Applied Mathematics

A J o u r n a l of Chinese Universit ies

Vol. 8 Ser. B No. 2 Dec. 1993

S E Q U E N T I A L Q U A D R A T I C P R O G R A M M I N G

M E T H O D S F O R O P T I M A L C O N T R O L P R O B L E M S

W I T H S T A T E C O N S T R A I N T S "

Xu C h e n g x i a n ( ~~,~)

(Department o f Mathematics Xi* an diaotong t nwersitg . 71 0 0 4 9 )

Jong de J. L.

(Technology ( n i v e r s i t y o f Eindhuvel~ , Itolland )

Abstract

A kind of direct methods is presented for the solution of optimal control problems with state

constraints. These methods are sequential quadrat ic programming methods. At every iteration a quadratic

programming which is obtained by quadratic approximation to Lagrangian function and linear

approximations to constraints is solved to get a search direction for a meri t function. The meri t function is

formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along

the search direction to determine a step length such that the meri t function is decreased. The methods

presented in this paper include continuous sequential quadrat ic programming methods and disereate

sequential quadratic programming methods.

Key Words Optimal Control Problems with State Constraints, Sequential Quadrat ic Programming,

Lagrangian Function, Meri t Function, Line Search.

l . Introduction

A state-constrained optimal control problem is to determine a control function )~ C- ( ' : ' ( R

R ~) . a s t a t e trajectory z E CZ( R ~ R' ) to minimize the functional

ho(x(O)) + [".ro(x(t). , , (n. t)dt + g~O.(r)) (1. 1)do

Received on May, 8. 1992.

164 Appl ied Mathemat ies - -A J o u r n a l o f C h i n e s e Unive r s i t i e s Vol. 8 Ser. B

s u b j e c t to f o l l o w i n g constraints

dx) 7 = f ( x ( t ) , u ( t ) , t ) , 0 ~ t ~ T , ( 1 . 2 )

D ( x ( O ) ) ---- 0 , ( 1 . . 3 )

E ( x ( T ) ) ---- O, ( 1 . 4 )

w h e r e h 0 : R ' - - ~ R , f 0 : R~ X Re X R--~ R , go: R'---~ R , f : R" X R"~ t R---~ R' , D: R'---~ Rp , E :

R ' --~ Rq a r e all t w i c e c o n t i n u o u s l y differentiable functions with respect to t h e i r a r g u m e n t s , a n d T is

the f i x e d f i n a l t i m e . The constraints are e q u a t i o n s o f the d y n a m i c b e h a v i o u r o f the s y s t e m

( e x p r e s s e d by ( 1 . 2 ) ) , initial state constraints ( g i v e n by ( 1 . 3 ) ) a n d t e r m i n a l s t a t e constraints ( i n

( 1 . 4 ) ) . This kind o f p r o b l e m s a r i s e in practice w h e n t h e r e is a d e m a n d to control a s y s t e m from

one state to another in some optimal s e n s e , for e x a m p l e , f l i g h t path optimization o f air p l a n e s a n d

s p a c e v e h i c l e s , econometrics or robotics.

N u m e r i c a l m e t h o d s for the solution of the state-constrained optimal control p r o b l e m s can be

d i v i d e d into t w o k i n d s d i r e c t a n d indirect m e t h o d s . The d i r e c t m e t h o d i s s t a r t e d with an initial

approximation to the solution a n d the approximation i s i m p r o v e d iteratively by minimizing the

objective f u n c t i o n a l , a u g m e n t e d with a p e n a l t y t e r m , a l o n g a s e a r c h direction w h i c h is o b t a i n e d by

approximation to the p r o b l e m . In d i r e c t m e t h o d s , u ( t ) i s treated a s v a r i a b l e s o f the minimization

p r o b l e m , x ( t ) can be t r e a t e d either a s a q u a n t i t y d e p e n d e n t on the control u ( t ) by u s i n g the

differential e q u a t i o n s o r a s v a r i a b l e s o f the p r o b l e m w h i l e the differential e q u a t i o n s are r e g a r d e d a s

e q u a l i t y constraints. I n d i r e c t methods use the optimality conditions to d e r i v e a multipoint b o u n d a r y

v a l u e p r o b l e m , a n d the n u m e r i c a l solution o f the multipoint b o u n d a r y v a l u e p r o b l e m y i e l d s a

c a n d i d a t e for the solution o f the optimal control p r o b l e m . F o r indirect m e t h o d s the convergence

r e g i o n is g e n e r a l l y s m a l l a n d the g e n e r a t e d n u m e r i c a l solution is a c c u r a t e . H o w e v e r , the indirect

methods need the k n o w l e d g e o f the s t r u c t u r e o f the solution. F o r d i r e c t m e t h o d s , a l t h o u g h the

generated n u m e r i c a l solution i s not so accutate a s for the indirect m e t h o d s , its convergence r e g i o n i s

l a rge a n d it does not n e e d the k n o w l e d g e o f the solution.

