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Application of a New Class of Augmented Lagrangians to the Control of Distributed Parameter Systems
Copyright © IFA C 3 rd Symposium Control of Distribu tf'd Paramcter Systems T oulo use . France. 1982 .
APPLICATION OF A NEW CLASS OF AUGMENTED LAGRANGIANS TO THE CONTROL OF DISTRIBUTED PARAMETER SYSTEMS
J.
F. Bonnans
Institut National de Recherche en Informalique et en Automatique (INRIA) , Domain e de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France
Abstract. This paper describe:; a new r:ethod to solve O?tir.lal control problers of distributed pararreter systaTIs. The study is restricted to problems with a convex criterion and a state equation affine with respect to the pair (state, control).
\']e
show the equivalence betvleen the initial problem and the rinini-
sation with respect to the control, the state and also the costate of sc.r-e augmented lagrangian, obtained by addition of the lagrangian and of b...a tern; penalizinq the state and costate equations. An exarple is treated, and nllJ"'"erical results are given. the problen and the hypothesis - including convexity of the problan and affinity of the The classical r.ethod to solve an optiI'lal con-
trol
probl~
state equation. '!hen we sho",- the identitv
consists in introducing a costa-
beb>JeeIl the set of saddle-points of the la-
te who allows the carputation of the crite-
<]rangian (who correspond to solutions of the
rion I s gradient. This can be interpretated
control problerr.) and the ar<)'lID'€Ilt of the ri-
as a reduced gradient rreth:xi
«('~y,
~
1979) i f
we consider the pair (state, control) as the
of the aUCTented lagrangian. 'Ihe
exaM-
ple of control of a systB". governed by the
unknown variable and the state equation as an
reat equation is treated, the observation bei!19
equality constraint. In this general fraMe,
distributed or final. Eventually nurerical re-
we can think to sore other ways to solve the
sults are presented.
control probleu : the state equation can be penalized (Yvon, 1970) or dealt with an augr.ented lagrangian approach (Di Pillo and ~1e
Grippo, 1979 a) .
present the problen in an abstract :"'rarre,
in order to inclooe state equations of elli!,Di Pillo and Grippo (1979 b) also introduced,
tic,parabolic or hyperlx>lic V7pe, with final
in a finite-dimensional frarre, a different
or distributed observation. All function spaces
kind of lagrangian, incltrling terms penali-
are Hilbert spaces. E' deootes the dual of the
zing the constraints and the dual equations.
space E. Let the control be u
The solution is then obtained by rn:inimizing
state Y
€
';'".
€
U, and the
'!he state equation is
the laSTangian with respect to the prinal and
Ay
dual variables. Di Pillo, Grippo and Larrpa-
=f
+ Bu in H,
(1)
riello (1980) applied that r.ethod to optinal the state equation space, where f
control problems.
ven, B
'Ihis paper extends
Scr.'e
€
-1
are continuous. Equation
an infinite dirrensional frar.e. First we set
237
is gi-
L (U,T'!) and l\ is an iscr:oD?hiSM ::rom
Y to \7, wich r-eans that A
of their results in
€ ~!
exists and A,A
(1) is well-posed
-1
J. F. Bonnans
238
every =ntrol u defin2s a unique state y (u)
the adjoint operators of A, Banc. ClG(u) the
=ntinoously deperrling on u. 'Ihe cptinal =n-
subdifferential of r, at r:oint u
trol problem is ClG(u) minimize J(y,u) = F(y) + G(u),
(2)
with (y,u) € Y x U such that (1) holds ,
{u*
€
U'
'IN
€
U}
,,(v)
~
Frcr.l the general studv of saddle-roints
(Ekeland arrl Te!"an, 1974) F and G are 1. s. c. =nvex
(2) is assurred to have a solution; that is, €
y x U such that
€
V(y,u)
€
€
Y
x
U is a solution of probler.. (2) €
W' such that \>Ie
have
(3)
Y x U
such that (1) holds ~€
(y,u)
(y, u,p) is a sac.dle-r:oint of L ; then
R
J(y,u) ~ J(y,u)
c.educe the
preposition 1 i f arrl only i f there exists p
(1) ho lds for (y,u)
J(y,u)
\~
functions frcrn,
respectively, Y to R and U to R. The problem there exists (y, u)
+ (T'P
r,(u)
ClG(u)
~
Ay - f -
also assume that
*- Bp
BU
(5)
in U',
(6)
= 0 in H,
A*P - F' (y) = 0 in Y'.
