time of

CO

² in the atmosphere is estimated as 10 years, from which it follows that the greenhouse eect would not be reduced immediately even if the production process was stopped. Preventive measures are being worked out, based on various data assessment, measurement and modeling methods.

A common way to identify key parameters of these measures is simulation of economic growth scenarios and their impact on climate change (8], 7]). Usually such approach requires extremely complex modeling, involving a variety of uncertain parameters and a number of components and links, some of which are yet not well understood. Our study follows the aggregated modeling approach suggested by endogenous economic growth the- ory 6]. Based on the simpli ed model of economic growth under an emission constraint, constructed in 3], we extend it in three aspects: we introduce a time-varying upper con- straint for annual GHG emission we represent the utility ow as an arbitrary convex function of the production output and we take into account the depreciation of the tech- nology stock. The model assumes that industrial GHG emission is positively related to the world's production output driven by the development of the \production" technology stock. \Cleaning" technology is developed in parallel to keep the annual GHG emission within a \safety" zone. The ratio between annual investment in \cleaning" technology and annual investment in \production" technology acts as a time-variable control parameter in the model.

The substantial part of this research, including a model calibration, an uncertainty analysis and economic interpretations is presented in 4]. In this paper we provide a rigorous mathematical justi cation for the methodology used in 4]. Under a set of natural assumptions we nd an optimal control which maximizes an integral utility characterizing the rate of economic growth over a given time period. In substantial terms, the optimal control strategy suggests that \production" technology is developed at a maximum rate until a critical point is reached, at which the annual emission hits the prescribed upper bound. In the subsequent period investment in \production" and \cleaning" technology is planned so that the annual emission \tracks" the prescribed upper bound. A rigorous proof of this statement whose mathematical formulation is Theorem 5.1 is the principal technical goal of this paper. One should note that the proposed control strategy optimal with respect to the chosen utility, is the most risky one since it assumes a minimum distance to the boundary of the \safety" zone.

2 Model

Let

Y

(

t

) stand for the total annual production output,

T

(

t

) stand for the \production"

technology stock used for production, and

C

(

t

) stand for the stock of \cleaning" technology used to reduce GHG emissions. Let

# >

0 be a xed time horizon. Here and in what follows,

t

is time varying from 0 to

#:

Using a simplest form of the standard Cobb-Douglas production function, we assume

Y

(

t

) =

aT

(

t

) (2.1)

where

a >

0

:

Let

u

² (0

1) be the total fraction of the production output, annually allocated for developing both \production" and \cleaning" technology stocks, and

u

(

t

)²0

u

] (2.2)

be the fraction of the production output

Y

(

t

)

allocated for developing the \production"

technology stock

T

(

t

)

:

The complementary fraction,

u

^;

u

(

t

)

of the production output

Y

(

t

) is allocated for developing the \cleaning" technology stock

C

(

t

)

:

Based on this, using (2.1) and introducing the depreciation of technology, we set

T

_(

t

) =

u

(

t

)

aT

(

t

)^;

T

(

t

)

C

_(

t

) = (

u

^;

u

(

t

))

aT

(

t

)^;

C

(

t

)

^(2.3) where

>

0 is a discount factor. We also denote

T

(0) =

T

⁰

C

(0) =

C

⁰

:

(2.4)

Besides, for system (2.3) we introduce the utility index

J

=^{Z #}

0

e

^;t

f

(

Y

(

t

))

dt

(2.5)

commonly used to assess the eectiveness of an economy 2], 8]. Here

f

() : 0

¹) ^7!

0

¹) is a continuous increasing function, and

>

0 is a discount rate.

We model the annual GHG emission as

E

(

t

) = ⁰

Y

(

t

)

C

(

t

) = ⁰

aT

⁽

t

)

C

(

t

) =

T

⁽

t

)

C

(

t

)

where ⁰

>

0

= ⁰

a

and impose the constraint

E

(

t

)

E

(

t

)

(2.6)

where

E

(

t

)

>

0 is a xed function describing a time-varying admissible level of emission.

