Some theorems about the solutions of Stieltjes differential equations and the existence of optimal Stieltjes controllers

Some theorems about the solutions of Stieltjes differential

equations and the existence of optimal Stieltjes controllers

Citation for published version (APA):

Janssen, A. J. E. M. (1975). Some theorems about the solutions of Stieltjes differential equations and the existence of optimal Stieltjes controllers. (Memorandum COSOR; Vol. 7505). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1975

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L~

1505

"''''~'.n'"""r''i'

ARC1

01 '

COS

TECHNOLOGICAL UNIVERSITY EINDHOVEN Department of Mathematics

STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 75-05

Some Theorems about the Solutions of Stieltjes Differential Equations and the Existence of Optimal Stieltjes Controllers

by

A.J.E.M. Janssen

· ,(

Some Theorems about the Solutions of Stieltjes Differential Equations and the Existence of Optimal Stieltjes Controllers

by

A.J.E.M. Janssen

O. Introduction.

Consider a differential equation of the following type

x(t)

=

A(x(t), t)u(t) (t E [O,T])

(1)

x(O)

=

X

_o

where x(t) E Rn (t E [O,T]) X

o

E Rn and T > 0 ( given constants), the control-ler U a non-negative integrable real function defined on [O,T], A : Rn x R -+ Rn• In this paper we consider what we call Stieltjes differential equations

(2) x(t)

=

X

_o

+

J

A(x(T),T)dA(T)

[0,t)

(t E [O,T])

where A is a finite measure on ( [e,T],B) (B is the set of Borelsets in [O,T]).

We may compare (1) and (2) by interpreting Aas a controller having the

character of a o-function. Such controllers will be called Stieltjes

control-lers.

In section of this paper we prove a number of lemmas. These lemmas will be

used to find sufficient conditions (restrictions on A) for the existence

and uniqueness of solutions of (2) (theorem E I). It should be remarked that

(2) has , in general, not a (unique) solution if we replace the integration

interval [O,t) by [O,t].

In section 2 we shall also deal with systems of the form

(3) _x(t)

₌

X

_o

+

J

[0,t) A(x(T),T)dA(T) +

J

B(X(T),T) [0,t) (t E [0, T]) where B: Rn x R -+Rn•

It is proved that (3) can be reduced to a system of the type (2).

In section 3 some theorems of the theory of ordinary differential equations will be generalized to theorems about Stieltjes differential equations.

In the last section of this paper, section 4, we shall discuss the existence of optimal Stieltjes controllers. We consider the case that A is an element

2

-of the class W -of measures ~ satisfying ~ ([O,TJ) $ M(where M is a given positive constant, something like the "total energy").

The notion of optimality is not quite obvious here; we use the following interpretation of optimal. Let AE Wand let x be the solution of (3).Define

YA (T) = X

_o

+

J

A(x(t), t)d).(t) +

J

B(x(t) ,t)dt.

[O,TJ [O,TJ

We prove, using the theorems of section 3, that the set

V : = _{YA(T)

_I

A _{E W}}

is closed and bounded if A en B satisfy certain conditions (theorem E II).

n

Let now h : R ~R be a continuous mapping. We shall call the Stieltjes controller A optimal for the 6-tuple (XO,T,A,B,M,h) if h(yA(T»is maximal.

1. Notation and some lemmas.

1.1. Remarks about the notation used. Let T > 0. The collection of Borelsets of [O,TJ is denoted by

B.

If we use the word measuraBle then this should be interpreted as B-measurable. In this section v is a finite measure on

([0,TJ, B). We consider Rn(n E :N) with the usual'.'inner product norm

Ixl :

= (

I

i=1

n

(x E lR ).

Furthermore we use the lambda-notation of Church: if a function f is defined on a set Xand its value is given by an expression E(x) for every x E X, we write f = Y; E(x)(or, if it is obvious that x EX, ,,/,E(x».

XEX

x

11.2. Lemma Let y : [O,TJ~lR be a bounded, measurable function. Define

f (t) =

J

y (T)d'v (T )

[0,t)

(t E [O,TJ).

Then f is of bounded variation and continuous from the left in [O,TJ. Proof: We can split yet) = y+(t) - y-(t) into a negative and a non-positive part on [O,TJ. We have

f(t) =

J

Y+(T)dv(T)

[0,t)

3

-Now f is written as the difference of two bounded, non-decreasing functions

on [O,TJ.lt follows that f is of bounded variation in [O,TJ.

