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J. SIAM CONTUOL

Vol. 4,

No. 1,

1066

Pdnted n U S

OPTIMAL PROCESSES

GENERAL

THEORY

OF

OPTIMAL

PROCESSES*

SHEL ON

8.

L. CHANGT

In the presait paper a general version of the maximum principle is formu..

lated and proved. Pontryagin's maximum principle

[l]

nd various

ex ten-

sions

121-[9]

thereof become special cases which can be readily der ived

from the general version. Of special interest are the following generaliza-

tions: 1) discrete systems,

2)

systems with multiple merit criteria, 3)

restriction of the control function u to a special class of functions, and

4 )

systems with bounded sta te variables.

Operative addition

and convexity.

Let c denote an infinitesimalqunntity

and G ( t ) a give11 fu~~ct ior i~ f i ~ ~ d11 2 = t tl

t

t J. A functiorr t )

is said to vary infiriitesimnlly from 4 t ) f

where A is

a

positive canstt~ut.Obviously if u(t) is different from Q t )

for

a finite amourit, it

can

be only for a11 irifiriitesililal interval of t i x x r e ,

In

the present paper, one 11eedsotily to eonsider illfiriitesirnal variations

o f the

following form:

where may range from

I

to any finite number, t E

T,

and

ai

i

nd

t( t) are finite nun~bers rid function, respectively.

Operative udditim is denoted by O and is defined as follows: let

6zc l t )

and 6~2(t) enote two infinitesimal v:triations from

zi t ) .

If the

t x s - o

sets

of ti have no element in romnlon, the two sets T

are

disjoint for su ff ic ie nt ly

smdI e Then

If finite vnrintions occur tit tlic

s:tnlci

iristililt, the vslriutions are rearrttrlged

in sequence in 6ul(t)

O

l iuq t ) . I:ur

instatlcc*,

giver1

Received by the editors Jwre 29,

1905

P r e~ e rl te d t t h e F ir st I n t e r n a t i o r r r ~ I on

ference on Programming and Control, held a t the Urlited States Air Force Academy

Colorado , April 15, 1965.

t

Depar tment of Electrical Sciences, Stat e I:rliversity of New Y o r k

t

tony

Brook, New York. This work

was

aporisored by the Oflice of Scientific

Research

Air

Research and Developmerl Command, Waahirlgto~~,. C. , under Grant X o . AF-

AFOSR-542-64.

then

aul(t) au2(t) = akl, tkl t < t i +

A

variation from a to

b

can be considered as a variation from

b

to

a

for a

ne ga ti ve interval.

A set of functions C s operatwely mve x if it has the following property:

given ny u and idnitesimal variations 6u1 and 6th such that all three

functi ons u, u + 6 ~ 1 , nd u + 6th belong to C, then

belongs to C for all values of

h in

the interval 0 < h < 1.

h e

control problem. The controlled system is described by the following

set of differential equations :

where x and u a re the sta te vector and the control vector, respectively, and

f

s a vector function having continuous and bounded first derivatives in

x and being continuous in u. From known existence theorems, given x(t1)

a n d u(t) on

T

= t: tl z) x(t) is completely determined.

The control function u(t ) is required to satisfy three conditions:

a)

u(t) belongs to an operatively convex set of functionsC

11 T,

(b ) the x( t) resulting from u(t) stays within an allowed region X, x t ) t X

for all t E T,

(c) the path terminates at a point x(b), where

t2

may

be

fixed

or urbi-

trary, t2

5

T

A

set u(t) satisfying (a) and

b)

is called an adnira le

control

When

al l three conditions are satisfied, u(t ) is called an allowed control. The nierit

of an allowed control

is

judged by a set of variables yi where

r

An allowed controI A is said to be inferior to B if

6 )

yj(t2)14

yl(t2) 9

. .

and the inequality sign holds for at least one value of i. An allowed rontml

is said

to

be noninferior if i t is not inferior to any other dIo\~e.edmrltrol

in the sense defined above.

The noninferior controls are generalizations of opti~ualm11tmIs for

system with multivalued criteria.

7/25/2019 zhang.pdf


OPTIM L PROCESSES 49

First variations

of

state

and

merit variables

D u e to the infiI l ike~i ~l~al

variation in u( t) (see a ) ) , x ( t ) :ind y(1) :ire different fro111 t ) and ~ t )

~ ( t ) Z ( t )

=

e A t t ) + O ( r ) ,

7)

~ ( t )

= eAv(t)

O ( E ) ,

where &(t) and

A y ( t )

are finite imd arc

srlled

the firvt variations

r ( t )

and y (t ), respectively.