2. Optimality Conditions

Optimality conditions play the central r o l e s in a n y solution m e t h o d s for optimization. S i n c e the

state-constrained optimal c o n t r a l problems a r e the special c a s e s of the f o l l o w i n g a b s t r a c t optimization

p r o b l e m ,

minimize f ( x ) ,r E X

s u b j e c t to h ( x ) ~ 0 , ( 2 . 1)

w h e r e f . X ~ R , h : X --~ Z , a n d X , Z are B a n a c h s p a c e s , the optimality conditions for p r o b l e m

No. 2 Optimal Control Problems with State Constraints I 6 5

( 1 . 1 ) - - ( 1 . 4 ) c a n be d e r i v e d from those of the p r o b l e m ( 2 . 1 ) .

( 1 ) The f i r s t o r d e r necessary conditions for problem ( 1 . 1 ) - - ( 1 . 4 ) can be s t a t e d a s f o l l o w s :

Let ( z , u ) be a solution to p r o b l e m ( 1 . 1 ) - - ( 1 . 4 ) , then there exist a real n u m b e r p ~ 0 , a

vector f u n c t i o n ~,,: [ O , T ] --~ R ' , vectors o" E Rp a n d ~ E /i~ s u c h that

/.(t:) ~ -- £(t~)~ = - - ~" I6EtJdt, for all 0 ~ t, < t~ ~ T, ( 2 . 2 )dq

w h e r e

,~.(0) * = - - pho,E0-I - 3~D, E0~,

~'.(T) r = pgo,E0-] + ;/E, ET],

( 2 . 3 )

( 2 . 4 )

Hit] = p f o ( x ( t ) , u ( t ) , t ) + ~.(t)rf(x(t),u(t),t), ( 2 . 5 )

is a I4amiltonian f u n c t i o n , the notation [ t ] s t a n d s for ( x ( t ) , u ( t ) , t ) o r ( z ( t ) , t ) , a n d the subscript

x d e n o t e s the derivatives respect to z . F u r t h e r m o r e , if

rank ( D , [ 0 ] ) = p , rank ( E , [ T ] ) = q , ( 2 . 6 )

a n d t h e r e is a pair ( 6 x , 5 , ) such that

D,[O-]6,(O) = 0, (2. 7)

E, ET]6,(T) = O, ( 2 . 8 )

6Jc(t) = f , EtJ6,(t) + f : E t ] 6 , ( t ) , 0 ~ t ~ T , ( 2 . 9 )

then the r e g u l a r i t y constant/~ is not zero a n d hence p = [ . Conditions (2 . 6 ) - - ( 2 . 9 ) a r e c a l l e d

constrained qualifications.

In o r d e r to give the s e c o n d o r d e r conditions, d e f i n e the set S o f feasible directions

S = {(6,,6,,)]D,U_0]6,(0) = 0 , E~[T]6~(T) = O,

6 ~ ( t ) = f : l - t ] 6 , [ t ] ± f,,7#Ja,,(t), 0 ~ t ~ T}.

( 2 ) The s e c o n d o r d e r n e c e s s a r y conditions

Let ( x , u ) be a l o c a l solution to problem ( 1 . 1 ) - - ( 1 . 4 ) a n d constrained qualifications ( 2 . 6 )

- - ( 2 . 9 ) be satisfied. If there e x i s t vector function ).(t) : [ 0 , T ~ - + R° , a n d vectors <~ ~ R~ a n d / )

Rq satisfying the f i r s t o r d e r n e c e s s a r y conditions at ( ~ ' , u ) , then

~/2L(x,u,~;,~,~)(6,,6,,)(6,.6,,) ~ O. V (6,,&,) E S , (2. 10)

w h e r e L(z,u,o-,).,/~) is the L a g r a n g i a n functional d e f i n e d by

t*T= f o ( x ( t ) , u ( t ) ,t)dt + go(x(T)) + . 'TrD(J'(0))L(x,u,cr,).,~u) h 0 ( x ( 0 ) ) T Io