Now let t E [O,TJ. For every s in [O,t) we have

i

I

y(T)dv(T)

[0, t) [0, s)

J

y(T)dV(T)

I

~

J

Kdv(T)

[s,t)

=

Kv([s,t»

where K is an upper bound for IY(T)

I

(T E. [O,TJ). Using

lim v([s,t»

=

°

stt

we obtain that f ~s continuous from the left in [O,TJ.

1.3. For the proof of theorem E I we shall need some estimates of the Stieltjes-Lebesgue integrals

o

J

g(F(t»dF(t) (T, I

J

resp.

I

(T,1J g(F(t-O»dF(t) (T E [O,IJ),

where F is a non-decreasing, non-negative bounded function, continuous from the right on [O,IJ, and g is a non-decreasing, continuous function onR.

I •4. Lemma E I Let F and g be as described in 1.3. Then we have

I

(T,1J

g(F(t-O»dF(t) g(u)du g(F(t»dF(t) •

Proof. We give the proof ~n three steps.

I) First assume that F is continuous on (T,I) We define

= F(T+O) F(t) F(l-O) at t

=

T for T < t < at t= I.

Then F is uniformly continuous on [T,IJ. Moreover we have

J

g(F(t» dF(t) (T,I)

I

g(F(t»dF(t) [T,1J

J

g(F(t-O»dF(t). (T,I)

Given £ > 0, there is a partition T =A

I< ••• < Ak = I such that

F(

A.) - yeA. 1)< £ (i=2, •.• ,k).From the fact that F and g are

non-1

L

4

-k-l

=

L

g(F(A.)) [F(A. I) - F(>'i)J

i=1 ~ ~+

~

J

[.,IJ k-I

~

L

g(F(A. I)) [F(A;+I) - F(A.)J = R,

i=1 ~+ • ~

and for i=I, ••• ,k-1

So we may conclude L

~

J

g(F(t))dF(t)

~

R [.,IJ Finally we have

F(

1) L

~

_J

g(u)du

~

R. F(-c) < E

I

l~(F()'i+l»

- g(F(A_i

»]

=

£~(F(l))

- g(F(O))].

We can take £ > 0 arbitrarily small, thus proving

-

-g(F(t))dF(t)

F(l)

=

_J

g(u)du.

F(.)

And from this it follows that

J

g(F(t))dF(t) = (.,1) F (1-0)

J

g(u)du = F(.+O) (.,1)

J

g(F(t-O))dF(t) •

, ,

5

-2) Now assume that F is continuous on (T,IJ with the exception of the points of discontinuity a

l <••• < am and define aO = T and (only in the

case that a

:f

I) a I = 1. Applying the result of 1) and using the fact

m m+

that F is continuous from the right we obtain

J

g(F(t»dF(t) (T,IJ = iI

o

{

J

g(F(t»dF(!)+

f

g(F(t»dF(t)} (a.,a·+_I) _{{a·+ I }} 1 1 1 . F(a. 1-0) m

{

f

1+

=

_L

g(u)du + g(F(ai+I»[F(a_{i +I)- F(ai +I- O)J}

i=O

F(a. )

1

m F(a·+I-O) F(a i+l)

}

~ ~.

{r

_g(u)du +

I

g(u)du i=O F(a. ) _F(ai+I-O) 1 F(J)

J

g(u)du. F(T)

Here we again used that F and g are non-decreasing on (T,lJ. It is easily seen that we can use the same argument to prove

J

g(F(t-O»dF(t)

(1, 1J

g(u)du.

(t E [T,l]).

3) Now let {a ) ~ be an enumeration of the discontinuities of F in

m m E .I.'