Let z denote the (n

+

N)-di~mnsio~~slector

and h ( x , u, t ) denote thth

(n

N)-di~l~t~twiotldcctor furletiorl

Equations 4) and

( 5 )

rlrl Ix*c*olnbinctiis

he

first variation in z for t if T

is

nadily shown tcr IK

=

C A ( t ,

h )[h(.t,4,

ik') h(*,a,

k P ) ] & .

9 )

Z) at1 w i t h

t k e < t

where

A ( t ,

t') is an (n N)-dim~asio~lillquarci lluitrix satisfying

a A ( t , f ) dhta',6

at1 82

A ( t , t ) ,

an

( t , f )

---. =

<?/ t.f, i t )

dl'

i t , fr l di --

A ( ,

l =

I ,

and

ah/& is

an

( n

N)-tlitnc~~siolu~lclti;rnbnult

ris

with

Since the vector function k i8

i~idrpi~tld lr~f

f

I,

t h ~t m b N

colurnrls of the

matrix ah/& are identically zc%ro.

The following theorem is obvious fro111 I 1.

THEOREM

Let

( 6 u ) l ,

( 6 u ) ~

nd

(6

rfprc~c'ni

njinitesimal

vat iaf

ons

about d ( t ) related by

Let

,

and

(&)3

denote the rst variations i n

z

resulting from

( ),

,

6 ~ ) ~nd ( 8 ~ ) ~espectively. Th en

COROLLARY

The set of admissible first variations about any terminal

P o i n t z ( t2 ) s convex.

General theorems on optimal coneol

EOREM

Given xed points x( tl ) and x(t z) , and letting X be the x-space

(~nbounded), necessary condit ion for a control and path pair O (t ), 3( t)

to b e a noninferior control is that there exists a set of vector functions ( t )

a n d H( ,,

ti

t ) sa t is fy ing

N

12)

H ( ,

5,

i

t )

a 1

i(t) fi(% 6,

)

- 1C ckgk(3, ir

t ) ,

a n d

w h e r e

1 5 )

ck

r o,

a n d the equality sign

in

( 1 5 ) cannot hold for all values of k ; ar is thc rat

v a r ia t i m of the subsequent integral due to an in nitesivtaZ variati(m of u ( t ) ,

w i t h

,

and

t

considered as ixed.

Note that there is no restriction on the infinitesimal variation 6u except

t h a t

O + 6u

belongs to

C .

In this and later theorems

a,

is interpreted

in

the

sa me way. From the definition of a

9)

can be written as

Before proving Theorem 2 some geometrical notions

in

z-space will

established. A point p in 2-space is called accessible if there exists

ad-

missible u ( t ) which brings the system from z ( h ) to p at some time t2 The

set of all accessible points at fixed t2 and at t2 5 T are denoted by ~ ( b )

and SZ respectively.

The set of allowed t e k n a l points is an N-diniensio.al plaW

P,

x =

~ ( t p ) .he intersection of

&th

P

is denoted by I

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5 SHELDON S. L. CHANG

OPTIMAL PROCESSES

LEMMA . Th e terminal point i t 2 ) of a noninferior contlal is a

boundary

point of I and

Q.

LEMMA. Let 0 denote the set i( t2 ) k & fo r all adm.issz%le rst veriatiom

.

Then i b ) s a boundar y point of

9,

Lenlma 2 follows intuitively fmln Lci~l~lla.

It ss

ak0 been proven

rigorously by previous authors [ l ,pp. 86-1061.

LEMMA

.

The re exists a vector

sz~ch

hat

for all allowed first variations hz,

and the eqtcality

sign

i n (17) can?tot

koltl

for all

r~nlt~csf

i

Proof.

Let

l,

denote

i

section ie I mliic.ll sntisfir .s

z i - & i ( t r ) SO, i = n + 1 , n + 2 , . . . , n + ~ .

Since l h ) s the terlllinal point of 2~ 1101lillf(~riorQ I ~ ~ M I ,

ie

and P a

do

not

have itlterior poirlts in c.on11uo11. urthcrnionh Q, is corlvcx because of the

Corollary of Theores1

1.

Ttiere rrists it s u l ~ p o r t l e ~ i eS which sepnrtites

4 and Pa

Let 1 be the rlorriial to 8. Poirlts

011

tlic

4

side of

S are

repre-

sented by ( 1 6 ) and all th e poin ts or1

P,

satisfy

By choosing

z

-

i

<

0

for

j = i

( 1 8 )

Icads to

( 1 7 ) .

Bec.~usc (*:LII (+oiia*idc

t

t~most

with

one bourldnry

pk nc of

8

ho fi11:~I ss rrt ion of t ilc I~bt~lnlit

s

oI)t,:linrtI.

Proof n j Tfimrenl 2. Let / rrpna(*ut i e ruw vcrt or ( h z , . .

,

,,+,,, .