We assume that at the initial time

t

= 0 constraint (2.6) is satis ed:

E

⁰

< E

⁰

where

E

⁰=

E

(0) =

T C

⁰⁰

E

⁰ =

E

(0)

:

(2.7) In our analysis

u

(

t

) acts as a control variable. The set of all admissible controls, denoted by

U

is the collection of all measurable functions

u

() on 0

#

] which satisfy (2.2).

Thus we consider an optimal control problem which can be represented in the following standard form 5]:

maximize

J

⁰ = ^{Z #}

0

e

^;t

f

(

Y

(

t

))

dt Y

(

t

) =

aT

(

t

)

T

_(

t

) =

u

(

t

)

aT

(

t

)^;

T

(

t

)

C

_(

t

) = (

u

^;

u

(

t

))

aT

(

t

)^;

T

(

t

)

(2.8)

T

(0) =

T

⁰

C

(0) =

C

⁰

u

() ²

U T

⁽

t

)

C

(

t

)

E

(

t

)

t

² 0

#

]

:

An admissible control process in (2.8) is a triple (

T

()

C

()

u

()) satisfying the dierential equations (2.3), initial condition (2.4) and the state constraint (2.6). We assume that the set of all admissible control processes is nonempty.

We assume that

(A1)

E

() is monotonically decreasing on 0

#

]

(A2) the rate of the decrease of

E

() is small enough, more accurately, for each

t

²0

#

] 0

<

^j

E

^_ (

t

)^j

< au E

²⁽

t

)

:

3 Equivalent problem formulation

In our analysis we use an equivalent formulation of problem (2.8). From (2.3) and (2.4) we get that for each

t

²0

#

]

T

(

t

) =

T

⁰

e

^ap(t);t where

p

(

t

) =^{Z t}

0

u

(

s

)

ds

(3.1)

and

C

(

t

) =

e

^;t

C

⁰+

au T

^{0Z t}

0

e

^ap(s)

ds

^;

T

⁰(

e

^ap(t);1)

:

Now constraint (2.6) takes the form

v

(

t

)0

(3.2)

where

v

(

t

) = 1 +

C

⁰

T

^{0 +}

au

^{Z t}

0

e

^ap(s)

ds

^;1 +

E

(

t

)

e

^ap(t)

:

(3.3)

Note that

v

_(

t

) =

au e

^ap(t);

au

(

t

)

e

^ap(t)1 +

E

(

t

)

+

E

_ (

t

)

E

²(

t

)

e

^ap(t) (3.4)

and

v

(0) =

v

⁰ =

C

⁰

T

^{0 ;}

E

⁰

:

(3.5)

Constraint (3.2) is satis ed for

t

= 0 by (2.7), therefore

v

⁰

>

0

:

Consider the utility index (2.5). Taking into account (2.1) and (3.1), we get

J

=^{Z #}

0

e

^;t

f

(

aT

⁰

e

^ap(t))

dt

=^{Z #}

0

e

^;t

(

p

(

t

))

dt

(3.6) where

() : 0

¹)^7!0

¹) :

p

^7!

(

p

) =

f

(

aT

⁰

e

^ap) is a continuous increasing function.

Summarizing, we represent the original optimization problem (2.8) in the following equivalent form:

maximize

J

(

u

) = ^{Z #}

0

e

^;t

(

p

(

t

))

dt p

(

t

) = ^{Z t}

0

u

(

s

)

ds

v

_(

t

) =

au e

^ap(t);

au

(

t

)

e

^ap(t)1 +

E

(

t

)

+

E

_ (

t

)

E

²(

t

)

e

^ap(t)

v

(0) =

v

⁰ 0

(3.7)

v

(

t

) 0

u

(

t

) ² 0

u

]

t

² 0

#

]

:

As usual, a solution to problem (3.7) is said to be an optimal control in this problem.

Remark 3.1

Further on, we use a more informative notation for variables (3.1) and (3.3):

p

(

t

) =

p

(

tu

)

v

(

t

) =

v

(

tu

), thus stressing their dependence on a control

u

=

u

().