166 Applied Mathernaties--A Journal of Chinese Universities Voi. 8 Ser, B

(3) The second order sufficient conditions

I f the point ( x , u ) i s feasible to p r o b l e m ( 1 . ] ) - ( 1 . 4 ) a n d t h e r e e x i s t multipliers ~ E R ' , / ]

/ ~ a n d 5.: [ 0 , T ] - ~ R~ s u c h that the f i r s t o r d e r n e c e s s a r y conditions a n d constrained qualifications a r e

satisfied a t ( x , u ) . M o r e o v e r , if t h e r e is a fl ~ 0 s u c h that

V'L(~,,, ,~,; . ,~)(¢,~,)(¢,~,,) >~ ~(:11¢1l~ + I1~,,11~) for all ( 6 . , 6 , , ) E S, (2. 12)

then ( x , u ) is a l o c a l solution to problem ( 1 . 1 ) - - ( 1 . 4 ) .

3. Continuous Sequential Quadratic Programming Method

B a s e d on the i d e a w h e n a n approximation is near the s o l u t i o n , a better approximation c a n be

o b t a i n e d by s o l v i n g a s u i t a b l e q u a d r a t i c p r o g r a m m i n g w h i c h i s o b t a i n e d by a q u a d r a t i c

approximation to the L a g r a n g i a n functional a n d l i n e a r approximations to the constraints. T h e r e f o r e ,

S Q P - m e t h o d s iteratively s o l v e a s e q u e n c e of q u a d r a t i c s u b p r o g r a m m i n g p r o b l e m s u n t i l a satisfactory

approximation to the solution is o b t a i n e d .

Let ( ~ , u ) be a n approximation to the solution ( x , u ) . The q u a d r a t i c s u b p r o g r a m m i n g o f

p r o b l e m ( 1 . 1 ) - - ( 1 . 4 ) at the point (~', u) is to d e t e r m i n e a correction d,, f o r the control f u n c t i o n u ,

a n d a correction d. for the state trajectory ~ to minimize

" ½ho.[O]d.(O) + jo[fO.[t]d.(t) + fo,,[t-]d,,(t)]dt -p go.[T]d.(T) + d.(O)rM~d.(O)

1 "T r ?MzEt] MaEt~ Fd.~(t)] l+ T , ) :]1 . d, + (T )L ' I I~d , (T ) (3. 1)

L M 3 E t ] M 4 F t ] J La'o(t)J y d ,

s u b j e c t to

w h e r e

D[O]+D.[O]d,(O) = O.

d x ( t ) = Y . E t ] d . ( t ) + y o E t ] d , , ( t ) - - ~ ( t ) ~ f[t],

SET]+~,[T]d,(T) = o,

O ~ t ~ T ,

M , = ho,,EOq + ~,, ~ ,D . , , 703 E n "~ ' ,3"1

M ~ E t q = f o , , E t ~ + ~-~ ;.,f.,,,Et~ E R °×"j I

j I

M , E t ] = fo,,,,Et] + ;'.;f.,,,oEt~ q n . . . . .j ~ l

( 3 . 2 )

( 3 . 3 )

( 3 . 4 )

( 3 . 5 )

( 3 . 6 )

( 3 . 7 )

( 3 . 8 )

N o . 2 O p t i m a l C o n t r o l Problems w i t h S t a t e C o n s t r a i n t s ] 67

Ms = go,,ET] + ~ .mE~,,ET] E R'~" ( 3 . 9 )j . 1

a r e s e c o n d derivative matrices.

