Time-Optimal Control for a Class of ODEs
Kazufumi Ito 1
Karl Kunish 2
February 12, 2010
1
Departmentof Mathematis,North CarolinaStateUniversity,Raleigh,North
Carolina, 27695-8205, USA; researh partially supported by the Army Researh
OÆe under DAAD19-02-1-0394 and the Air Fore OÆe of Sienti Researh
under FA9550-09-1-0226
2
Institutfur Mathematik,Karl-Franzens-Universitat Graz, A-8010 Graz,
Aus-tria, supported in part by the Fonds zur Forderung der wissenshaftlihen
Forshung under SFB 32, "Mathematial Optimization and Appliations in
Time optimalontrol problems for a lass of linear multi-inputsystems are
onsidered. Theproblems are regularizedand theasymptotiand monotone
behavior of the regularisation proedureis investigated. For the regularised
problems the appliabilityofsemi-smoothNewton methodsisveried. First
numerialtestsarepresented whihshowthattheproposedapproah,
dier-entlyfromothermethods,doesnotrelya-prioryinformationoftheswithing
This paper addresses time optimal ontrol for a lass of linear multi-input
ontrols systems for ordinary dierential equations. Due to their pratial
relevane and inherent strutural diÆulties, time optimal ontrol has been
reeiving a onsiderable amount of attention for deades. Muh of the
lit-erature up to the late sixties is overed in [HL℄. Many reent results an
be found or are referened in [BPW , KLM, MO℄. Time optimal ontrol for
innite dimensionalsystems isonsidered in [Fa℄, for example.
The optimalitysystem assoiatedto time optimalontrolproblems with
pointwiseonstraintsontheontrolsisompliatedduetolakofsmoothness
of the optimal ontrols. In fat, the rst order optimality system for time
optimalontrolproblemsontainsamultivaluedoperationwhihimpedesthe
use of fastnumerialmethods. Forthis reasonwe introduea regularization
to the time optimal problem. In setion 2 the behavior of the solutions of
the regularized problems as the regularization parameter " tends to zero is
investigated. In partiularmonotoni struture ofthe solutionswith respet
to " is shown. An optimality system for the regularized problems is derived
under a ondition whih is stronger than ontrollability and weaker than
normality. TheoptimalontrolsoftheregularizedproblemsareW 1;1
regular
and onverge to a minimum norm solution of the original problem as the
regularization parametertends to zero.
The optimalitysystem of the regularized problems is stillnot C 1
sothat
seondordermethodswithloalquadrationvergeneorderarenotdiretly
appliable. However, suÆientonditionswillbeobtainedinsetion3whih
imply that semi-smooth Newton methods [IK2℄ are wellposed and loally
superlinearly onvergent.
Setion 4 ontains a brief desription of numerial results. We ompare
the hosen regularizationto an alternative one, whih has stronger
regular-ization properties. Sine the optimal ontrols of the original time optimal
problems are typially not ontinuous, it appears that our hoie of
regu-larization whih leads to W 1;1
regularized ontrols is preferable over other
regularizationstrategieswhihprovidesmootherontrols. More detailed
nu-merial tests are availablein [XK℄.
Let us note that the approah that we propose for solving time optimal
problemsdeviatesfromtraditionalapproahes, whiharefrequentlygrouped
intodiret andindiretmethods. Indiretmethodsbasedonmultiple
unknowns,inludingthe swithingfuntion,theshootingmethodisreported
to onverge fast and to generate very aurate solutions. The methods that
weproposealsooriginatesfromtherst orderondition,butdierentlyfrom
the shooting methoditdoes not requireaurate informationonthe
swith-ingstruture inadvane.
Diretmethods on the other hand, onsider time optimal problems as a
genuinenonlinear programmingproblems. Theyare usedinseveral variants,
whihfrequently involvereparametrization ofthe ontrols asthe unknowns.
Thenewunknownsanbetheswithingtimesasin[MB℄ortheardurations
as in[KN℄.