We denote the optimal value in problem (3.7) as

J

⁰

and the set of all optimal controls in problem (3.7) as

U

⁰

:

4 Discrete approximation

In this section we introduce a discrete approximation to problem (3.7). Take a uniform time grid

t

_i=

i

(

i

= 0

1

:::m

)

t

_m=

#

and introduce approximate piece-wise constant controls

u

(

t

) =

u

_i ^2f0

u

^g (

t

²

t

_i

t

_i⁺¹)

i

= 0

1

:::m

^;1) identi ed with the vectors

u

= (

u

⁰

u

¹

:::u

_m^;1)^2f0

u

^gm

:

For each approximate control

u

() we de ne the following piece-wise approximation

p

(

u

) to

p

() =

p

(

u

) (3.1):

p

(

tu

) =

p

⁰(

u

) = 0 (

t

²

t

⁰

t

¹))

p

(

tu

) =

p

_i(

u

) = ^iX;1

j⁼⁰

u

_j (

t

²

t

_i

t

_i⁺¹)

i

= 1

:::m

^;1)

identi ed with the vector

p

(

tu

) = (

p

⁰(

u

)

p

¹(

u

)

:::p

_m^;1(

u

)) (

t

²0

#

])

:

Remark 4.1

Note that the proposed approximation allows us to choose such an approx- imate control

u

() that for each

t

²0

#

]

j

p

(

tu

)^;

p

(

tu

)^j

u

and

j

p

(

tu

)^;

p

(

tu

)^j

u :

This leads to

j

p

(

tu

)^;

p

(

tu

)^j2

u :

(4.1) We de ne an approximation

v

(

u

) to

v

() =

v

(

u

) (3.3) by

v

_(

tu

) =

au e

^{ap (tu );}

au

(

t

)

e

^{ap (tu )}1 +

E

(

t

)

+

E

_ (

t

)

E

²(

t

)

e

^{ap (tu )} (

t

²0

#

]) or

v

_(

tu

) =

au e

^{api(u );}

au

_i

e

^{api(u )}1 +

E

(

t

)

+

E

_ (

t

)

E

²(

t

)

e

^{api(u )} (4.2) (

t

²

t

_i

t

_i⁺¹)

i

= 0

1

:::m

^;1)

:

Following (3.5), we set

v

(0

u

) =

v

⁰

:

Finally, for each approximate control

u

() we de ne an approximate utility value by

I

(

u

) =^{Z #}

0

e

^;t

(

p

(

tu

))

dt

=^mX;1

i⁼⁰

h

_i

(

p

_i(

u

)) where

h

_i=^Z_t^ti+1

i

e

^;t

dt

(

i

= 0

1

:::m

^;1)

:

Introduce a parameter

>

0 and an approximate constraint

v

(

t

_i

u

)^;

(

i

= 0

1

:::m

^;1)

:

The approximate optimization problem (further called the (

) -problem) can be repre- sented as follows

maximize

I

(

u

) = ^mX;1

i⁼⁰

h

_i

(

p

_i(

u

))

u

= (

u

⁰

:::u

_m^;1)^2f0

u

^gm (4.3)

v

(

t

_i

u

) ^;

(

i

= 0

1

:::m

^;1)

:

We assume that the set of all admissible controls in (4.3) is nonempty. Denote the optimal value by

I

⁰

and the set of all optimal controls by

U

⁰

:

Lemma 4.1

Let k

>

0 (

k

= 0

1

:::

)

k ^! 0 (

k

^{! 1})

:

Then there exist

k

>

0 (

k

= 0

1

:::

) k ^!0 (

k

^!1)

such that

(i)

I

^k0!

I

⁰

(ii)

u

^{k0 !}

U

⁰ weakly in

L

²0

#

]

where

u

^k0 =

u

^k0() ²

U

^k0

U

and

I

^k0 =

I

^k0 are, respectively, an optimal control and the optimal value in the ( k

k)-problem.

Proof.