(.,lJ. We define for mE:N

Fm(t) : = F(t) -

L

[F(~) - F(~ - O)J

~:5t,

k>m

Notice that every F is continuous from the right, non-decreasing, and,

m

moreover, has a finite number of discontinuities. Furthermore we observe that

L

[F(~) - F(~ - O)J

~:5t,

k>m

6

-~s non-decreasing and continuous from the right. So we find

I

J

g(F(t±O»d(F(t) - Fm(t» \:0; (T,1J

J

(T,1J MOd(

l:

[F ('it) 'it:o;t, k>m - F(a

_{k - 0)])=}

MO

L

[F(a_{k) -} F('it - 0)J-+ 0 'it:o; I, k>m (m+<>o) • Here M

O is an upper bound for Ig(x)I(XE[O,F(I)Jl. It

show that F (t±O)-+ F(t±O) (m -+00) uniformly on (T,IJ

m

that g is uniformly continuous on [O,F(I)J, we find uniformly on (T,IJ. We conclude

is not difficult to and, using the fact

that g(F (t±O» -+ g(F(t±O»

m lim m+oo

J

(T,1J g(F (t±O» - g(F(t±O»\dF (t)

=

O. m m

Now, using the result of 2), we obtain

J

g(F(t)dF(t) "lim

r

g(Fm(t»dFm(t)

(T,IJ m+oo (T:IJ

~ lim m+oo F (1) m

f

g(u)du F (T) m

FJ I)

=

g(u)du. F(T) In the same way we prove that

J

g(F(t-O»dF(t)

·

,

7

-1.5. Remark: The inequalities of 1.4. are no longer valid if we replace the interval of integration by [T,IJ.

Example: Let g : R+R be a continuous non-decreasing function and define

{

0 if tE [0,

D

F(~) =

1 if tE [~,IJ.

If we choose T = ~ we find successively

J

g(F(t»dF(t) = g(l)

U,

1

J

J

g(F(t-O»dF(t) = g(O) [~, 1

J

Fr)

g(u)du -

0

F(D

By making suitable choices for g, it ~s easily seen that both inequalities can be violated.

2. A theorem about the existence and uniqueness of the solution of Stieltjes differential equations (Theorem E1) •

2.1. We consider the following Stieltjes differential equation

x(t) = x +

o

J

H(x(T),T)dA(T) [0,t)

(t E [O,T]),

where x(t) E ~n (t E [O,TJ), H : ~n x [O,TJ +Rn, Aa finite measure on .

n

([O,TJ, B), and X

o

E Rand T >

°

given constants. In section 4 we shall

consider Stieltjes differential equations of the form

(**)

k

x(t) = X

_o

+

L

i ...1

(t E [O,TJ),

where A. : ~n x [O,TJ +~n (i = 1, ••• ,k) and A. a finite measure on

~ ~

·

,

8

-2.2. Lemma: It LS possible to reduce system (**) to system (*).

Proof: Define a new measure A.on ([O,T],B) by

A(A) k

=

L

i=I A. (A) L (AEB) •

Notice that every A. is absolutely continuous with respect to A. Therefore L

we can apply the Radon Nikodym theorem (see [SG] page 189). For i=I, ••• ,k

there exists a measurable function f., non-negative and A-summable, such

L that for each AE'.1.>

A. (A)

L

=

f

A f .dA.

L

Furthermore it follows that for each AEB

A(A)

If

A

f.dA :::

L

f. = I holds almost everywhere in the sense of A.By choosing L

k

so that

L

i=1

k

f i =

°

on the set where

L

i=1

f. i I we achieve that O~f.~

L L on [O,T]

(i=I, .•• ,k). Now we are able to write (**) formally as a system of the

form (*): x(t) = X

_o

+

J

[0,t) k

L

i= 1 A.(X(T),T)f.(T)dA(T) 1 L (t E [O,T]).

o

2.3. From now on we consider the system (*).

Definition: For t~[O,T] we have the A-generated function

F(t) : = A([O,t]).

It is easily seen that F is non-decreasing, continuous from the right and non-negative. Therefore we can write the integral

, ,

9

-as a Stieltjes-Lebesgue integral viz.

J

g(-r)dF(T).

A

We also replace A in (*) by F.

2.4. Remark: In the case that A is a signed measure, its generated function F will be continuous from the right and of bounded variation. We can split

F into a non-decreasing part G

I and a non-increasing part G2:

I f we define

=

lim GI(T),

G

₂(t)

=

TH (t E [O,TJ). (t E [O,T».

-then G

I and G2 are continuous from the right and F(t)

=

G

I(t) +

G

2(t) (t E [O,TJ).

We can consider G

I and G2 as generated functions of the measures Al and

A

2 defined by (0 ~ to ~ tl ~ T)

AI ( {O} )

=

GI(O),At«to,ttJ)

=

Gt(t_{I) - Gt(to);}

A2( {O} )

=

G2(0),A2«tO,ttJ)

=

G

2(tt) - G2(tO).