(7)
F is Gateaux-differentiable on Y arrl there exists a > 0 such that for every Yl' Y2
€
Y :
THE Al.n1ENTED
(4)
APProACH
Let us define the aU<]IT'ented lacrrangian S
F(Y2) ~ F(Yl) +Y'Y + + al lF' (Yl) - F' (Y2)
IAGRA~I1\N
yxu x W-+Rby:
I I ~,
2
S(y,U,P)=L(y,U,P)+c21IAY-f-Bullw +
where F' (y) denotes the G-derivative of F at
+ c 2 11A*P - F' (y) Ily. where cl and C
r:o int y.
2
gian and of:
€ R+.
b'~
S is a sum of the lagran-
terr.s ;,:enalizinc; t..':e state
equation (6) anc. the =state equation ( 7 ).
Remark 1
Here is tl-e r.ain result
(4) does not ir.ply that F is =ercive or
strictly =nvex : for instance (4) holds i f F is an affine function, arrl also if
Under assUI'ptions ( 3 ) arrl (4), for c
1 2 F(y) ="2 I ICy - zd l l x where X is a l:ilbert space, zc1
'n1eorern 1
there exists c* €
X
>
0 such that, if Cl
0,
>
2 >
c*,
the set of (v, u, p) Minilr.izincr S is identical
arc.
to the set of saddle-points of L.
C " L(Y, X) .•;e have then :
F(Y2) = F(Yl) +Y'Y +
i li
C (Y2 - Y l)
Proof
II ~
~'€ just give the main steps. Let (y,u,p)be
whicn, using the =ntinuity of C, inplies
(~).
a saddle r:oint of L arrl define
Notice that i f (4) holds for F 1 and F 2' (4) also holGs for P +F . \~ will see later that 1 2 (4) holds i f F is scree ~ization of the positive part of y.
t:, = S(y+6y , u+6u, 1)+6p)- SIy,ti,p). The problem is : show that t:,
saddle-r:oint of L. Define 'lhe lagrangian L : Y x U x W -+ R defineC: by L(y,u,;:»
= P(y)+G(u) -
~
0, t:, being
null only if (y+6y , u+6u, p+6p) is also
Bu>H'~";
is associated to problffG (2). ~ ;ote A*, B*
D( 6y) = F' (y + 6y) - p' (y) Fran (4) to (7)
\>le
deduce :
Appli c ation o f a New Class of Augmented Lagran gi a ns
f::,
2
'
~ 01IO(6y) Il y ' - < 6p, My - B6u>\V'Vi + )
+ clllMy -
B6ull~ + c 2 11A*6p -
239
A REIATIOO WITH EXAcr PENALIZATIOO
, (8)
O(6y)
II~,')
As a consequence of the theorem it is possible
to exhibit an exact penalty function for proFor any S > 0 we have :
blem (2). Define
S 12H' - <6p, My - B6u>W'v] ~ - 2"116PI
* -1 F'(y), p(y) = (A)
-
~SIIMY
-
E6ull~,
R(y,u)
S(y,u,p(y))
=
, fron the theorem we deduce
and the ooerciveness of (A*)-l implies that for
> 0
SCIre ID
Proposition 2 If Cl is great enough, the solutions of problem (2) can be obtained
b~
minimizing the
exact penalty function R(y,u) = J(y,u) -WI'] + Fron (8) we deduce that 2
f::, ~
(0 - rnj3) IIO(6y) Il y ' +
+(c l -
~S)
(9)
liMy - B6ull;: +
+(c 2 - mS ) IIA* 6p - O( 6y)
Same results of Fletcher (1970), in a finite
II~,
dirrensional frarre, are similar to proposition
2, we choose
APPLICATIOOS \'7e
All coefficients in (9) are positive. I t is then easy to show that i f /':, = 0 then (y+6y, u+6u) is a solution of the original problem
study a classical case when the state is
solution of a parabolic equation.
v~
first
oonsider the case of a distributed observation, then the one of a final observation.
(2) •
Let It be a bounded open set of RN with reguRemark 2
r.
lar boundary
'!he preceding proof shows that hypothesis (4)
-
has to be checked only for y 1 = y.
FEmark 3
S includes no term penalizing the optirrality condition:
Z= r
Denote Q
.EY at
/':,V
-
=
f in Q,
~ = u on Z, an y(O,x)
to add such a term. Numerically we found
J 0, 'If and
system :
=
) (l0)
)
0 in ll .