1. We use simpli ed notations:

u

^k() =

u

^k() =

u

^k

p

^k(

u

^k) =

p

^k(

u

^k)

v

^k(

u

^k) =

v

^k(

u

^k)

:

Put

k = sup_u

2U

v

^k(

u

^k)^;

v

(

u

)_C (

k

= 0

1

:::

)

:

(4.4) We will show that sequence (

k) satis es the conditions of the lemma. Suppose the contrary: there exist a subsequence (

k

j) and an

" >

0 such that

k^j

> "

for each

j

= 0

1

::::

Without loss of generality we set

k

> "

for each

k

= 0

1

::::

Consider elements

u

k ²

U

that provide the supremum in (4.4) at least with accuracy

"=

2

i.e.

usup²U

v

^k(

u

^k)^;

v

(

u

)_C^;

v

^k(

u

_kk)^;

v

(

u

k)_C

"

2 (

k

= 0

1

:::

)

where

u

_kk is the approximation to

u

k on the k-grid.

Hence,

v

^k(

u

_kk)^;

v

(

u

k)_C

"

(

k

= 0

1

:::

)

:

(4.5) We will complete the rst part of the proof by arriving at a contradiction to the latter inequality.

2. As

u

k()²

U

and

U

is a weak compact in

L

²0

#

]

there exist subsequence of (

u

k()) weakly converging to some control

u

()²

U:

Without loss of generality we set

u

k()^!

u

() weakly in

L

²0

#

]

:

From (4.1) we have

j

p

(

t u

k)^;

p

^k(

t u

_kk)^j2

u

k

and while k ^!0

Z t

0

u

k(

s

)

ds

^{;Z t}

0

u

_kk(

s

)

ds

^!0

:

The latter fact together with the weak convergence of (

u

k()) to

u

() yields that

u

_kk()^!

u

() weakly in

L

²0

#

]

:

3. Using (3.4), we obtain

v

(

t u

k) =

v

⁰+^{Z t}

0

ae

^ap(suk)

"

u

^;

u

k(

s

)1 +

E

(

s

)

+

E

_ (

s

)

aE

²(

s

)

#

ds:

From the approximation condition (4.1) we have

e

^ap(uk)!

e

^ap(u) in

L

²0

#

]

:

From the weak convergence of (

u

k()) to

u

() we have

u

^;

u

k()1 +

E

()

+

E

_ ()

aE

²() ^!

u

^;

u

()1 +

E

()

+

E

_ ()

aE

²() weakly in

L

²0

#

]

:

Consequently

v

(

t u

k)^!

v

⁰+^{Z t}

0

ae

^ap(su)

"

u

^;

u

(

s

)1 +

E

(

s

)

+

E

_ (

s

)

aE

²(

s

)

#

ds:

(4.6) Similarly we nd that

v

^k(

t u

_kk)^!

v

⁰+^{Z t}

0

ae

^ap(su)

"

u

^;

u

(

s

)1 +

E

(

s

)

+

E

_ (

s

)

aE

²(

s

)

#

ds:

(4.7) Thus, (4.6) and (4.7) lead to

v

^k(

u

_kk)^;

v

(

u

k)_C ^!0

which contradicts (4.5). We proved that

k ^!0

:

4. Consider a control

u

⁰() ²

U

⁰

U

optimal in problem (3.7). For each k-grid we consider its approximation on each

u

^0k()

:

According to 1-3 control

u

^0k() is admissible in problem (4.3) with

=

k

= k

:

Let

k =^j

J

^0;

I

(

u

^0k)^j

:

Obviously,

k = ^j

J

^{0 ;}

I

(

u

^0k)^j

=

Z #

0

e

^;t

(

p

(

tu

⁰))

dt

^;mX;1

i⁼⁰

h

i

(

p

_ki(

u

^0k))

=

m^X;1 i⁼⁰

Z t^ki+1

t^ki

e

^;t

(

p

(

tu

⁰))

dt

^;mX;1

i⁼⁰

Z t^ki+1

t^ki

e

^;t

dt

(

p

_ki(

u

^0k))

m^X;1 i⁼⁰

Z t^ki+1

t^ki

e

^;t

(

p

(

tu

⁰))^;