2 The time-optimal problem and its
regularization
Consider the time-optimalontrolproblemfor the linearmulti-inputsystem
(P) 8
>
>
>
<
>
>
>
: min
0 R
0 dt
subjet to
d
dt
x(t)=Ax(t)+Bu(t); ju(t)j
` 1
1; x(0)=x
0
;x()=x
1 ;
where A 2 R nn
; B 2 R nm
; x
0 2 R
n
; x
1 2 R
n
are given, u(t) 2 R m
, u is
measurable, and jj
`
1 denotes the innity-norm on R m
. The olumns of B
are denoted by b
i
. It is assumed that x
1
an be reahed in nite time by an
admissible ontrol. Then (P) admits a solution with optimal time denoted
by
, and assoiated state x
and ontrolu
.
The rst order optimality system for (P) an be expressed in terms of
the adjointp and the Hamiltonian
H(x;u;p
0
;p)=p
0 +p
T
(2.1)
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: _
x=Ax+Bu; x(0) =x
0
; x()=x
1 ;
_ p=A
T
p;
u=argmin
jvj
` 11
H(x;v;p
0
;p); a.e. in(0;);
p
0
+p(t) T
(Ax(t)+Bu(t))=0; p
0 0;
where the supersript T denotes transposition,see e.g. [MS ℄, hapter V, pg.
109, 110. Further p is not identially 0, so that there exists a nontrivial
vetor q2R n
suh that
p(t)=exp(A T
( t))q:
Due to the speial struture of H the optimal ontrol an be expressed
as
(2.2) u
i
= (b
T
i
p)= b T
i
exp( A T
( t))q
;
where denotes the oordinate-wise operation
(s) 2 8
>
<
>
:
1 if s <0
[ 1;1℄ if s =0
1 if s >0: (2.3)
The last equation in (2.1) holds everywhere rather than a.e. on [0;℄. In
fat, p and xare ontinuous and p(t) T
Bu(t)= P
m
i=1 jp(t)
T
b
i j.
Let us reall the notions of ontrollability and normality, whih will be
referred to below.
(2.4)
(
The pair (A;B)is alled ontrollable if
rank fB;AB;:::;A n 1
Bg=n
(2.5)
(
The pair (A;B)is allednormal if (A;b
i )
isontrollable for allolumnsb
i of B:
Normality of (A;B) implies ontrollability. Moreover, if (A;B) is normal,
then the optimalontrolu
to (P)is unique, it is bang bang, and pieewise
(2.6) p
0 >0
is referred to as strit transversality. In this ase it an be assumed that
p
0
=1,whihanbeahievedbysalingq. Ifstrittransversalityholdsthen
(x
;u
;
) is a strit loal minimum, in the sense that there exists Æ > 0
suh that x
1
is not inthe attainable set for t2(
Æ;
), [HL℄, pg.89.
With(2.6) holding,we an express the optimality ondition as
(2.7)
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: _
x=Ax+Bu; x(0) =x
0
; x() =x
1 ;
_ p=A
T
p;
u=argmin
jvj
` 11
H(x;v;p); a.e. in(0;);
1+p() T
(Ax()+Bu())=0:
Here we eliminatethe variable p
0
fromthe notation forH sine it wasxed
to be 1.
Introduing the transformation ^
t= t
and setting
^ x(
^
t)=x( ^
t )=x(t); p(^ ^
t )=p( ^
t)=p(t); u(^ ^
t ) =u( ^
t)=u(t);
we obtain the following equivalent system to (2.7), where for the ease of
presentation we omit the supersripts ^
:
(2.8)
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: _
x=(Ax+Bu); x(0)=x
0
; x(1) =x
1 ;
_ p=A
T
p;
u=argmin
jvj
` 11
H(x;v;p); a.e. in(0;);
1+p(1) T
(Ax(1)+Bu(1))=0:
The non-dierentiable operation involved in haraterizing the optimal
ontrol,
u= (B T
p);
ompare (2.2), prohibits the use of Newton-type methods for solving (2.8)
(P " ) 8 > > > < > > > : min 0 R 0 (1+ " 2 ju(t)j 2 )dt subjet to d dt
x(t)=Ax(t)+Bu(t); ju(t)j
` 1
1; x(0)=x
0
; x()=x
1 ;
with " >0 is onsidered. The norm jj used in the ost-funtional denotes
theEulidean norm. Itisstraightforwardtoarguetheexisteneofasolution
(u " ;x " ; " ).