Oi Pillo, Grippo and Larnparielle (1980) already noticed it. Of course it is possible
= It x
JO,T[ . '!he state y is solution of the
x
"'le suppose f
E
L
2
(Q), u
E
L2 (Z); then (IC)
has a unique solution in
that it did not irrprove the results. Z(O,T) = {y
~ eps -
I
2 I L (O,T,H W ));
E
E
2 L (O,T, H1( m ') L
J. F. Bonnans
240
F (y) = ..!cl Iy _ y
Let K, the set of ad'nissible controls, be a 2 closed convex set of L (Ll am ~ the function 2 fron L (Ll onto R defined by
2
G(u) =
d
112
N 2'l l u ll 22
L2(0)
+ :s( (u).
L (2: )
~ 0 if u
(-to:>
E
K, F and G are clearly los.c. convex functions
i f rot.
and the existence of an optir.Bl control is a
2 Yd being given in L (Q), the control problan
classical result (Lions, 1968) ; "le can apply remark 1 to F. 'lherefore we can apply the
is :
theorem and the solution of (11) can be obminimize J(y, u) =
il IY-Yd l 12 2
tained by a minimization of
+
L (Q)
+
N 2 u ll 2 2'll
LW
(ll)
+ IK(U)
with respect to (y,u)EZ(O,T)
S(y,u,p) = J(y,u) -
Bu>~'l\I'
+
x K SUdl
t11at (10) holds. In (ll), N ?: 0 and to insure the existence
of a solution to problem (ll), \...e suppose N > 0 i f K is not bounded. Let us write the
state equation as in (1) and ckeck the hypothesis on the criterion to awly the theoran. \'€ define the spaces
~ =
{y
E
2
l
PE Z (0, T)
and p(T,x) = 0, so that
A*P - (y - Yd)
E
L2 (O,T, H1 (S"l) ')
2
2
1
W = L (0,T,H (st) ') and so H' = L (0,T,H W)) U
HI (rl)), under the constraint u E K. NoN, \...e
and \...e can penalize the costate equation
Z(O,T) ; y(O,x) = O} 1
2 with respect to (y,u,p) E Y x U x L (0,T,
= L2 m
with the norm of L (O,T,H 1 (S"l ) '); as 2 2 1 1 y e L (0,T,H (S"l )), the norm of L (O,T,H (S"l) ') is "stronger" than the norm of y ' . Eventually we have
Y is a Hilbert space. Define A fron Y onto ~:
and B frail U onto \'l by
T
proposition 3 c
a
W\'l' = J [ <....x z > + o at' HI (n) 'HI (n)
> 0 being set, if Cl is qreat enough, the 2 solutions of optir.Bl control probleo (11)
are obtained bv the mininization of
+ J V'yV'z.dx]dt S"lW\';' = J u z dI 2:
for any y E Y, u E U,
T dt + J HI (st) H1 (m I 0 T 2 dt + + Cl JII Ay-f-Bu I I o HI (m' T 2 + c dt 2 J I IA*p - (y-Yd) 11 1 0 H (m '
-
Z
EH'.
I t is \...ell known that the equation
Ay = f + BJ in H
with respect to (y ,u,p) E Z(O ,T) is fonnally interpreted as (10), and that A
Z(O,T), under the constraints
is an iscrrorphism fron Y onto \ I. Let us check the hytX)thesis on J. \'€ have :
y(O,x) = 0 in S"l , U E
K,
p(T,x) = 0 in
[I .
(12)
x
L2( S"l )
x
Application of a New Class of Augmented Lagrangians Re!:\ark 4
+ c~
We chcosed an h:r.ogeneous intitial condition
T
.,
0
H (Il)'
L
JIIA*pl!,,\
241
dt +
to make the proof rrore sinple. The result 2 still holds if y(O,x) = Yo(x) € L (n).
+ c211 p (T, .) - y ('1', ; ) + 1122
Consider
with respect to (y,u,p)
roN
the case wNan the observation is
the positive part of the final state. Let the
L
(Ill
€
Z(O,T) x L2( l: ) x
Z (0, T) anC with the constraints
problem be :
n,
~ y(O,x) = 0 in
+ 2 mi.ni.rnize J (y, u) = '2 J [y(T ,x) ] dx + 1
~ u
n +
Nil u 11 2 2 L
+ ~ (u), (y, u)
€
Yx U
K.
€
(13)
Ntl1ERICAL RESULTS
(1"1 )
being such that (10) holds
'1lle problem considered here is a particular
case of (11), in which 1"1 = ~e
] O,l ~
and the
boundary conditions on y are
checks the hypothesis on F. F is l.s.c.
convex and its G-derivative F' is defined ¥n(t,O) = u(t) ,
by
t
~(t
an ' 1)
Y'Y = J y(Tx,) +z(T,x)dx.
€
JO,'11:
= O.