(

p

_ki(

u

^0k))

dt:

Since

() is continuous and (4.1) holds we nd that

k ^!0

:

Then for each

k

= 0

1

:::

I

^k0

J

^0;

"

k

:

Letting

k

^!1, we get

klim^!1

I

^k0

J

⁰

:

(4.8)

Now we consider controls

u

^k0() ²

U

optimal in problems (4.3) with

=

k

= k

:

Since

U

is a weak compact in

L

²0

#

] there exist a subsequence of (

u

^k0())

which converges to some ~

u

()²

U

weakly in

L

²0

#

]

:

Without loss of generality we set

u

^k0()

u

~( ) weakly in

L

²0

#

]

:

Reasoning as in 1-3 we nd that control ~

u

() is admissible in problem (3.7). Hence,

J

(~

u

)

J

⁰

:

Similarly to a previous reasoning we get

I

^{k0 !}

J

(~

u

)

:

Therefore

klim^!1

I

^k0

J

⁰

:

(4.9)

We see that (4.8) and (4.9) prove (i). Statement (ii) follows from the weak compactness of

U

in

L

²0

#

]

:

5 Solution of

^{( )}

-problem

In this section we nd a solution to the (

) -problem (4.3). Let

u

⁰() be an arbitrary control optimal in (4.3). By (4.2) we have

v

_(

tu

⁰) =

(

⁺(

t

)

e

^{api(u 0)} if

u

_i⁰ = 0

^;(

t

)

e

^api(u0) if

u

_i⁰ =

u

⁽

t

²

t

_i

t

_i⁺¹)

i

= 0

1

:::m

^;1) (5.1) where due to assumptions (A1) and (A2)

⁺(

t

) =

au

+

E

_ (

t

)

E

²(

t

)

>

0

(5.2)

^;(

t

) = ^;

au

E

(

t

) +

E

_ (

t

)

E

²(

t

)

<

0

:

(5.3)

Note that from (5.1) and (5.2), (5.3) we have

v

(

t

_i⁺¹

u

⁰) =

v

(

t

_i

u

⁰) +

(

_i⁺

e

^{api(u 0)} if

u

_i⁰ = 0

_i^;

e

^{api(u 0)} if

u

_i⁰=

u

⁽

i

= 0

1

:::m

^;1) (5.4) where

_i⁺ = ^Z_t^ti+1

i

⁺(

t

)

dt

(5.5)

_i^; = ^Z_t^ti+1

i

^;(

t

)

dt:

(5.6)

As

u

⁰() is admissible in (4.3), we have

v

(

t

_i

u

⁰)^;

(

i

= 0

1

:::m

)

:

(5.7) We call a point

t

_j

< t

_m ^criticalif the replacement of

u

_j⁰ with

u

leads to the violation of constraint (5.7) at point

t

_j⁺¹

i.e.

v

(

t

_j⁺¹

u

⁰)

<

^;

(

j

= 0

1

:::m

^;1) or

v

(

t

_j

u

⁰) +

e

^{apj(u 0)Z}_t^{tj +1}

j

;

E au

(

t

) +

E

_ (

t

)

E

²(

t

)

!

dt <

^;

(

j

= 0

1

:::m

^;1)

:

Each point

t

_j

< t

_m that is not critical is called ^regular.

Remark 5.1

Let

t

_j

< t

_m be a critical point. Then

u

_j⁰ = 0

: Lemma 5.1

Let

t

_j

< t

_m^;2 be a regular point. Then

u

_j⁰=

u :

Proof.