Convergeneofthesolutions(x
" ; p " ;u " ; " )of(P
"
)toasolution(x ;p ; u ; )
of (P) is onsiderednext. Notethat
isunique.
Proposition 2.1. For every 0 < "
0 < "
1
and any solution (
; u
) of (P)
we have (2.9) " 0 " 1 (1+ " 1 2 ); (2.10) ju " 1 j L 2 (0; " 1 ) ju " 0 j L 2 (0; " 0 ) ju j L 2 (0; ) : If u
is a bang-bang solution, then
(2.11) 0ju j 2 L 2 (0; ) ju " j 2 L 2 (0;")
meas ft2[0;
℄:ju
"
(t)j<1g
for every ">0.
Proof. From the denition of and " wehave "
for every ">0;
and " + " 2 Z " 0 ju " j 2 dt + " 2 Z 0 ju j 2 dt; hene ju " j L 2 (0;") ju j L 2 (0; ) ; and " (1+ " 2 ):
For 0<"
" 0 " 0 " 0 1 2 (" 1 " 0 ) R " 1 0 ju "1 j 2
dt on both sides implies that
(2.12) "0 + " 1 2 Z " 1 0 ju "1 j 2 dt+ " 0 2 Z " 0 0 ju "0 j 2 dt Z " 1 0 ju "1 j 2 dt Z " 1 0 (1+ " 1 2 ju " 1 j 2 )dt " 0 + " 1 2 Z " 0 0 ju " 0 j 2 dt:
Estimatingthe rst by the last expression in (2.12)implies that
" 1 Z " 1 0 ju " 1 j 2 dt Z " 0 0 ju " 0 j 2 dt " 0 Z " 1 0 ju " 1 j 2 dt Z " 0 0 ju " 0 j 2 dt ; and hene (2.13) ju "1 j L 2 (0; " 1 ) ju "0 j L 2 (0; " 0 ) :
Estimatingthe rst by the seond expression in(2.12) we obtain
" 0 + " 0 2 ju " 0 j 2 L 2 (0; " 0 ) " 1 + " 0 2 ju " 1 j 2 L 2 (0; " 1 )
and by (2.13)
" 0 " 1 :
These estimates imply (2.9) and (2.10).
Ifu
is bang-bang,then
0ju j 2 L 2 (0; ) ju j 2 L 2 (0;") R ft2(0; ):ju"(t)j<1g (1 ju " (t)j 2
)dt measft 2(0;
):ju
"
(t)j<1g;
so that (2.11) holds.
Theorem 2.1. For " ! 0 +
we have
"
!
and every onvergent
subse-queneofsolutionsf(u
" ; x " )g ">0 to(P "
)onvergesinL 2 (0; ;R m )W 1;2 (0; ; R n )
to a solution (u
; x
) of (P), where u
Here onvergene of u
"
to u isdened as
Z 1 0 ju " ( " t) u ( t)j 2
dt !0
and analogously forfx
"
g, and for weak onvergene.
Proof. The rst laim follows from Proposition 2.1. Sine fu
" ( " )g ">0 and fx " ( " )g ">0
areboundedinL 2
(0; 1;R m
)andW 1;2
(0; 1;R n
),thereexistweak
aumulation points u 2 L 2 (0; ;R m
); andx 2 W 1;2 (0; ; R n ).