Ineyu:~lit~y1 6 ) car1 hc writf

r r ~

Since

,Y

is the critirr

x - S ~ : I C C ,

hc (.I:Lss of :Idt~lissiblc .ontroIs is iderltica1

with C. Froln (9),

12

Az(t2)

=

a.

J

2

t ? ,

t ) h ( 9 ,

i,

l )

dt.

I

I , $ ( t ) denote the row vector l l A ( t e , ) , and let H( , , u, t ) be de-

fined

as

22)

H( ,, u, ) = ' ( t ) h ( x , u 1).

I n e q u a l i t y (21) is identical with ( 1 3 ) . Multiplying ( 1 1 ) on the left by I

gives

23)

The first n components of

23)

give ( 1 4 ) . The last

N

components of (2 3 )

give

Let

C h

Equat ion

( 2 2 )

s then identical with

(1 2 ) .

THEOREM

. Given jked points x( t l ) and x( t2) ,and letting X be an m

dinzensionaE smooth region in x-sp ace, a necessary con ditio n for a control

and

p a t h p ai r 8 ( t ) , 2 ( t ) o be a n a i n fe r im con tro l s that there exist (f ), { f,

t )

a n d H( &,3, ti ) satis fy ing (12) , 13) ,

15),

nd the following:

w h e r e

2 5 )

= O i f

5

i s a n interior point of X ,

>=

0

i f

5 i s o the boundary of X ,

a n d

s

is the normal to

X

at 2 pointing away

from

X .

Proof. The pmof of Theorem 3 is identical with that of Tlrwrenl 2 up

t o (19). Inequality (19) hol ds on ly f or in finitesim al admissible v:tri:~t ions

of U.Condition (2) defining admissible variations can be written

8s

w h e r e

r'

i s the part of the path d t ) lying on the boundary of S. hug

19) is replaced by itself together with the side condition 26). r* t

represent the ( n

+

N)-component row vector with the

n as

its f i ~ t1

Components and or the remaining N components. I nequ c~Iit~ '

i

ident ical wi th

( 2 7 )

( 2 ) ( t 0 on r .

7/25/2019 zhang.pdf


52

SHELDON S. L. CHANG

From a well-known result

in

variation calculus ( 1 9 ) together with the

side condition ( 2 7 ) is equivalent to the existence of a { ( t ) such that

The integral s taken over periods of time

in

which x

lies

on the boundary

of

X.

LEMMA

.

A

necessary condition for d ( t ) and ( t ) to be a noninferior

control and path pair i t h t there exis ts a I( t ) such that ( 28 ) is satis ed by

all first variations A( z) resulting from 6u (t ) with

i

+

u

belonging to

C

Let { d, t ) = { ( t ) when 2 ( t ) is a boundary point and equal zero when

f t is an interior point. Prom 9),

=

a i: l:2 , ) q J ( ) ~ ( t ,' ) h ( a ,

t )

dt' dt.

Changing the order of integration but integrating over the same area gives

Interchanging the notations t and t one finally obtains

Substituting ( 2 0 )and ( 2 9 ) into ( 2 8 ) gives

Let ' ( t ) be defined as

and let H(+,, u, t ) be defined the same way as

n

( 2 2 ) ; hen ( 3 0 ) is

identical with ( 1 3 ) .Fmm ( 3 1 ) ,

OPTIMAL

PROCESSES

53

The first

n

components of 32) give ( 2 4 ) . The last N components give

and

( 1 2 ) .

The condition ( 2 5 ) is proved in a previous paper for a less general pmb

lem ( 1 0 ) .

t

also follows from the intuitive notion that y i ( h ) can be re-

duced

if

the path

x ( t )

is allowed an excursion beyond X .

The following theorem is the well-know transversality condition and

will be stated without a proof.

THEOREM

.

Let u ( t ) , t 5 L e a n admissible control which transfer s

the phase point from some position %( ti ) S i to the position x( ) E 8 2

where

Sl

a n d S 2 are smooth regi om of points satisfying the following equations

and inequalities:

s2

F / ( x )

=

0 ,

i

=

1,

2,

k

n,

GI( , )

5

0 ,

i

=

I 2,

m .

I n

order

that u ( t ) and ~ ( t )ield the solution of the noni nfm 'or problem

with v ariable endpoi nts, i t i s necessary that there exists a nonzero continuous

vector func tion )( t) which satisfies the conditions of Th em em 3 and , in nddi-

tion, the Iranmersality condition at both endpoints of the hajedory z (t ),

where aj and

a

are arbitrary constants, and b j and b / are nonnegative con-

stants such that

b j =

i f

G j ( x ) < 0 or if x( tl ) is an interior point of

36)

l~ 0 i G j ( % )= 0 and the equation actually defines the ~ou*-

dary of S1 at x( tl ),

and s imilar c d d s old for b /. I n 34), (3 5) and (3 6) the value8 of

the func tions and partial derivatives are evaluated at the ca te sp on di q end-

points.