So we have written a system with a signed measure as a system (**).

2.5. Definition: Let L be the set of n-vector functions that are measurable

and A-summable on [O,TJ.

2.6. Definition: Let a ~ O. For every f E L we define

II f II

a

=

J

exp(-aF(t» •

I

f(t)

I

dF(t).

, ,

10

-It is easily seen that II II is well defined on L.Furthermore i t is not hard a

to prove that II II is a semi-norm in L. a

2.7. For every a >

°

and every f E L we have exp ( -aF (T

»).

II f 110 ::; II f lI

a ::; II f 110 • So II lI

a and II 110 are equivalent on L for a ~O.

2.8. From now on we identify elements f, gEL if II£-gllo

=

0. Then, using 2.7, we obtain II f-gII = 0 (a ~ 0). We shall call the space constructed this way

a

(£ ,Ill! ) on which II II is a positive norm.

a a

2.9. Theorem: For every a ~

°

(£, II II ) is a Banach space. a

Proof: We remark that (£ ,II 110 ) is a Banach space and according to 2.7.

II 110 and II lI

a are equivalent on £. [

2.10. Let H ~n x [O,T] + ~n be a function satisfying the following conditions

1) 3_L >

°

Vx,y

E~n

"It E [o,TJ[IH(x,t) - H(y,t)1 ::; Llx-ylJ

2) V n [

If

H(X,t)E£]

x E~ tE[O,T]

3)

y.;

H(O,t) is bounded.

tt[O,T]

2.11. Lemma: If H satisfies the conditions 2.10. 1), 2) and 3) we have 1.

. ) . -mn

there exists a posit1.ve number Msuch that for every X o E ~ (t E [O,T])

ii) if f :[O,T] +~n is measurable then so is

r

H(£(t),t)

tdO,T]

iii) if f E£ then

l'

H(f(t) ,t) E £ • tE[O,T]

Proof: (i) For every X

o E ~n a~d t E [O,TJ we have

.

"

II

-wli~re M is an upper bound for !H(O,t)

I

(t E [O,T]).,

ii) Let f: [O,T] +Rn be a measurable function. There exists a sequence

of elementary functions

(lQ

f =

La.

(k) A. (k)

k i=I ~ ~

for which f

k + f holds pointwise. The sets (A. (k».~ ~ E

~

.I.~ are, for every

k E :N,measurable and mutually disjoint. Using the fact that H is continuous with respect to x we have

H(fk(t),t) + H(f(t),t)

For every k E :N we can write

(t E [O,T]). 00 = H(

I

i=1 ai'(k) A. (k) (t) , t) ~ ~ 00 = \' L i= 1 ( (k) ) A. (k) (t) H a. ,t ~ ~ (t E [O,T]).

So ~ H(fk(t),t) is measurable (k E:N) and therefore ~ H(f(t),t).

tEeO,T] tE[O,T]

iii) This follows immediately from i) and ii). []

k

2.12. Remark: If H(x,t) =

I

. i=1

A.(x,t) f:(t) as ~n 2.2., then H satisfies the

~ ~

conditions 2.10. 1), 2) and 3) if every A. does. This follows from the fact

~

that f. is measurable and bounded for i = I, ...,k.

~

2.13. Now we define on £ the following operator

cp

\fJ

_(x

O +

tdO,T]

J

[0, t)

H(xer),T)dF(T».

We observe that cp maps £ into £ •

2.14. Theorem: If a > _{L then} cp is a contraction of (£,11 II ).

a Proof: Let y,z E £ .• For every t E [O,T] we have

·

"

-

12-I(ljiy)(t) - (cp z)(t) I

~

L

J

IY(T) - z(,)/dF(,)

[0t t) hence

IIcp y - cpzlla =

f

exp(-aF(t»

I

(cpy)(t) - (cpz)(t)I dF(t)

[O,T]

~

L

f

[OtT]

exp(-aF(t» [

J

[0tt)

Now we apply Fubini's theorem and find

II cp y - cp zll ~ L

J{

J

\y(,) - z(,)Texp(-aF(t»dF(t)}dF(,) a [O,T] ("T]

=

L

f

_{!y(T) - z(,)1 {}

J

exp(-aF(t»dF(t)}dF(T). [O,T] (T, T]

Lemma E I (1.4.) applied to the inner integral with g

=

f

-exp(-ax) gives

x E lR

r

J exp(-aF(t»dF(t)

~

a-I [exp(-aF(T» - exp(-aF(T»]

(" T]

-1

~ a exp(-aF('».