Subse-quently we avoid subsequential indies. Passing to the limit in x_
" ( " ) = " (Ax " ( "
)+Bu
" (
"
)) and x
"
(0) = x
0 ; x
" (
" ) = x
1
it follows that x
is
admissible. Due toweaklosedness of fu2L 2
(0; 1;R m
): ju(x)j
` 1
1a:e:g
we have that u
isadmissibleas well. Sine
lim "!0 + " + " 2 Z " 0 ju " j 2 dt= ;
the triple (
; u
; x
)is optimalfor(P). By Proposition 2.1andweaklower
semi-ontinuity of norms
(2.14) lim "!0 sup ju " j L 2 (0; " ) ju j L 2 (0; ) lim "!0 inf ju " j L 2 (0; " )
and hene lim
"!0 ju " j L 2 (0;";R m ) = ju j L 2 (0; ;R m )
. As a onsequene u
" and
x
"
onverge strongly in L 2
(0;
"
),respetively W 1;2
(0;
" ;R
n
), tou
and x
.
Letu^denoteanother optimalontrolfor(P)withj^uj<ju
j. Thenby (2.10)
and (2.14) lim "!0 sup ju " j L 2 (0;";R m ) j^uj L 2 (0; ;R m ) <ju j L 2 (0; ;R m ) lim "!0 inf ju " j L 2 (0;";R m ) ;
whihis aontradition. Consequently (P)has a minimalnorm ontroland
the laimedstrong onvergene properties hold.
Corollary 2.1. If (2.5) holds, then the solution u
to (P) is unique, it is
bang-bang, and u
" !u
in L 2
as "!0 +
.
Proof. (2.5) implies that the solution to (P) is unique and it is bang-bang.
"
" (s) 2
8
>
<
>
:
1 if s "
s
"
if jsj< "
1 if s ": (2.15)
If
"
is appliedto avetor, then it ats oordinate-wise.
We shall use aontrollabilityassumption whih isstronger than
ontrol-lability and weaker than normality.
(H1) Thereexists i
suh that (A;b
i )
is ontrollable:
Theorem 2.2. Assume that (H1) holds and let (x
" ;u
" ;
"
) be a solution of
(P
"
). If there exist >0 and an interval I
i
(0;1)suh that
(2.16) j(^u
" )
i
(t)j
` 1
1 for a.e. t 2I
i ;
then there exists an adjoint state p
"
suh that
(2.17)
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: _ x
" =Ax
" +Bu
" ; x
"
(0)=x
0 ; x
" (
" )=x
1
_ p
" =A
T
p
"
u
"
=
" (B
T
p
" )
1+
2 ju
" (
" )j
2
R m
+p
" (
" )
T
(Ax
" (
"
)+Bu
" (
"
))=0:
Proof. We use a Lagrange multiplier argument for the reparameterized
for-mulationof (P
"
) whih isgiven by
(2.18)
8
>
>
>
<
>
>
>
: min
0 R
1
0 ( +
"
2 j^u(t)j
2
)dt
subjet to
d
dt ^
x (t)=(A^x (t)+Bu(t));^ u^2C; x (0)^ =x
0
; x(1)^ =x
1 ;
where C = f^u 2 L 2
(0;1;R m
) : juj^
` 1
1g. Here (^u;) are treated as
0
onstraint e(^u;)=x (1)^ x
1
=0. The resulting Lagrangianis
L(^u;;
0 )=
Z
1
0 ( +
"
2 j^u(t)j
2
)dt+ T
0
(^x (1) x
1 );
where x (1)^ isdened through the dierentialequation and the initial
ondi-tion.
We now argue that e : C R L 2
(0;1;R m
)R ! R n
satises the
regular point ondition in the sense of Maurer-Zowe [MZ, IK2℄. Thus we
have toverify that
(2.19) 02intfe
0
(^u
" ;
"
) (C u^
" )R
g;
where e 0
(^u
" ;
"
) denotes the linearisation of e at(^u
" (
" );
" ).
Consideringe 0
(^u
" ;
"
) indiretions Æ =0and Æu satisfying(Æu)
i
=0for
i6=i
and (Æu)
i
=0 in(0;1)n( ;+Æ),with ( ;+Æ):=I
i
, we nd
e 0
(^u
" ;
"
)(Æu u^
" ;0)=
R
+Æ
e
A(1 t)
b
i
(Æu(t) u^
" (t))
i dt
= R
Æ
0 e
A(Æ t)
~
b
i
(Æu(t+ ) u^
"
(t+ ))
i dt;
where ~
b
i
=e
A(1 Æ )
b
i .