THEOREM. C orn.& a control problem satisf ying the jollowing conditio ns:

( 3 7 )

f ( x ,

u,

t )

=

A ( t ) x B ( t ) u

+

( t ) ,

( 3 8 ) g ( x ,

U ,

t ) = p (5 , t ) t ) ,

7/25/2019 zhang.pdf


54 SHELDON

S.

L

CHANG

where A( t) and B (t ) are matr ices , f ( t ) , p(x , t ) , and q(u ,t ) are vector func -

tions, p(x, t )

i s

convex in x, and q(u , t ) is convexin

u,

X , 81 Sz and the class C are convex.

If

for a n allowed control and path pair, & ( t)and I ( t ) , a set of functions,

H( ,

2 ,

.d,

t ) , ( t ) , l ( 2 , t ) , a n d C i > 0

i =

1 , 2 ,

- - - ,

can befound

such that (12 ) , 13 ) , (24 ) , 25 ) ,and the transversality condition are satisfied,

then ( t) s a nonin ferior control among all ad missible controls whic h transfer

the phase point from a point onS1at tl to a woint on

SP

at te .

-

Ptoof. Let C' denotethe row vector (Cl; 2 ,

.

C N ) .From ( 1 2 ) , 37),

and ( 3 8 ) ,

( 3 9 )

H +,,U,

t ) =

P 'A (~ ) x

' B ( t ) u P ' f ( t ) C ' p ( z, t )

-

f q ( u , )

From ( 2 4 ) ,

Consider any other allowed control and path pair

u ( t ) , x ( t )

which satisfy

the same terminal conditions. Evaluate the following total derivative:

Subtracting C 'P(2 , t ) c t q ( d , ) C ' P ( X , ) - cfq(u,

)

from both sides

of

41)

and integrating from tl to give

4 ' ( 2 - >l:;

-

C f [ i ( t z >- (tz)l

2 - 2 -

C f p (2 , ) C f p x , t )

dt

OPTIMAL PROCESSES

' (2 - x )

is nonpositive because of the transversality condition and convexity of

l and

S Z.

Therefore

~ [ ~ ( h )

(tz)l

0.

Examples of application of the theorem to discrete systems and other

special cases are given in

a

companion paper.

R E F E R E N C E S

[ I ]L S . P O N T R Y A Q I N , G. OLTYANSILI I ,. V. GAMKRELIDZEN D E. F. s c a

CHENXO, The Mathematical Theo ry of Optimal Processes, Interscience, Ne w

Yo rk , 1962.

121

S.

S . L. CHANQ, omputer optimization of nonlinear control systems by means of

dioit i zed mazimu m princ iple, IR E Internat ional Convent ion Record, Ne w

~ o r k ,961, part 4,-pp. 48-55.

B1 Synthesis of Op timu m Control Systems, McGraw-Hill, New Yo rk , 1961,

Cha p. 12.

[ I

S .

KATZ,

discrete version of Pontryagim's maxim um principle, J. Electr onics

Co nt ro l, 13 (1962), pp. 179-184.

151

L

A N

A N D

C. S . W AN G , he Discre te Mazimum Princ iple , John Wil ey , New

.

Yo rk , 1964.

161 R .

GAMKRELIDZE,ptimal control processes with restricted phase coordinates,

I Z V . A k a d . N a u k S S S R S e r . M at . ,

24

(1960), pp. 315-356.

171 S . S .

L

HA NQ , inimal time control with multiple saturation limits, IE EE Tran s.

.

~ u t o m a t ' i c o n t r ol , AC-8 (1963), pp. 35-42.

a] Optimal control

in

bounded phase space, Automatics, 1(1963),pP. 55-67.

[91 A

modijied maxi mu m principie for optimum control of &m s with bounded

phase coordinates, Second IFA C Congress, Basel, Switze rland, 1963, pp.

405/1-4.

la A n eztension o( Asm li'r theorem and it8 application to the theom of op ti m l

control, Tran s. Ame r.

Math.

Soc., (1965), t o appea r.

(111 General theory of optim al pro~ esseswith applications, Proceedings o f P A C

To kyo S ymposium , 1965, to appear .

On the right-hand side of ( 4 2 ) , he first integral

is

nonnegative because

of ( 1 3 ) and the convexity of the class C. The second and third integrals

are nonnegative because of the convexity of the functions p ( x , t ) and

q ( u , ) , and Ci

>

0 = 1, 2 ,

. .

N. The last integral is nonnegative be-

cause of

25)

and the convexity of X . On the left-hand side of ( 4 2 ) ,

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