1"1
To check (4) it is sufficient to prove that,
for sare
Cl
so that K can be identified to L2 (0,1). S is quadratic, and its minimization was perfor-
> 0,
rred with a conjugate gradient al<]Orithm. '.Ihe speed convergence depenjs on the choosed
discrete mm for Y : details abalt it can ~'1e
be fo1m:1 in Bonnans ( 1981).
J Yl (T,x) + (Y2 (T,x) - Yl (T,x))dx +
+
derote :
1"1
NI' number of t.ir.le steps, Nx nur..ber of spaoe steps,
NI nlll'i:ler of iterations needed to satis-
and this holds for
Cl
:> 1/2.
fy sa;e convergence test, CT cc.tIputing t.ir.le (on HB68 Mul tics fron
A study similar to the one of the distribu-
INFIA)
ted observation case irrluces the
'.Ihe convergenoe tests is Proposition 4 solutions of problem (13) are obtained by
I
the minimization of
= '21
the
value of S at step tl, i f
c 2 > 0 being set, if Cl is great enough, the
S(y,u,p)
Er denoting
L
< 10
-3 ,
then stop. l.1nklnm variables are first set
Nil u 112 2 J [y(T,x) +2 ] dx + '2 1"1
d'l-ll
f11 -f11
(l:)
to zero.
T
- J dt + o HI (1"1 ) HI (1"1 ) , T 2 + Cl J I IAy-f-&l I I 1 dt +
o
H (Il)'
a) ~ibili!y....£f_the co!l.~~~ wi~
respect to cl
an:;
c2.
We set N = Nx = 10 and here c
a function of cl'
2 = 80. NI is
242
J. F. Bonnans
1
NI
I'b convergence
5
20
30
60! 200
33
25
23
28
in sane cases.
45 '!he theory of Di Pillo and Grippo (1979 b)
~Je noN
set cl to 30, and give NI as a func-
tion of c
has been exterrled to sare distributed systan
control probler-s. An exarrple has been ntr.eri-
2
cally treated. '!he solution has been obtaired
5
10
30
with much rrore CCIlpUting t.il!e than i f the
200
classical rrethod was used. One can see there rP convergeoce
NI
29
23
27
a confirMation of the efficacity of the reduced gradient rrethod. !1ay be a better choice
As expected there is no convergence if cl or
is too small. '!he convergence 3peed is 2 less sensitive to c than to cl. 2
c
of the norm could inprove the conditioning of the
a~ted
lagrangian, and therefore in-
crease the convergen:::e speed. One can also think to a nurrerical use of the exact penalty
b) ~!eg~EL2L~_~n~~_~~~
function exhibited in this paper.
~~E_ to _m:_~_Nx.
= 80. For Nx = 10 2 give NI and er as functions of NT.
\;e set cl = 30 and c
NT
10
30
50
NI
23
54
77
er !11.5 s
72s
\\lE!
165 s
Bonnans J.F.
(1981). ArPlication d't.Jn3 nou-
velle classe de lagrangiens augmentes en NoN NT is set to 10
NI and
er are functions
of Nx.
NI
contrOle optimal (le systerres distribues. ~rt
10
30
50
23
28
29
37 s
63 s
er !11.5 s
nffiIA · n° 102.
Di Pillo G., Grippo L. (l979a). '!he multiplier r.ethod for optimal oontrol problems of parabolic systems. Appl. r1ath.
~tirl.
5, 253-269. The cx::nputing t.il!e is approximately a linear function with respect to the number of space steps : this is satisfactory. On the oontrary it grows dangerously when the number of tirre steps increases. In Bonnans (1981) a a::rrpari-
Di Pillo G., Grippo L. (l979b). A new class of augrrented lagrangians in nonlirear prograIT.!ing. SIN' J. of Control and
~tim.
Vol. 17, nO 5, 618-628 .
son is made with the classical rrethod which consists in a::rrputing the state, the costate and the gradient of the criterion with res-
pect to the oontrol, to apply a gradient rrethod. It is
s~
conj~ate
that the classi-
cal method converges much quicker . Experir.ents have been also made when the state equation is not affine : the rrethod proposed here .....orKs
Di Pillo G., Grippo L., I...anpu"iello F. (1980). A carputing technique for solving discrete time optim3l. oontrol probler:s. 2nd IFAC
\orkshop on oontrol applications of nonlirear prograITl!ling and optimisation. Hunich.
Application of a New Class of Augmented Lagrangians
Ekeland 1., Teman R. (1974). Analyse oonvexe et problerres variationnels. Dunod, Paris. Fletcher R. (1970). A class of rrethods for non linear programning with teIlTlination and convergence prcperties ; in Integer and non lirear prograrmring. ed. J. Abadie, North Holland.