Then
e 0
(^u
" ;
"
)(Æu u^
" ;0)=
Z
Æ
0 e
A(Æ t)
~
b
i (Æ~
u(t) u~
" (t))
i
dt;
where Æu~
i (t)
=Æu
i (t
+ );u~
";i (t)
=u^
";i (t
+ ). Note that by (2.16)
f(Æu~ u~
" )
i
:[0;Æ℄!R 1
j j(Æu)~
i j
1gS :=fv :[0;Æ℄!R 1
; jvj
2 g:
Observe that ontrollabilityof (A;b
i
) impliesthat (A; ~
b
i
) isontrollableas
well. Controllabilityof the singleinput system (A; ~
b
i )
implies that
(2.20) 02intf
Z
Æ
0 e
A(Æ t)
~
b
i v
dtjv 2Sg:
In fattheset ontherightof (2.20) ontains 0andithas nonemptyinterior,
see e.g. [LM℄, page 77, 133. Moreover, if 0 was a boundary point of this
set, then the orresponding ontrol v = 0 is an extremal ontrol, whih is
arity properties (2.21) L (^u " ; " ; 0 )=0;
L u (^u " ; " ; 0
)(Æu u^
"
)0for allÆu with jÆuj
` 1
1:
From the seondproperty in (2.21)we have,
Z
1
0 ("u^
" +B T e A T (1 t) 0
)(Æu u^
"
)dt0:
Setting
(2.22) p(t)=e
A T
t
q with q=e A T 0 ; this implies Z 1 0 ("^u " +B T ^ p "
)(Æu u^
"
)dt 0;
for all Æu as in (2.21). The seond and third laim in (2.17) follow with
p
"
(t)=p^
" (
1
t).
Exploitingthe rst property in (2.21) impliesthat
L (^u " ; " ; 0
)=1+ " 2 R 1 0 j^u " j 2 dt + T 0 Ae A x 0 + R 1 0 e A(1 t)
Bu^
"
(t)dt+ R
1
0
A(1 t)e A(1 t)
Bu^
" (t)dt
=1+ " 2 R 1 0 j^u " j 2 dt+ T 0 (Ae
((1 t)+t)A
x 0 + R 1 0 e A(1 t)
Bu^
"
(t)dt+ R 1 0 Ae A(1 t) R t 0 e A(t s)
Bu^
"
(s))dsdt=0:
This implies (2.23) L (^u " ; " ; 0
)=1+ " 2 Z 1 0 j^u " j 2 dt+ Z 1 0 p T
(t) Ax^
"
(t)+Bu^
" (t)
dt =0:
From u
" = " (B T p "
) weonlude that u
" 2W 1;1 (0; " ;R m ).
We introdue the Hamiltonianfor (P
" )as
H
"
(x;u;p)=1+ " 2 juj 2 R m +p T
on (0;1)
d
dt H
" (^x
" ;u^
" ;p^
" )="^u
T
" d
dt ^ u
" +
1
" d
dt ^ p T
" d
dt ^ x
" +p^
T
" A
d
dt ^ x
" +p^
T
" B
d
dt ^ u
"
=("^u
" +B
T
^ p
" )
T d
dt ^ u
" =0:
Combined with (2.23)this impliesthat
1+ "
2 ju^
" j
2
R m
+p^ T
" (A^x
" +Bu^
"
)=0on [0;1℄:
This implies the laim.
The proof revealed extra regularity of u
" :
Corollary2.2. UndertheassumptionsofTheorem2.2wehaveu
" 2W
1;1
(0;
" ;R
m
).
Remark 2.1. Condition (2.16) requires that the modulus of atleast one of
the oordinates of u^
"
is not almost everywhere equalto1. One it isknown
from Corollary 2.2 that u^
"
is ontinuous this amounts to requiring that at
least one of the oordinates of u
swithes from1 to 1 or vieversa.