Let (2 , ,d=) be a solution o f the problem ( 3 . 1 ) - - ( 3 . 4 ) , then there e x i s t a real n u m b e r } ~ 0 ,

a vector function ). E R~ , v e c t o r s 3 E R P a n d ~ E / F , not a l l z e r o s , s u c h that

~ . ( t ) r = - - ) . ( t ) r f , [ t ] - - p f 0 , [ t ] - - ~ , ( t ) r M 2 [ t ] - - Pd, ,( t)rM~[t], O ~ t ~ T , ( 3 . 10)

f~(O) r - = - - pho,[-O] - - ~ro,[-O] - - ~ , ( O ) r M , , (3 . 1 1 )

f~ (T) T = pgo,[T] + ~ r F ~ ( T ) + ~ l , ( T ) r ] l t 3 , (3 . 12)

f ~ ( T ) r f , [ t ] Jr- pf0~[t] + pd~(t)rM3[t] -+- ~9~l, ,( t)rM4[4] = O, O ~ t ~ T . ( 3 . 13)

It f o l l o w s from the f i r s t o r d e r n e s e s s a r y condition ( 3 . I O ) - - ( 3 . 13) a n d the constraints (3 . 2 ) -

( 3 . 4 ) that if problem ( 3 . 1 / - ( 3 . 41 h a s a solution for w h i c h the constrained qualifications are

satisfied then the solution of the p r o b l e m ( 3 . 1 ) - ( 3 . 4 ) c a n be o b t a i n e d a s the solution o f the

f o l l o w i n g set of e q u a t i o n s .L

,L,(t)|,~. j L-- M2Et] -- f , E t ] r L~( t ) L - - M~Et] r ]

r f r t ] - ~ ( t ) ]+ L - So,-J] ' J ' o < , ~< :~, (3. t,~1

_ r ~ , ( t ) ]

L ] + M,E~]~.(~/--/o.1-~], o ~ t < 7, (3.1,~)E,~IZj ] ' L L t ] ' ] ;.(t/

+ o : = - - , ( 3 . 16 )L M, /J L}.(0) J D,[01" Lh~,E01U

o o .

L Ms - - I L, ; . (T)J L£',['/']rJ Lg0,E:r]'J

This system of equations becomes a standard linear multipoint boundary value problem, when the

equations in ( 3 . |5 ) are used to eliminate d,(t l , while the expressions in (3. 16) and (3. ]7)

constitute b o u n d a r y equations.

For the solution of the l i n e a r multipoint b o u n d a r y v a l u e p r o b l e m , approximation m e t h o d can

be e m p l o y e d , that i s , the t ime functions are approximated by a finite-dimensional base ( g e n e r a l l y

p i e c e w i s e polynomials o n [ 0 , T ] ) , a n d a la rge a n d s p a r s e system of i i n e a r e q u a t i o n s is g e n e r a t e d .

w h i c h usua l ly has the f o l l o w i n g form

I i:1(" 0J

168 Applied Mathematics--A Journal of Chinese Universities Vol. 8 Ser. B

w h e r e M a n d C are s p a r s e a n d b a n d e d m a t r i c e s , a n d d a n d ~ a r e coefficients to be determined for the

t ime functions. The s y s t e m (3. 18) m a y be s o l v e d with a n y s p a r s e t e c h n i q u e s , for e x a m p l e r a n g e

s p a c e or null s p a c e m e t h o d s .

The derivation of the q u a d r a t i c s u b p r o g r a m m i n g p r o b l e m is b a s e d on the fact that the

approximation h o l d s o n l y in a neighbourhood o f the s o l u t i o n ( x , u , o', 5.,/~). H e n c e it i s a s s u m e d that

the c u r r e n t iterate i s c o l s e to ( x , u , a , k , / ~ ) . In o r d e r to f o r c e a g l o b a l c o n v e r g e n c e , the solution o f

the q u a d r a t i c p r o g r a m m i n g problem is u s e d a s a s e a r c h direction of a m e r i t f u n c t i o n , w h i c h a s s i g n s

a real v a l u e to each p o i n t ( x , u , o ' , Z , a ) , a n d w h i c h approaches its m i n i m u m a t the point ( z , u , ~ , ) . ,

/~). A s u i t a b l e choice for the m e r i t function i s the a u g m e n t e d L a g r a n g i a n f u n c t i o n , introduced by

P o w e l l a n d Hestence ( 1 9 7 6 ) on finite-dimensional nonlinear p r o g r a m m i n g .

M(x,u,cx,).,g,p) = L ( x , u , c + , ; , . , / ~ ) -F- -~EIID(~(o))ll ~ -+- , u ( t ) , t )

dz r dx; i ] [ f ( x ( t ) , u ( t ) , t ) - - ~ ] d t + i l g c z < V ) ) I I ~, <3. 19)

With sufficiently l a rge v a l u e o f p , the ( ~ r , ~ r ) o b t a i n e d in the solution of the p r o b l e m ( 3 . 1 ) - - ( 3 .