Therefore we conclude

II cpy - cpzll a

~

a-I L

J

IY(,) - z(,)

I

exp(-aF(,»dF(,) [O,T]

-1

= ex, Lily - z'lI •

a

2.15. Theorem E I:

i) There exists an ~n A-sense un~que solution of (*).

ii) The solution of (*) is pointwise unique.

Proof:

i) Let a > L and apply the Banach theorem to the operator CjJ ~n

, ,

- 13

-<.c,

II II ); remark that II II and II 11

0 are equivalent on £ (2.7.).

ex ex

ii) Assume that (for t E [O,T]) ~x

=

x and ~y

=

y:

x(t) = X

_o

+

J

H(x(T),T)dF(T)

[0, t)

yet) = X

_o

+

J

H(y(T),T)dF(T).

[0,t)

Then x and yare equal in the sense of A, thus for t E [O,T]

J

H(x(T),T)dF(T) =

[0,t)

J

H(y(T),T)dF(T).

[0,t)

This means that x = y pointwise in [O,TJ.

3. Some theorems about the solutions of (*).

o

3.1. The theorems proved in this section will be applied in section 4 with the

exception of theorem 3.7 •• This last theorem is included here because it

can serve as a point of departure to extend the theory of optimal Stie1tjes

controllers (eog. finding necessary conditions for optimal Stieltjes

control-lers) •

3.2. Theorem: Let x be the solution of (*). There exist constants C₁ and C₂

(only depending on H, X

o

and F(T» such that

I)II x11

0 ::;; C1

2) \x(t)

I ::;;

_G2 (t E [O,T]).

Proof:

I)Let ex ~ 2L and let ~ be the operator as defined in 2.13 •• Then ~ is a contraction of (!,II II ) with contraction constant s ~. Therefore we have

ex

IIxII = II lpxII ::;; II qlx

·

,

- 14

-~ !II xII + !1I\jl x011 + II 'P(ty xO)11 .

a t a t a

Now using 2.11. we find

IIx II_a ~. IIfXOIl_a + 211 'P (fx_o)II_a t t + 211

'Y

(x O+ L

I

xol F(T) + MF(T))~~a ' t + 2 Ll x oIF 2 (T) + 2MF2(T). The first part of the theorem now follows from 2.7 •. 2) For every t E [O,T] we have, again using 2.11.,

Ix(t)!

~

Ixol +

J

IH(x(,),,) !dF(,)

[0, t)

~ Ixol + L II x 11₀ + MF(T).

The theorem follows immediately from 1).

0

3.3. Theorem: Let x

o' YOE R n

• The solutions of (*) corresponding to X_o resp.

Yo are indicated by x resp. y. There exist numbers M_I, M

2 ~ 0 ( only depending on Hand F(T» such that

I) II x - y II 0 ~ MI IX_{o - YO}

I

2) Ix(t) - y(t)\ ~ M21xo - yol (t E [O,T]).

Proof:

I) Let 'PI resp. 'P2 be the operator as defined in 2.13. corresponding to Xo

resp. YO' We choose a

=

2L and find

IIx - ylla = _{II 'PIX - 'P 2ylla} ~ _{II 'PIX - 'P 1ylla} + II _{('PI - 'P 2) ylla} ~ ~ II x - y II + 1I~(xO - YO) II •

a t a

We may complete the proof in a manner similar to 3.2.1). 2)For every t E [O,T] we have according to 2.10.

Ix(t) - yet)

I

~

Ix

o - yol + L

f

I xC') - y(,)

I

dF(,)

[0,t)

~ IX_o - Yo I + L II _{x - y 110•}

·

,

- 15

-3.4. Remark : Theorem 3.3. states that the solution of (*) depends continuously

on the initial value x

_o.

This theorem also occurs in the theory of ordinary

differential equations.

n .

3.5. Theorem: Let XOE ~ • There exists a constant M>

°

(only depending on H,

X

_o

and F(T)) such that the total variation of the solution of (*) is at

most M.