UndertheassumptionsofTheorem2.2therstorderneessaryoptimality
ondition for (P
"
) afterthe transformation t! t
isgiven by
(2.24)
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: _
x=(Ax+Bu); x(0)=x
0
; x(1)=x
1
_ p=A
T
p
u=
" (B
T
p)
1+
2 ju(1)j
2
+p(1) T
(Ax(1)+Bu(1))=0;
where for onveniene of notationthe dependene on " and the supersript
hat were dropped.
Inthefollowingsetionweshallinvestigatesemi-smoothNewtonmethods
for solving (2.24).
Welose this setionwith asimpleexample whihillustratessomeof the
simple ontrolsystem
(
_ x
1 =u
1
_ x
2 =u
2 ;
with so that A is the zero, and B the identity matrix, with initialondition
(1; 1
2
)and terminalonditionthe origin. Thissystem isontrollablebut itis
not normal. The optimal time is
= 1, the rst oordinate of an optimal
ontrol is uniquely determined u
1
= 1 , with assoiate state x
1
= 1 t.
There are innitely many hoies for optimalsolutions u
2
of bang-bang and
non bang-bang type. The assoiated onstant adjoints are (p
1 ;p
2
) = (1;0).
They satisfy
u= (p)2
1
[ 1;1℄
:
The transversality ondition 1+p T
Bu=0 issatised.
For the regularized problem we nd
"
= 1. Dierently from the
un-regularized problem the solution to the regularized problem is unique. The
optimal ontrol and trajetory are given by
(u
1 ;u
2
)=( 1; :5) with (x
1 ;x
2
)=(1 t;:5(1 t)):
In this partiular example the solution of the regularized problemdoes not
depend on ". Note that this solution is also one of the minimum norm
solutions of the unregularized problem. The adjoint is p
"
= (1+ 3"
8 ;
"
2 ). It
satises
u=
" (p
T
" )=
1
1
2
and the transversality ondition 1+ "
2 juj
2
+p T
"
Bu=0:
3 Semi-smooth Newton method
In this setion the semi-smooth Newton method for solving the regularized
optimalitysystem (2.24) is desribed and analyzed. It willallowthat (2.24)
an besolved eÆiently inspite of the fat that
"
(x
" ; u
" ;
"
) 2 W 1;2
(0;1) L 2
(0;1)R a solution to (P
"
) with assoiated
adjoint p
" 2W
1;2
(0; 1). It is assumed that
(H2) there exists s2(0;1) suhthat j 1
" b
T
i
p
"
(s)j=j(u
" )
i
(s)j<1;
and
(H3) jb
T
i p
"
(1)j6="; for alli=1;:::;m:
Assumption (H2)orresponds to(2.16),wherewe nowuse thefat thatasa
onsequeneofTheorem2.2theontrolu
"
isontinuous. With(H2)and(H3)
holdingthereexists aneighborhoodU
p" ofp
" inW
1;2
(0;1;R n
);
t2(0;1),and
a nontrivialinterval( ;+Æ)(0;1)suh that for p2U
p"
we have
jb T
i
p(t)j6=" for allt 2[
t; 1℄; and i=1;:::;m
and
(3.1) jb
T
i
p(t)j<" for t2( ;+Æ):
We set U =fu 2 L 2
(0;1;R m
) :uj[
t; 1℄ 2 W 1;2
(
t;1;R m
)g endowed with the
norm
juj
U
=(juj 2
L 2
(0;1) +juj_
2
L 2
(
t ;1) )
1
2
;
and introdue
F :D
F
X !L 2
(0;1;R n
)L 2
(0;1;R n
)U R n
R
where
D
F =W
1;2
(0;1)U
p"
U R;
X =W 1;2
(0;1;R n
)W 1;2
(0;1;R n
)U R;
and
(3.2) F(x;p; u; )= 0
B
B
B
B
B
B
B
B
_
x Ax Bu
_ p A
T
p
u+
" (B
T
p)
x(1) x
1
1+ "
2 ju(1)j
2
+p(1) T
(Ax(1)+Bu(1)) 1
C
C
C
C
C
C
C
C
1 5 1 2
F
3
. For F
4 ;F
5
it follows from the fat that W 1;2
(0;1) embeds ontinuously
into C(0;1). Morevover F(x
" ;p
" ; u
" ;
"
) = 0: We shall keep x
"
(0) = x
0 as
an expliit onstraint.