4 ) is a d e s c e n t direction o f the m e r i t function M ( x , u , o ' , ) , , g , p ) at the point ( x , u ) , then a step

length 3 ~ 0 can be t a k e n a l o n g the direction s u c h that

w h e r e

M ( h ) < M ( O ) ,

M ( a ) = M ( x --~ a d , , u -q- ctd,,o" + a(o" - - a ) , ) , q- a ( ~ - - A) , p q- a ( ~ - - g ) , p ) ( 3 . 2 0 )

is a function of s i n g l e v a r i a b l e a , a n d g , L a n d ~ a r e o b t a i n e d in the solution of p r o b l e m ( 3 . 1 ) - -

( 3 . 4 ) . Then next approximation to the solution a n d multiplers a r e c a r r i e d out by setting

xi+: = xi Jr- ad~,, ui+l = u, q- a d : , ,

cr~+~ = a, + a(a, - - c~,), ;'.~+~ = L ÷ a ( L - - L ) ,

It i s p r o v e d t h a t w h e n

/l~+, = u, + a(~+ - - /~i).

i s s a t i s f i e d , ( d , , d , ) i s a d e s c e n t direction o f the m e r i t f u n c t i o n , w h e r e

= Z(d:,d=)/ll(d:,,d:,)[l~, '/=- II(d=,d,)ll~/ll(~-- -,,~-- z , ~ - ~)11~,

f r r M2 d~M ( d , , d ~ ) = d . ( O ) r M , d g O ) + o(d, , d ,r ) dt + d . ( T ) r M s d = ( T ) .M r M4 [ d ~

H e n c e the penalty f a c t o r p c a n be a d j u s t e d a t e v e r y iteration to s a t i s f y the i n e q u a l i t y ( 3 . 2 1 ) .

( 3 . 2 1 )

N o . 2 O 0 t i m a l C o n t r o l P r o b l e m s w i t h S t a t e C o n s t r a i n t s 169

4. Discrete sequential quadratic programming

The state-constrained optimal control p r o b l e m c a n be converted into a finite-dimensional

nonlinear p r o g r a m m i n g by discretizing the t ime f u n c t i o n s , that i s , the interval [-0, T] i s d i v i d e d into

s u b i n t e r v a l s , the i n t e g r a l is r e p l a c e d by n u m e r i c a l i n t e g r a l , a n d the differential is s u b s t i t u t e d by

difference. The r e s u l t i n g finite-dimensional nonlinear p r o g r a m m i n g i s s o l v e d by g e n e r a l s e q u e n t i a l

q u a d r a t i c p r o g r a m m i n g m e t h o d s , a n d by m a k i n g use of the characteristic o f the problem.

Let the i n t e r v a l [ - 0 , T ] be d i v i d e d into _Vs u b i n t e r v a l s w i t h e q u a l d i s t a n c e h = T ~ N , a n d h~ =

i h , i : 0 , 1 , . . . , N . Then the discretized state-constrained o p t i m a l control p r o b l e m h a s the f o l l o w i n g

form

m i n i m i z e

s u b j e c t to

ho(xo) -~- S hf°(xi'ui'hi) -~- gO(Xv)' ( 4 . l )i~O

x,+l - - x i : h f ( x i , u . , h i ) , i = 0 , I , " ' , N - - 1 , ( 4 . 2 )

D ( x o ) = 0 , ( 4 . 3 )

E ( z ~ ) = 0 , ( 4 . 4 )

w h e r e x~ ---- z ( h , ) E R", u, = u ( h , ) E R m, a n d there a r e total N ( n + m ) + n decision v a r i a b l e s a n d

Nn q- p t q e q u a l i t y constraints.

Let ]',, i : 0 , 1 , . " , N , u , , i---- 0 , 1 , - " , N - - 1 be a solution o f p r o b l e m ( 4 . 1 ) - - ( 4 . 4 ) , then

t h e r e e x i s t vectors 5.0,~1,'",,~.x E R " , ~ E R' a n d / ~ E / ~ , not all z e r o , s u c h that

^ A ^)r ).r 3 H ' ( z , , u , , ) ~ _ ~ )^ - - : - - , i----- 0 , 1 , . . . , N - - 1 ,

Oxi

~T go,(~,) +/.TE,(_;-,.),A3 =

w h e r e

H ' ( x i , u , , ) . i + j ) = A ( x , , u , ) "q- f i (x , ,u~)rzi+J

is a H a l m i t o n i a n function with

f o ( x , , u , ) = & o ( x i , u ~ , h i ) ,

f ' ( x i , u D = hf(z~,u~,h~).