Suppose the contrary:

u

_j⁰ = 0

:

Consider an admissible control

u

= (

u

⁰

::: u

_m^;1) ^{2 f}0

u

^gm and complete the proof by arriving at a contradiction with the optimality of

u

⁰ in problem (3.7), namely we will prove that

I

(

u

)

> I

(

u

⁰)

:

1. First we suppose

u

_i⁰= 0 (

i

=

j

+ 1

:::m

^;2)

:

(5.8) We de ne control

u

by

u

_i =

u

_i⁰ (

i

= 0

:::j

^;1)

u

_j =

u

(5.9)

u

_i =

u

_i⁰ (

i

=

j

+ 1

:::m

^;2)

:

Then

p

⁰(

u

) = 0 =

p

⁰(

u

⁰)

p

_i(

u

) = ^jX;1

l⁼⁰

u

_l= ^jX;1

l⁼⁰

u

_l⁰ =

p

_i(

u

⁰) (

i

= 1

:::j

)

p

_j⁺¹(

u

) = ^jX;1

l⁼⁰

u

_l+

u >

^jX;1

l⁼⁰

u

_l⁰ =

p

_j⁺¹(

u

⁰)

p

_i(

u

) = ^jX;1

l⁼⁰

u

_l+

u

+ ^mX;1

l⁼j⁺¹

u

_l

>

^jX;1

l⁼⁰

u

_l⁰+ ^mX;1

l⁼j⁺¹

u

_l⁰=

p

_i(

u

⁰) (

i

=

j

+1

:::m

^;1)

:

Thus nally we have

p

_i(

u

) =

p

_i(

u

⁰) (

i

= 0

:::j

^;1)

(5.10)

p

_i(

u

)

> p

_i(

u

⁰) (

i

=

j

+ 1

:::m

^;1)

:

(5.11) Now consider

v

(

u

)

:

From (5.9), (5.4) we have

v

(

t

_i

u

) =

v

(

t

_i

u

⁰)^;

(

i

= 0

:::j

)

:

Furthermore

v

(

t

_j⁺¹

u

) =

v

(

t

_j

u

) +

_j^;

e

^{apj(u )} =

v

(

t

_j

u

⁰) +

_j^;

e

^apj(u0)

:

By assumption

t

_j is regular. Hence,

v

(

t

_j

u

⁰) +

_j^;

e

^apj(u0);

from which it follows that

v

(

t

_j

u

)

:

Since

v

_(

t u

) =

⁺(

t

)

e

^{api(u )}

>

0 (

t

²

t

_i

t

_i⁺¹)

i

=

j

+ 1

:::m

^;2)

function

t

^7!

v

(

t u

) is monotonically increasing on

t

_j⁺²

t

_m^;2]

:

Therefore

v

(

t

_i

u

)

> v

(

t

_j⁺¹

u

)^;

:

(

i

=

j

+ 2

:::m

^;1)

:

Summarizing, we have

v

(

t

_i

u

)^;

(

i

= 0

:::m

^;1)

:

In other words,

u

is an admissible control in (4.3).

For the corresponding value of the utility index we get

I

(

u

) = ^mX;1

i⁼⁰

h

_i

p

_i(

u

)

>

^mX;1

i⁼⁰

h

_i

p

_i(

u

⁰) =

I

(

u

⁰)

(see (5.10), (5.11)). Thus we obtained a contradiction proving the statement of the lemma in case (5.8).

2. Now let (5.8) do not hold, i.e.

u

_i⁰ =

u

(

i

=

j

+ 1

:::m

^;2)

:

Set

u

_i⁰= 0 (

i

=

j

+ 1

:::k

^;1)

u

_k⁰ =

u :

We put

u

_i =

u

_i⁰ (

i

= 0

:::m

^;2

i

⁶=

jk

)

u

_j =

u

(5.12)

u

_k = 0

:

Then

p

⁰(

u

) = 0 =

p

⁰(

u

⁰) (

i

= 1

:::j

)

p

_i(

u

) = ^jX;1

l⁼⁰

u

_l= ^jX;1

l⁼⁰

u

_l⁰ =

p

_i(

u

⁰)

p

_j⁺¹(

u

) = ^jX;1

l⁼⁰

u

_l+

u >

^jX;1

l⁼⁰

u

_l⁰=

p

_j⁺¹(

u

⁰) (

i

=

j

+ 2

:::k

)

p

_i(

u

) = ^jX;1

l⁼⁰

u

_l+

u

+ ^Xk

l⁼j⁺¹

u

_l

>

^jX;1

l⁼⁰

u

_l⁰+ ^Xk

l⁼j⁺¹

u

_l⁰ =

p

_i(

u

⁰) (

i

=

k

+ 1

:::m

^;1)

p

_i(

u

) = ^jX;1

l⁼⁰

u

_l+

u

+ ^kX;1

l⁼j⁺¹

u

_l+ ^mX;1

l⁼k⁺¹

u

_l = ^jX;1

l⁼⁰

u

_l⁰+ ^kX;1

l⁼j⁺¹

u

_l⁰+

u

+ ^mX;1

l⁼k⁺¹

u

_l⁰ =

p

_i(

u

⁰)