Proof: If x ~s the solution of (*) and

°

=

partition of [O,T], we have

= T is a k

L

Ix(t,) - X(tJ'_I)I j=l J k

=

L

I

j=l

J

[t, 1,t,)

r

J

J

IH(x(T),T) IdF(T). [0, T)

By using 2.11. and 3.2. the theorem is easily proved.

3.6. We now choose in (*)

o

H(X,T) = A(T)X (x ERn, t E [O,T]).

Here A is a bounded, measurabl~matrixvalued function on [O,T].

It is easily seen that H satisfies the conditions 2.10. and therefore we

have for every X

_o

E ~n a unique solution of (*) which will be indicated

by ~(t,xO) (t E [O,T]). We define (just as in the theory of ordinary

differential equations) a fundamental solution

~~

by

F

J

F (T)dF(T) (t E [O,T]).

~A(t) I + _A(T)~A

[0,t) For every X

_o

IE ~n we have

,"

- 16

-3.7. We now assume that H in (*) has a continuous derivative with respect to x

(notation for this derivative VxH(c,t»and ~urher we assume

(i) V mn [

If

V H(c. t)

C E= tE[O,T] x

is measurab le]

It is a matter of routine (see 2.11.) to show that

LV VH(f(t),t)

x

tE[O,T]

is measurable and A-summable if f is. We denote the solution of (*) with

initial value x

o by ~(t,xO) (t

ative of ~(t,xO) with respect

following

E [O,T]) and we are interested in the

deriv-to x (notation: V ~(t,c». We prove the

°

x

o

Theorem: ~(t,xO) (t E [O,T]) has a continuous derivative with respect to

x o and V ! t;(t,c) = Jf:₀ Where

~~

satisfies

~~(t)

=

I +

f

[0,t) (t E [O,T]). (t E [O,T]).

Proof: Take a fixed c E :Rn and let x

o E :R n • Consider vet) :

=

~(t,xo) - ~(t,c) and ACr) : = V H(~(T,C),T) X

We have for every t E [O,T]

(i E [O,T]).

(t E [O,T]).

vet)

=

x - c +

·

,

(1)

17

-+

'f

'{[H(~(T,XO),T)

-

H(~(T,C),T)

J - A(T)v(T)}dF(T). [0, t)

We wish to compare v with the function z that satisfies

(2) z(t) = x - c +

°

I

[0, t)

and therefore we have to estimate the third term of (1). Let K be a bounded

set in ~n and x,y E K. For T E [O,TJ we consider

R: =[ H(X,T) - H(y,T)J - V H(y,T) (x - y)

x

and define for s E [O,IJ

h(s) = [H(X,T) - H(x + s(y - X),T)J - sV H(y,T) (x - y).

x

We observe that h(O) =

°

and h(l) = R. Now we have

I

/RI = Ih(l)

I

~

f

lli(s) Ids.

°

Using the fact that V H(c,t) ~s continuous with respect to c we find

x n(s) = - V H(x+s(y - X),T) (y - x) + V H(y,T) (y - x) x x [V H(y,T) - V H(x + s(y - X),T)J (y - x) x x =o(ly-xl) (y-x-rO) uniformly in x,y E K.

From the theorems 3.2. and 3.3. we know that

uniformly on [O,TJ, and therefore we have for every T E [O,TJ

Applying the theorem of Lebesgue on dominated convergence we finally find

v(t) = x - c

°

+

J

A(T)v(T)dF(T) + o(/xO- cl) [0, t) (x

o -

c+O) uniformly in [O,TJ.

'.

- 18

-a. such that q> ~s a contraction of (£,II II ) with contraction constant

a. ~

!.

We can conclude II v - zII = II v - q> zll ~ II q>v - cpzII +II v - cpvII a. a. a. a. (x O+ c), and by 2.7. we obtain

By the boundedness of A we have as a final result F

~( t , x

O) - ~( t , c) - CPA(t) (x

o -

c)

=

v ( t) - z ( t)

=

=

J

A(.)[ v(.) - z(.)]dF(.) + o(lx

_{o -}

cl)

[0, t)

This completes the proof of the first part of the

to prove that V ~(t,c) is continuous with respect

x

_o

uniformly in t E [O,T]. theorem. We still have to xO'

The proof is analogous to the proof of 3.3. and we shall omit it.