The L a g r a n g i a n function for p r o b l e m (4. 1 ) - - ( 4 . 5 ) i s

L ( x , u , a , ) . , # ) = h 0 ( x 0 ) + trrD(xo)

+ ~ Efo(~,,,,,) + z+,(:'(~,,,,,) - x,+, + ~,)3i = O

( 4 . 5 )

( 4 . 6 )

( 4 . 7 )

( 4 . 8 )

( 4 . 9 )

( 4 . 1 0 )

1"70 Applied Mathematies--A Journal of Chinese Universities Voh 8 Ser. B

-k- g 0 ( z , ) q- ~ r E ( z x ) . ( 4 . 1 1 )

The s e c o n d derivative m a t r i x of the L a g r a n g i a n function L with respect to z a n d u is a b l o c k d i a g o n a l

m a t r i x

V 2 L ( z , u , a , ) . , u ) ---- d i a g ( M [ 0 ] " , M [ I ] , . . . , M [ . V - - 1 ] , M s ) ( 4 . 12)

w i t h

LM~D- I T M,Ez]..I ,,,~

M:0j.[M, + ,E0 lL M ~ E 0 Y M, EOL! ,×,

Ml = ho,~(xo) ~ ~ ajD:.~(xo),1 - i

MZ,3 = H~£i3,

M3[i] = I l l , [ i - ] , i-.~ 0 , 1 , . . . , _ x - 1 ,

M, D3 = H,.,D],

Ms = go:,(z,,-) + 2 u ~ E ~ , ( z , ) .j = l

i = 1 , 2 , . - - , N - - 1 , ( 4 . 1 3 )

(r = n - ~ - m ) , ( 4 . 1 4 )

( 4 . 1 5 )

( 4 . 1 6 )

( 4 . 1 7 )

( 4 . 1 8 )

( 4 . 1 9 )

d = [dxo,duo ,dZl , d u , , . . . ,dz:~_, , d u ~ - ~ , d x ~ ] r ,

~") - - [-h0,(x0) + ~,(zo,uo),fS~(z~,uo),f~,(x~ , ~ ) , . . . ,

u ~ * " - ~ z . . . , . _ , ) , ~ . ( z , ) J ~fL-~(z~-,, . , - ~ , j o . ~ .,--~

c t*) ~- [ D ( x o ) , f f ( x o , u o ) - - xl "t- z o , f I ( x ~ , u : ) - - Xz + x i , " ' ,

y " - ~ ( z , - _ , , u , _ , ) - r , + z~_, , E ( z , ) ] ~

( 4 . 2 0 )

w h e r e

u*) c a n be expressed a s

m i n i m i z e

s u b j e c t to A " / 6 -~- c (~) = 0 ,

With these expressions the q u a d r a t i c s u b p r o g r a m m i n g of p r o b l e m ( 4 . 1 ) - ( 4 . 4 ) at the point ( x (~) ,

No. 2 Optimal Control Problems with State Constraints 171

A(')r :

- D , ( x o )

r ~ [ o [ - I

r , D ] r_ .D] - I

. . , , . .

r,E_v - t]

( 4 . 2 2 )

that is,

du, = - - [ M , Ei] ]-- 'E(A,,) T 4- f~TZ,--, 4- M3D]rdx,] ,

- - A ( ~ ) ' 6 = c <'). ( 4 . 29 )

i = 0 , 1 , . . . , X - - 1 . ( 4 . 3 0 )

r , [ _ \ ~ - 1 ] _ j lj"

E,(z,)

with I 1 [ i ] = I q - f ' , ( x , , u , ) , I ' 2 [ i ] = f ' o ( x , , u , ) , a n d 1t(k) is either ~ 2 L ( ' ) or its approximation.