Remark 3.1. The need for introduing U in suh a way that its elements
are more regular at 1 is due to the fat that we use here the point-wise
transversality ondition rather than the integrated form (2.23). - Condition
(H3)willbeneeded toprovesuperlinearonvergeneoftheNewtoniteration.
ApplyingNewton'smethodtoF =0isimpededbythenon-dierentiability
of
"
. We use
G
" (s):=
(
1
"
if jsj< "
0 if jsj " (3.3)
as a generalized derivative and argue that the resultingNewton iterationis
semi-smooth and hene loallysuperlinearly onvergent. The Newton
itera-tion step isgiven by
(3.4) DF(x;p; u; )(Æx; Æp;Æu; Æ)= F(x; p;u; )
whereÆx(0)=0andDF denotes the Frehet-derivativeinalltermsofF
ex-eptforp!
" (B
T
p),forwhihthegeneralizedderivativeistaken aording
to (3.3). Forfurther referene wegive the detailedform of (3.4):
(3.5)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
: d
dt
Æx AÆx BÆu Æ(Ax+Bu)= F
1
; Æx(0)=0
d
dt
Æp A T
Æp ÆA
T
p= F
2
Æu+G
" (B
T
p)B T
Æp= F
3
Æx(1) = F
4
p(1) T
(AÆx(1)+BÆu(1))+Æp(1) T
(Ax(1)+Bu(1))
+"u(1) T
Æu(1)= F
5 ;
where the oordinatesof G
" (B
T
p)B T
Æpare given byG
" ((B
T
p)
i ))(B
T
Æp)
i .
Apossibleinitializationmayonsistinhoosing((u)
0 ;
0
),setting(x)
0 as
the linear interpolation between x
0
and x
1
, and determining (p)
0
suh that
whih are relevant for this paper. Let X and Z be Banah spaes and let
F :D
F
X !Z be a nonlinear mappingwith open domain D
F .
Denition3.1. ThemappingF :D
F
X !Z isalledNewton-dierentiable
on an open subset U D
F
, if for eah x 2 U there exists a generalized
derivative DF(x)2L(X; Z) and
(3.6) lim
h!0 1
jhj
X
jF(x+h) F(x) DF(x+h)hj
Z =0:
Theorem 3.1. Suppose thatx
2U isa solution to F(x)=0and that F is
Newton-dierentiableinanopensetU ontainingx
. IffurtherfkDF(x) 1
k:
x2 Ug is bounded, then the Newton-iteration x
k+1 =x
k
DF(x
k )
1
F(x
k )
onverges q-superlinearly tox
, provided thatjx
0 x
j
X
issuÆiently small.
For the statement and proof of superlinear onvergene of the time-optimal
ontrolproblem,some furthernotationisrequired. For(x; p; u;)2D
F we
dene A2R
(n+1) (n+1)
by
A= A
11 A
12
A
21 0
!
;
where
(3.7) A
11 ="
1
Z
1
0 e
A(1 t)
B
I B
T
e A
T
(1 t)
dt2R nn
;
(3.8)
A
12 ="
1
R
1
0 e
A(1 t)
B
I B
T R
1
t e
A T
(t s)
A T
p(s)dsdt
R
1
0 e
A(1 t)
(Ax+Bu)dt2R n
;
(3.9) A
21
=(Ax(1)+Bu(1)) T
(p T
(1)B+"u T
(1))G
" (B
T
p(1))B T
2(R n
) T
;
with
I
=diag(
I1
;:::;
Im
) and
I
i
the harateristi funtionof the set
I
i =I
i
(p)=ft:j(B T
p)
i j<
1
"
whih is nonempty for p 2 U
p
"
and i = i . The ontrollability assumption
(H1) together with (H2) imply that the symmetri matrix A
11
is invertible
with uniformly bounded inverse with respet to p 2 U
p"
and in ompat
subsets of (0;1). In fat,sine I
i
(p)( ; +Æ) weobtain forsome >0
A 11 =" 1 Z 1 0 e A(1 t) m X i=1 b i I i b T i e A T (1 t) dt " 1 Z +Æ e A(1 t) b i b T i e A T (1 t) dt =" 1 e A(1 ) Z Æ 0 e At b i b T i e A T t dte A T (1 )
dt>;
where we use that the ontrollabilityGramian
Z Æ 0 e At b i b T i e A T t dt;
is uniformlypositive denitefor inompat subsets of (0;1).