4. A theorem on the existence of optimal Stieltjes controllers.

4.1. Let X

o

ERn and M, T > 0 be g~ven. In this section we consider systems of

the following type

o

x(t) = X

_o

+

J

A(X(.),T)dA(T) (I) { [O,t) ;\([O,T]) ~ M. +

I

B(x(.),.)d. (t E [O,T]) [0, t)

We assume that A and B satisfy the conditions 2.10. so that, according

to 2.12. and 2.15.,system (I) has a unique solution. Notice that the the-orem 3.2., 3.3. and 3.5. are applicable to system (I). We again define

F(t) : = ~ ([O,t]) (t E [O,T]).

4.2. We now restrict ourselves to functions A for which there exists a k E E,

,

..

- 19

-(1) the first k components of A(x,\) are identically zero for x E~n,

T E [O,TJ

(2) the last n - k components of A(X,T) only depend on the first k components

n

of x for x E R , T E [O,TJ

n

(3) A(X,T) is a continuous function of x ElR ,T E [O,TJ.

4.3. Definition':Let W be the set of functions defined on [O,TJ that are

con-tinuous from the right, non-decreasing, and for which

a

=:; F(t) =:; M (tE[O,T]).

Furthermore we define for FEW

=

X

o +

I

A(x(T),T)dF(T) +

J.

B(x(T),T)dT

[O,TJ [O,TJ

where x ~s the solution of (I) corresponding to F, and finally we define

V :

=

{YF IF E W}.

4.4. For the proof of theorem E II we need ReIly's selection principle (see

[NJ page 222)

If (Sm)m E n is a sequence of ly bounded variation on [O,TJ that converges pointwise to

functions, uniformly bounded and of uniform-, then there exists a subsequence of (g )

mmEn a function g of bounded variation on [O,TJ. If (gm)m E n are continuous from the left resp. on the right, it is easily

verified that there is a limit function

g,

continuous from the left resp.

from the right, su~h that the subsequence converges to g in the points of

continuity of

g

(it only requires the definition' of g(t_{n )}

=

lim g(t) (to E (O,TJ)

_ _ _ v ttt

g(O)

=

g(0) resp. g(t

O) - lim g(t)(to E [O,T», g(T)

=

g(T».

a

. tft

a

A second theorem we use is the theorem of Arzela (see [RSJ page 226)

If (h) ~ is a sequence of functions that are equicontinuous and

m m E ...~

uniformly bounded on [O,TJ, then there exists a subsequence that converges to a continuous function g uniformly in [O,TJ.

The third theorem we use is a special form of the theorem of Relly-Bray

(see [NJ page 233):

If F and (Fm)m E n are non-decreasing, non-negative, uniformly bounded

functions which are continuous from the right on [O,TJ and F + F on a

m

" - 20 -lim

J

g(T)dF.(T) . 1 1-+00 [0,t]

f

g(T)dF(T) [0,t]

in every point t E [O,T] for which lim F.(t) = F(t).

i-+oo 1

4.5. We are able now to prove the following

Theorem: If A satisfies the conditions 4.3. then V is closed and bounded.

Proof: Let y be a limitpoint of V. Hence there exists a sequence (F.). E ~

( . ) 1 1 ~~

in Wand a corresponding sequence of solutions (x 1). ~ of (I) such that

1 E ~~

~im YF. = y.

1-+00 1

h f f · · . h h « (i) (i)))

From t e orm 0 A 1t 1S easy to see t at t e sequence xl , ••• ,~ i E E

is equicontinuous on [O,T]. Therefore there exists a subsequence ( again

(i) (i) ) )

denoted by «xl ,~•• ,xk ))i € ~and a continuous function (~l""'~ such that (i) (i) lim (xl , .•• ,x k ) = (xl' .... xk) uniformly in [O,T]., i-+oo (i:) (i)

On the other hand we may apply theorem 3.5. to the sequence «xl " " , x_n ))iiN and we find that it is of uniformly bounded variation. We conclude that

there exists a subsequence (again denoted by

«x~i)

, •.•

,x~i)))iiN')

and a function (xI, ••• ,x

n) continuous from the left in [O,T] such that

in the points of continuity of x and in t = 0.