W e s o l v e the p r o b l e m ( 4 . 2 0 ) to get a solution 6c,) a n d n e w estimates of L a g r a n g i a n multipliers

, # a n d ). , then 6 (~ is u s e d a s a s e a r c h direction to minimize a m e r i t f u n c t i o n w h i c h is o b t a i n e d by

a u g m e n t i n g the L a g r a n g i a n function w i t h a penalty t e r m ,

M ( x , u , q , ) . , ~ t , p ) = L ( x , u , q , Z , / . z )

3 - - 1

+ [l lD(xo) IIN + -- + z, + ( 4 . 2 3 )i . I)

Once a step l e n g t h , say ak , i s d e t e r m i n e d s u c h that M(ctk) < M ( O ) , a n e w approximation to the

solution a n d n e w estimates of L a g r a n g e multipliers a r e o b t a i n e d a s

x~ '~ ' ; = z~ ') + akdxp~, i = 0 , 1 , ' " , X - - 1 , ( 4 . 2 4 )

Z/~ ' q - l ) = H!' ) - ] - ( l kd l l~k) ,

a ~+') = a (') ÷ a ~ ( ? , - - a<~>), ( 4 . 2 5 )

Z~'-1) = ).~) 4 - a,(~., - - Z}~)), i = 1 , 2 , - " , N , ( 4 . 2 6 )

/ ~ ( ' - ' ) = /1Ck) 4 - a ~ ( ~ - - /1(~)). ( 4 . 2 7 )

It f o l l o w s from the f i r s t o r d e r n e c e s s a r y conditions o f p r o b l e m ( 4 . 2 0 ) that the solution of

problem ( 4 . 2 0 ) can be o b t a i n e d by s o l v i n g the f o l l o w i n g s y s t e m of e q u a t i o n s .-!= . ( 4 . 2 8 )

- A <')~ 0 i L c ") J

W h e n 1t-(~) t a k e s the s e c o n d derivative m a t r i x o f the L a g r a n g i a n f u n c t i o n , d,,, c a n be eliminated

from the s y s t e m

1 7 2 Appl ied Mathemat ics - -A J o u r n a l o f C h i n e s e Unive r s i t i e s V o l . 8 Ser. B

S u b s t i t u t i n g them into the f i r s t system of ( 4 . 2 8 ) , we g e t , a f t e r some m a n i p u l a t i o n , a s y s t e m o f

e q u a t i o n s

y,+~ - - y, = A,y,+~ + B,y, + f , , i = 0 , 1 , ' " , . V - - 1 , ( 4 . 3 1 )

Myo + Xy~ = b

w i t h

~-dx,

' b , [ i ]

, f , = i0

iLo

, b ~ I-- D(zc) 1- - E(z~ ) / '

- hL(xo)!

~oS (x,) j

b , [ i ] = f - - x,+, + x~ - - f : , (M4r - i ] ) - ' ( f~oo) T,

b~[ i ] ' '= - (fo,) + M ~ E i ] ( M , [ i ] ) -'(foo)" ' ,

( 4 . 3 2 )

( 4 . 3 3 )

( 4 . 3 4 )

A i

/11=

-0 - - M T [ - i ] 0

o z - M,D] o

0 0 0

0 0 0

-IL(zo) 0 0

0 0 o

ML I D,(zo) r

0 0 0

:lO;

O_ ~Xr

O

0

0

0 r'/:,~

, B i =

- M 6 [ i ] - - I

- - M, Ei]

0

L 0

0 0

E,(x, ) 0

0 0

- - 1113 I

0 0

0 0

0 0

0 0

0

0

0

0

O.

0

0

O. r ¢ , r

0

0

0

E , ( x x )7 r × r

M~D] = J + f ; - - £ , ( M , [ Q ) ' M , [ i y .

/117[i] = f '~(M4[i])- ' ( f i , ) r

( 4 . 3 5 )

( 4 . 3 6 )

M s E i ] = I + (J~L) r - - M 3 E i l ( / 1 1 4 E i ] ) - ' ( f . ) r ,

MgEi] = M ~ [ i ] - - M 3 r - i l ( M , r i ] ) - ' M 3 [ i ] r ,

( 4 . 3 7 )

( 4 . 3 8 )

The solution o f the s y s t e m ( 4 . 3 1 ) can be g e n e r a t e d by the f o l l o w i n g f o r m u l a ei--I

u, = ~o(i,O)vo + ~ o ( i , j + 1)3"~, i = 1 , 2 , . . . , _ v ,j = 9

( 4 . 3 9 )

w i t h

3. -- 1

Yo = [-M + N q g ( N , O ) J - ' ( b - - ~ X c p ( . \ , j + 1 ) f j ) ,]=6

(4.4O)

No, 2 Optimal C o n t r o l P r o b l e m s with S t a t e C o n s t r a i n t s 173