Gabay D. ( 1979). Methodes nl.llTEriques pour l' optimisation non lineaire. 'Ihese lh1iversi te Paris VI. Lions J.L. (1968). ContrOle optimal de systares gouvernes par des equations aux derivees partielles. Dunod, Paris. Yvon J.P. (1970). Application de la penalisation
a
la resolution d' un problare de oon-
trOle optimal. Cahier de l' IRIA n O 2.
243
APPLICATION OF A NEW CLASS OF AUGMENTED LAGRANGIANS TO THE CONTROL OF DISTRIBUTED PARAMETER SYSTEMS
J.
F. Bonnans
Institut National de Recherche en Informalique et en Automatique (INRIA) , Domain e de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France
Abstract. This paper describe:; a new r:ethod to solve O?tir.lal control problers of distributed pararreter systaTIs. The study is restricted to problems with a convex criterion and a state equation affine with respect to the pair (state, control).
\']e
show the equivalence betvleen the initial problem and the rinini-
sation with respect to the control, the state and also the costate of sc.r-e augmented lagrangian, obtained by addition of the lagrangian and of b...a tern; penalizinq the state and costate equations. An exarple is treated, and nllJ"'"erical results are given. the problen and the hypothesis - including convexity of the problan and affinity of the The classical r.ethod to solve an optiI'lal con-
trol
probl~
state equation. '!hen we sho",- the identitv
consists in introducing a costa-
beb>JeeIl the set of saddle-points of the la-
te who allows the carputation of the crite-
<]rangian (who correspond to solutions of the
rion I s gradient. This can be interpretated
control problerr.) and the ar<)'lID'€Ilt of the ri-
as a reduced gradient rreth:xi
«('~y,
~
1979) i f
we consider the pair (state, control) as the
of the aUCTented lagrangian. 'Ihe
exaM-
ple of control of a systB". governed by the
unknown variable and the state equation as an
reat equation is treated, the observation bei!19
equality constraint. In this general fraMe,
distributed or final. Eventually nurerical re-
we can think to sore other ways to solve the
sults are presented.
control probleu : the state equation can be penalized (Yvon, 1970) or dealt with an augr.ented lagrangian approach (Di Pillo and ~1e
Grippo, 1979 a) .
present the problen in an abstract :"'rarre,
in order to inclooe state equations of elli!,Di Pillo and Grippo (1979 b) also introduced,
tic,parabolic or hyperlx>lic V7pe, with final
in a finite-dimensional frarre, a different
or distributed observation. All function spaces
kind of lagrangian, incltrling terms penali-
are Hilbert spaces. E' deootes the dual of the
zing the constraints and the dual equations.
space E. Let the control be u
The solution is then obtained by rn:inimizing
state Y
€
';'".
€
U, and the
'!he state equation is
the laSTangian with respect to the prinal and
Ay
dual variables. Di Pillo, Grippo and Larrpa-
=f
+ Bu in H,
(1)
riello (1980) applied that r.ethod to optinal the state equation space, where f
control problems.
ven, B
'Ihis paper extends
Scr.'e
€
-1
are continuous. Equation
an infinite dirrensional frar.e. First we set
237
is gi-
L (U,T'!) and l\ is an iscr:oD?hiSM ::rom
Y to \7, wich r-eans that A
of their results in
€ ~!
exists and A,A
(1) is well-posed
-1
J. F. Bonnans
238
every =ntrol u defin2s a unique state y (u)
the adjoint operators of A, Banc. ClG(u) the
=ntinoously deperrling on u. 'Ihe cptinal =n-
subdifferential of r, at r:oint u
trol problem is ClG(u) minimize J(y,u) = F(y) + G(u),
(2)
with (y,u) € Y x U such that (1) holds ,
{u*
€
U'
'IN
€
U}
,,(v)
~
Frcr.l the general studv of saddle-roints
(Ekeland arrl Te!"an, 1974) F and G are 1. s. c. =nvex
(2) is assurred to have a solution; that is, €
y x U such that
€
V(y,u)
€
€
Y
x
U is a solution of probler.. (2) €
W' such that \>Ie
have
(3)
Y x U
such that (1) holds ~€
(y,u)
(y, u,p) is a sac.dle-r:oint of L ; then
R
J(y,u) ~ J(y,u)
c.educe the
preposition 1 i f arrl only i f there exists p
(1) ho lds for (y,u)
J(y,u)
\~
functions frcrn,
respectively, Y to R and U to R. The problem there exists (y, u)
+
r,(u)
ClG(u)
~
Ay - f -
also assume that
*- Bp
BU
(5)
in U',
(6)
= 0 in H,
A*P - F' (y) = 0 in Y'.