:

Thus nally we have

p

_i(

u

) =

p

_i(

u

⁰) (

i

= 0

:::j

and

k

+ 1

:::m

^;2)

(5.13)

p

_i(

u

)

> p

_i(

u

⁰) (

i

=

j

+ 1

:::k

)

:

(5.14)

Now consider

v

(

u

)

:

From (5.9), (5.4) we have

v

(

t

_i

u

) =

v

(

t

_i

u

⁰)^;

(

i

= 0

:::j

)

:

Furthermore,

v

(

t

_j⁺¹

u

) =

v

(

t

_j

u

) +

_j^;

e

^{apj(u )} =

v

(

t

_j

u

⁰) +

_j^;

e

^apj(u0)

:

Due to the regularity of

t

_j we get

v

(

t

_j

u

⁰) +

_j^;

e

^apj(u0);

from which it follows that

v

(

t

_j⁺¹

u

)^;

:

For

i

=

j

+ 1

:::k

v

_(

t

_i

u

)^;

v

_(

t

_i

u

⁰) =

⁺(

t

i)(

e

^{api(u );}

e

^{api(u 0)})

:

Using (5.14) and the admissibility of

u

⁰ in problem (4.3) we get

v

_(

t

_i

u

)

> v

_(

t

_i

u

⁰)^;

(

i

=

j

+ 1

:::k

)

:

Consider a

t

²

t

_k

t

_k⁺¹) :

v

_(

t u

)^;

v

_(

tu

⁰) =

⁺(

t

)

e

^{apk(u );}

^;(

t

)

e

^apk(u0)

>

(

⁺(

t

)^;

^;(

t

))

e

^{apk(u 0)} here we used (5.14). Notice that

v

(

t

_j

u

) =

v

(

t

_j

u

⁰)

v

_(

t

_i

u

)

> v

_(

t

_i

u

⁰) (

i

=

j

+ 1

:::k

)

which together with (5.13) leads to

v

(

t

_k

u

)^;

v

(

t

_k

u

⁰)

> v

(

t

_j⁺¹

u

)^;

v

(

t

_j⁺¹

u

⁰)

=

^;_j

e

^{apj(u );}

⁺_j

e

^apj(u0)

= (

^;_j^;

⁺_j)

e

^apj(u0)

:

Taking into account (5.4) we have

v

(

t

_k⁺¹

u

)^;

v

(

t

_k⁺¹

u

⁰)

> v

(

t

_k

u

)^;

v

(

t

_k

u

⁰) + (

_k^+;

_k^;)

e

^{apk(u 0)}

>

(

_k^+;

_k^;)

e

^{apk(u 0)})^;(

_j^+;

_j^;)

e

^apj(u0)

>

(

_k^+;

_k^;;

_j⁺+

_j^;)

e

^{apj(u 0)}

:

Consider the round brackets in the latter inequality. Obviously we have:

_k^+;

_j⁺ = ^Z_t^{tk +1}

k

au

+

E

_ (

t

)

E

²(

t

)

!

dt

^;Z_t^{tj +1}

j

au

+

E

_ (

t

)

E

²(

t

)

!

dt

=

au

+ ^Z_t^{tk +1}

k

dE

(

t

)

E

²(

t

) ^;

au

^{; Z}_t^{tj +1}

j

dE

(

t

)

E

²(

t

)

=

0

@ 1

E

(

t

)

t^{k +1} t^{k ;} 1

E

(

t

)

t^{j +1} t

1

A