~ o ( i , j ) = I ] l-, = ( t r . . ~ . . . r , _ , )

I ' : = ( 1 - - A : ) - ' ( I J r - B j ) , j = O , I , ' " . . V - - 1

f , = ( I - - A j ) - l f ~

( 4 . 4 1 )

is factorized into the form

W h e n the s e c o n d derivative m a t r i x of the L a g r a n g i a n function L is not e v a l u a t e d , but just

approximated by a positive definite m a t r i x , a n d the approximation is u p d a t e d by u s i n g u p d a t i n g

f o r m u l a e for e x a m p l e BFGS f o r m u l a e . The positive definite approximation, s a y t l-'~k), is stored in

L D L r form

w h e r e

11"(') ~--- d i a g (II~~) , i,l-~k), . . . , , , tr v - 1e) ,--w¢*)).', = L D L r ' ( 4 . 4 4 )

L ---- d i a g ( L 0 , Z ~ , ' " , L , ) ( 4 . 4 5 )

is a b l o c k d i a g o n a l a n d l o w e r t r i a n g u l a r m a t r i x , a n d [,,, i : 0 , ] , . . . , N are unit l o w e r t r i a n g u l a r

m a t r i c e s , D is a d i a g o n a l m a t r i x with positive d i a g o n a l elements.

The s y s t e m ( 4 . 2 8 ) is s o l v e d by u s i n g L D L r factorization o f the coefficient m a t r i x , that i s , the

m a t r i x

_ _ A(k) r

This can be done by just f o r m i n g the m a t r i x

= - D - I L - 1 A (*) ( 4 . 4 7 )

a n d u s i n g Q R factorization m e t h o d to the m a t r i x B to get L D L r factorization o f the m a t r i x

A ( ~ : [ , - r ] ) - I L - 1 A (~). In the processes o f f o r m i n g B a n d Q R factorization of B , the sparsity of the

m a t r i x A (k) is preserved. The solution o f ( 4 . 2 8 ) is then o b t a i n e d by b a c k w a r d a n d f o r w a r d

substitutions.

A f t e r a n e w iterate i s o b t a i n e d , e v e r y b l o c k W~~) , i ---- 0 , 1 , " ' , N in the m a t r i x lJ=(~) i s u p d a t e d

( i f n e c e s s a r y ) by u s i n g BFGS f o r m u l a e

r / ? ) ~ ) T W?)6?)5¢~:W(? )i = 0 , 1 , . . - , N . ( 4 . 4 8 )W~+I) _-- B'[~) + 6(~)T~(,) 6(kTW(k)6(~) '

w i t h

( 4 . 4 6 )

( 4 . 4 2 )

( 4 . 4 3 )

174 Applied Mathematics--A Journal of Chinese Universities Vol. 8 Ser. B

[z~,+~ _ xP'1= . i = O , ] , . . . . X - - 1 ,

a!~~ = ~ , + ~ __ ~,!~,

/

' ' " ~ \ - 1(t~) T

and d <~r = (a<]' K . d ~ ~ ,d~ ). The p a r a m e t e r ~9 is determined in such a way that

T . T , •

d{~> q t) ~ 0 .2d ~' II ~',5C'}

so that the posi t ive defini te property of the m a t r i x 11-¢~} is guaranteed.

( 4 . 1 9 )

( , t . 50)

( 4 . 5 1 )

( 4 . 5 2 )

References

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1982.

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constraints I: Necessary conditions for extremal solutions. AIAA .lo~rnal , 1 (1963).

3] Dickmans, E. D . . and Wel l , K. H. , Approximate solution of optimal control problems using third order

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E4! Fletcher, R. , Practial Methods of Optimization Vol. 2. Constrained Optimization. John Wiley ~,. tons. New

York, 1981.

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[ 6 ] Jacobson. D. H. , and Lele, M. M . . A transformation technique for optimal control problems with a s ta te

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[_7] Jong de J. L. and Machielsen, K. C. P. , On the application of sequential quadrat ic programming to state-

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[8] Lorentz, J . , Numerical solution of the minimum-time flight of a glider through a thermal by use of multiple

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Eg] Maurer, H. , On the minimum principle for optimal control problem with s ta te constraints, Schriftenreihe des

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El0] Norr is , D. O. , Nonlinear programming applied to s ta te constrained optimization problems, Journal o f

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