(7)
F is Gateaux-differentiable on Y arrl there exists a > 0 such that for every Yl' Y2
€
Y :
THE Al.n1ENTED
(4)
APProACH
Let us define the aU<]IT'ented lacrrangian S
F(Y2) ~ F(Yl) +
IAGRA~I1\N
yxu x W-+Rby:
I I ~,
2
S(y,U,P)=L(y,U,P)+c21IAY-f-Bullw +
where F' (y) denotes the G-derivative of F at
+ c 2 11A*P - F' (y) Ily. where cl and C
r:o int y.
2
gian and of:
€ R+.
b'~
S is a sum of the lagran-
terr.s ;,:enalizinc; t..':e state
equation (6) anc. the =state equation ( 7 ).
Remark 1
Here is tl-e r.ain result
(4) does not ir.ply that F is =ercive or
strictly =nvex : for instance (4) holds i f F is an affine function, arrl also if
Under assUI'ptions ( 3 ) arrl (4), for c
1 2 F(y) ="2 I ICy - zd l l x where X is a l:ilbert space, zc1
'n1eorern 1
there exists c* €
X
>
0 such that, if Cl
0,
>
2 >
c*,
the set of (v, u, p) Minilr.izincr S is identical
arc.
to the set of saddle-points of L.
C " L(Y, X) .•;e have then :
F(Y2) = F(Yl) +
i li
C (Y2 - Y l)
Proof
II ~
~'€ just give the main steps. Let (y,u,p)be
whicn, using the =ntinuity of C, inplies
(~).
a saddle r:oint of L arrl define
Notice that i f (4) holds for F 1 and F 2' (4) also holGs for P +F . \~ will see later that 1 2 (4) holds i f F is scree ~ization of the positive part of y.
t:, = S(y+6y , u+6u, 1)+6p)- SIy,ti,p). The problem is : show that t:,
saddle-r:oint of L. Define 'lhe lagrangian L : Y x U x W -+ R defineC: by L(y,u,;:»
= P(y)+G(u) -
~
0, t:, being
null only if (y+6y , u+6u, p+6p) is also
Bu>H'~";
is associated to problffG (2). ~ ;ote A*, B*
D( 6y) = F' (y + 6y) - p' (y) Fran (4) to (7)
\>le
deduce :
Appli c ation o f a New Class of Augmented Lagran gi a ns
f::,
2
'
~ 01IO(6y) Il y ' - < 6p, My - B6u>\V'Vi + )
+ clllMy -
B6ull~ + c 2 11A*6p -
239
A REIATIOO WITH EXAcr PENALIZATIOO
, (8)
O(6y)
II~,')
As a consequence of the theorem it is possible
to exhibit an exact penalty function for proFor any S > 0 we have :
blem (2). Define
S 12H' - <6p, My - B6u>W'v] ~ - 2"116PI
* -1 F'(y), p(y) = (A)
-
~SIIMY
-
E6ull~,
R(y,u)
S(y,u,p(y))
=
, fron the theorem we deduce
and the ooerciveness of (A*)-l implies that for
> 0
SCIre ID
Proposition 2 If Cl is great enough, the solutions of problem (2) can be obtained
b~
minimizing the
exact penalty function R(y,u) = J(y,u) -
f::, ~
(0 - rnj3) IIO(6y) Il y ' +
+(c l -
~S)
(9)
liMy - B6ull;: +
+(c 2 - mS ) IIA* 6p - O( 6y)
Same results of Fletcher (1970), in a finite
II~,
dirrensional frarre, are similar to proposition
2, we choose
APPLICATIOOS \'7e
All coefficients in (9) are positive. I t is then easy to show that i f /':, = 0 then (y+6y, u+6u) is a solution of the original problem
study a classical case when the state is
solution of a parabolic equation.
v~
first
oonsider the case of a distributed observation, then the one of a final observation.
(2) •
Let It be a bounded open set of RN with reguRemark 2
r.
lar boundary
'!he preceding proof shows that hypothesis (4)
-
has to be checked only for y 1 = y.
FEmark 3
S includes no term penalizing the optirrality condition:
Z= r
Denote Q
.EY at
/':,V
-
=
f in Q,
~ = u on Z, an y(O,x)
to add such a term. Numerically we found
J 0, 'If and
system :
=
) (l0)
)
0 in ll .