In the same way we find that there exists a subsequence of (F.). ,. (again

1 1 € A~

denoted by (Fi) i EE) and a function F which is nO,n-decreasing and

contin-uous from the right such that lim F. = F

i-+oo 1

in the points of continuity of F., and, according to 4.4., lim F. (T)

=

F (T) ;

. 1 .

- 21

-this F satisfies F(T) $ M, so FEW.

We shall prove the following statements

(a) X(t)

=

J

A(x(T),T)dF(T) +

f

B(x(T),T)dT (t E [O,T])

[O,t) [O,t)

(b) _{y = ~im YF. =}

~~ ~

A(x(T),T)dF(T) +

f

B(x(T),T)dT.

[O,T]

It is easy to see that x. + x dominated ~n [O,T] with the exception of

~

a set of measure zero. Therefore we conclude that

and lim i~

J

B(x(i)(T),T)dT [O,T]

J

B(x(i) (T) ,T)dT = [0, t)

J

B(X(T),T )dT [O,T]

J

B(x(T),T)dT [0,t) (t E [O,T]).

Using the fact that A only depends on the first k components of x we find

(1) lim A(X(i) (T),T) = A(X(T),T)

i~

uniformly in T E[O,T]. Moreover we have in the points of continuity of F and -a.f the F. 's, us~ng the theorem of Helly-Bray,

~

(2) lim

i~

J

A(x(T),T)d(Fi(T) - F(T» =

~im

J

A(x(T),T)d(Fi(T)-F(T»=O.

[O,t) ~~[O,t] .

Combining (I) and (2) and recalling that (F.). ~ is uniformly bounded

~ ~ E ~~ on [O,T] we find lim ~~ lim i~

J

A(x(T),T)dF(T) [O,T]

J

A(x(T),T)dF(T) [0, t)

22

-~n the continuity points of F and of the F. 'so This proves (b) and furthermore

l.

we can conclude that (a) holds in a set that is dense in [O,TJ. We observe how-ever, that both sides of equation (a) are continuous from the left and

therefore statement (a) holds everywhere in [O,TJ.

This proves the first part of the theorem: V is closed. It is trivial (by

3.2.) that V is bounded. C

4.6. Remark: In 4.5. we have to take YF instead of x(T).It is easily verified

that the set {x(T)

I

FEW} is not necessarily closed.

4.7. We have restricted ourselves in theorem 4.5. to functions satisfying the

conditions 4.2 •• The necessity of the restriction is illustrated in the

following

Example: Consider the system

maximize

[

X(t)]

yet)

I

[0, t)

l

fCr) y(T).] dF(T) 1 (t E [0,1 J)

I

f(T)y(T)dF(T) [0, I

J

under the condition

°

~ F(t) ~ I (t E [O,IJ).

Here f is a continuous, non-negative real function on [O,IJ for which f(t) < f(l) =

Using lemma E I (1.4.) we conclude

(t E [O,IJ,t ~ 1).

I

f(T)y(T)dF(T) =

I

f(T)F(T - O)dF(T)

~

!.

[O,IJ [O,IJ

Consider the following sequence of controllers (n EE)

F = n

a

1 n(x - 1 + - ) n 1 (0 ~ x ~ 1 - -) n 1 (1 - - ~ x ~ 1) n

23

-From lemma E I (1.4.) it easily follows that

lim

I

f(T)y (T)dF (T) = , .

n n

n+W[O,IJ

Now suppose we have a controller F for which

J

f(T)Y(T)dF(~)

= ,.

[O,lJ

From the fact that the max~mum of f is uniquely achieved in t = I we

conclude that F

=

°

on [0,1) and F(I)

=

I. But then we have

J

f(T)y(T)dF(T) =

°

[O,lJ

Hence the set V is not closed. Acknowledgement

This paper is a result of a senior project at the Technological University of Eindhoven. Therefore the author owes many thanks to the supervisor of

this project Prof. Dr. Ir. M.L.J. Hautus for his inspiring support and remarks.

References

[SGJ Shilov, G.E. and Gurevich, B.L. ; Integral Measure and

Deriv-ative: A Unified Approach; Prentice-Hall, Inc., Englewood Cliffs, N.J.; Revised English Edition, 1966.

[NJ Natanson, I.P. ; Theory of Functions of a Real Variable

(Chapter I to IX) ; Frederick Ungar Publishing Co., New York; American Edition, 1955.

[HSJ Heider, L.J. and Simpson, J.E.; Theoretical Analysis;