Oi Pillo, Grippo and Larnparielle (1980) already noticed it. Of course it is possible
= It x
JO,T[ . '!he state y is solution of the
x
"'le suppose f
E
L
2
(Q), u
E
L2 (Z); then (IC)
has a unique solution in
that it did not irrprove the results. Z(O,T) = {y
~ eps -
I
2 I L (O,T,H W ));
E
E
2 L (O,T, H1( m ') L
J. F. Bonnans
240
F (y) = ..!cl Iy _ y
Let K, the set of ad'nissible controls, be a 2 closed convex set of L (Ll am ~ the function 2 fron L (Ll onto R defined by
2
G(u) =
d
112
N 2'l l u ll 22
L2(0)
+ :s( (u).
L (2: )
~ 0 if u
(-to:>
E
K, F and G are clearly los.c. convex functions
i f rot.
and the existence of an optir.Bl control is a
2 Yd being given in L (Q), the control problan
classical result (Lions, 1968) ; "le can apply remark 1 to F. 'lherefore we can apply the
is :
theorem and the solution of (11) can be obminimize J(y, u) =
il IY-Yd l 12 2
tained by a minimization of
+
L (Q)
+
N 2 u ll 2 2'll
LW
(ll)
+ IK(U)
with respect to (y,u)EZ(O,T)
S(y,u,p) = J(y,u) -
Bu>~'l\I'
+
x K SUdl
t11at (10) holds. In (ll), N ?: 0 and to insure the existence
of a solution to problem (ll), \...e suppose N > 0 i f K is not bounded. Let us write the
state equation as in (1) and ckeck the hypothesis on the criterion to awly the theoran. \'€ define the spaces
~ =
{y
E
2
l
PE Z (0, T)
and p(T,x) = 0, so that
A*P - (y - Yd)
E
L2 (O,T, H1 (S"l) ')
2
2
1
W = L (0,T,H (st) ') and so H' = L (0,T,H W)) U
HI (rl)), under the constraint u E K. NoN, \...e
and \...e can penalize the costate equation
Z(O,T) ; y(O,x) = O} 1
2 with respect to (y,u,p) E Y x U x L (0,T,
= L2 m
with the norm of L (O,T,H 1 (S"l ) '); as 2 2 1 1 y e L (0,T,H (S"l )), the norm of L (O,T,H (S"l) ') is "stronger" than the norm of y ' . Eventually we have
Y is a Hilbert space. Define A fron Y onto ~:
and B frail U onto \'l by
T
proposition 3 c
a
> 0 being set, if Cl is qreat enough, the 2 solutions of optir.Bl control probleo (11)
are obtained bv the mininization of
+ J V'yV'z.dx]dt S"l
for any y E Y, u E U,
T dt + J
-
Z
EH'.
I t is \...ell known that the equation
Ay = f + BJ in H
with respect to (y ,u,p) E Z(O ,T) is fonnally interpreted as (10), and that A
Z(O,T), under the constraints
is an iscrrorphism fron Y onto \ I. Let us check the hytX)thesis on J. \'€ have :
y(O,x) = 0 in S"l , U E
K,
p(T,x) = 0 in
[I .
(12)
x
L2( S"l )
x
Application of a New Class of Augmented Lagrangians Re!:\ark 4
+ c~
We chcosed an h:r.ogeneous intitial condition
T
.,
0
H (Il)'
L
JIIA*pl!,,\
241
dt +
to make the proof rrore sinple. The result 2 still holds if y(O,x) = Yo(x) € L (n).
+ c211 p (T, .) - y ('1', ; ) + 1122
Consider
with respect to (y,u,p)
roN
the case wNan the observation is
the positive part of the final state. Let the
L
(Ill
€
Z(O,T) x L2( l: ) x
Z (0, T) anC with the constraints
problem be :
n,
~ y(O,x) = 0 in
+ 2 mi.ni.rnize J (y, u) = '2 J [y(T ,x) ] dx + 1
~ u
n +
Nil u 11 2 2 L
+ ~ (u), (y, u)
€
Yx U
K.
€
(13)
Ntl1ERICAL RESULTS
(1"1 )
being such that (10) holds
'1lle problem considered here is a particular
case of (11), in which 1"1 = ~e
] O,l ~
and the
boundary conditions on y are
checks the hypothesis on F. F is l.s.c.
convex and its G-derivative F' is defined ¥n(t,O) = u(t) ,
by
t
~(t
an ' 1)
€
JO,'11:
= O.
1"1
To check (4) it is sufficient to prove that,
for sare
Cl
so that K can be identified to L2 (0,1). S is quadratic, and its minimization was perfor-
> 0,
rred with a conjugate gradient al<]Orithm. '.Ihe speed convergence depenjs on the choosed
discrete mm for Y : details abalt it can ~'1e
be fo1m:1 in Bonnans ( 1981).
J Yl (T,x) + (Y2 (T,x) - Yl (T,x))dx +
+
derote :
1"1
NI' number of t.ir.le steps, Nx nur..ber of spaoe steps,
NI nlll'i:ler of iterations needed to satis-
and this holds for
Cl
:> 1/2.
fy sa;e convergence test, CT cc.tIputing t.ir.le (on HB68 Mul tics fron
A study similar to the one of the distribu-
INFIA)
ted observation case irrluces the
'.Ihe convergenoe tests is Proposition 4 solutions of problem (13) are obtained by
I
the minimization of
= '21
the
value of S at step tl, i f
c 2 > 0 being set, if Cl is great enough, the
S(y,u,p)
Er denoting
L
< 10
-3 ,
then stop. l.1nklnm variables are first set
Nil u 112 2 J [y(T,x) +2 ] dx + '2 1"1
d'l-ll
f11 -f11
(l:)
to zero.
T
- J
o
H (Il)'
a) ~ibili!y....£f_the co!l.~~~ wi~
respect to cl
an:;
c2.
We set N = Nx = 10 and here c
a function of cl'
2 = 80. NI is
242
J. F. Bonnans
1
NI
I'b convergence
5
20
30
60! 200
33
25
23
28
in sane cases.
45 '!he theory of Di Pillo and Grippo (1979 b)
~Je noN
set cl to 30, and give NI as a func-
tion of c
has been exterrled to sare distributed systan
control probler-s. An exarrple has been ntr.eri-
2
cally treated. '!he solution has been obtaired
5
10
30
with much rrore CCIlpUting t.il!e than i f the
200
classical rrethod was used. One can see there rP convergeoce
NI
29
23
27
a confirMation of the efficacity of the reduced gradient rrethod. !1ay be a better choice
As expected there is no convergence if cl or
is too small. '!he convergence 3peed is 2 less sensitive to c than to cl. 2
c
of the norm could inprove the conditioning of the
a~ted
lagrangian, and therefore in-
crease the convergen:::e speed. One can also think to a nurrerical use of the exact penalty
b) ~!eg~EL2L~_~n~~_~~~
function exhibited in this paper.
~~E_ to _m:_~_Nx.
= 80. For Nx = 10 2 give NI and er as functions of NT.
\;e set cl = 30 and c
NT
10
30
50
NI
23
54
77
er !11.5 s
72s
\\lE!
165 s
Bonnans J.F.
(1981). ArPlication d't.Jn3 nou-
velle classe de lagrangiens augmentes en NoN NT is set to 10
NI and
er are functions
of Nx.
NI
contrOle optimal (le systerres distribues. ~rt
10
30
50
23
28
29
37 s
63 s
er !11.5 s
nffiIA · n° 102.
Di Pillo G., Grippo L. (l979a). '!he multiplier r.ethod for optimal oontrol problems of parabolic systems. Appl. r1ath.
~tirl.
5, 253-269. The cx::nputing t.il!e is approximately a linear function with respect to the number of space steps : this is satisfactory. On the oontrary it grows dangerously when the number of tirre steps increases. In Bonnans (1981) a a::rrpari-
Di Pillo G., Grippo L. (l979b). A new class of augrrented lagrangians in nonlirear prograIT.!ing. SIN' J. of Control and
~tim.
Vol. 17, nO 5, 618-628 .
son is made with the classical rrethod which consists in a::rrputing the state, the costate and the gradient of the criterion with res-
pect to the oontrol, to apply a gradient rrethod. It is
s~
conj~ate
that the classi-
cal method converges much quicker . Experir.ents have been also made when the state equation is not affine : the rrethod proposed here .....orKs
Di Pillo G., Grippo L., I...anpu"iello F. (1980). A carputing technique for solving discrete time optim3l. oontrol probler:s. 2nd IFAC
\orkshop on oontrol applications of nonlirear prograITl!ling and optimisation. Hunich.
Application of a New Class of Augmented Lagrangians
Ekeland 1., Teman R. (1974). Analyse oonvexe et problerres variationnels. Dunod, Paris. Fletcher R. (1970). A class of rrethods for non linear programning with teIlTlination and convergence prcperties ; in Integer and non lirear prograrmring. ed. J. Abadie, North Holland.
Gabay D. ( 1979). Methodes nl.llTEriques pour l' optimisation non lineaire. 'Ihese lh1iversi te Paris VI. Lions J.L. (1968). ContrOle optimal de systares gouvernes par des equations aux derivees partielles. Dunod, Paris. Yvon J.P. (1970). Application de la penalisation
a
la resolution d' un problare de oon-
trOle optimal. Cahier de l' IRIA n O 2.
243