Forour analysis we shall utilize the fat that the Shur omplement
A 21 A 1 11 A 12 2R
of A for (x; p;u; ) in a neighborhood of (x
" ; p " ;u " ; "
) is nontrivial. If
A 21 A 1 11 A 12
6=0 at (x
" ; p " ; u " ; "
),weannotonludethatA
21 A 1 11 A 12 6=0
for (x;p; u; ) in a neighborhood of (x
" ; p " ; u " ; "
), sine, while with (H2)
holding, A
21
is ontinuous with respet to (x;p; u; ) 2 X, this is not the
ase for A 1
11
and A
12
due to the term
I
. We thereforeassume that
(H4) 8 > > > < > > > :
there exists a bounded neighborhood
U D
F
X of (x
" ; p " ; u " ; "
) and >0 suh that
jA 21 A 1 11 A 12
jfor all (x;p; u; )2U:
Theorem 3.2. If (H1){(H4) hold and (x
" ; u
" ;
"
) denotes a solution to
(P
"
) withassoiated adjoint p
"
, then thesemi-smooth Newton algorithm
on-verges superlinearly, provided that the initialization is suÆiently lose to
Lemma 3.1. Suppose that (H1) { (H4) hold. Then there exists a onstant
C suh that for every (x;p; u; )2U, and F 2L 2
(0; 1)L 2
(0;1)U R
DF(x; p; u; )(Æx; Æp;Æu; Æ)= F
admits a unique solution (Æx;Æp;Æu; Æ)2X and
(3.10) j(Æx; Æp;Æu;Æ)j
X
C jFj
L 2
L 2
UR :
Proof. Let (x;p; u;)2U and note that system (3.5) is equivalent to
(3.11)
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
Æx(t)= R
t
0 e
A(t s)
( F
1
+bÆu+Æ(Ax+Bu))ds
Æp(t)=e A
T
(1 t)
Æp(1)+ R
1
t e
A T
(t s)
(ÆA T
p F
2 )ds
Æu+G
" (B
T
p)B T
Æp= F
3
Æx(1) = F
4
p(1) T
( AF
4
+BÆu(1))+Æp(1) T
(Ax(1)+Bu(1))+"Æu(1)= F
5 :
On I
i
we have (G
" (B
T
p))
i = "
1
. The third equation in (3.11) an be
expressed as
(3.12) Æu= F
3 "
1
I B
T
Æp a.e. in (0;1):
Let usset
^
F
1 =
Z
1
0 e
A(1 t)
F
1 (t)dt;
^
F
2 ="
1
Z
1
0 Z
1
t e
A(1 t)
B
I B
T
e A
T
(t s)
F
2
(s)dsdt;
^
F
3
= B Z
1
0 e
A(1 t)
F
3 (t)dt:
From (3.12), and the rst and fourth equation in(3.11) we have
(3.13)
F
4
=Æx(1)
= " 1
R
1
0 e
A(1 t)
B
I B
T
Æpdt+Æ R
1
0 e
A(1 t)
(Ax+Bu)dt+ ^
F
1 +
^
F
F
4 = "
1
Z
1
0 e
A(1 t)
B
I B
T
e A
T
(1 t)
Æp(1)dt
" 1
Æ Z
1
0 e
A(1 t)
B
I B
T Z
1
t e
A T
(t s)
A T
p(s)dsdt+ ^
F
2
+Æ Z
1
0 e
A(1 t)
(Ax+Bu)dt+ ^
F
1 +
^
F
3 ;
whih involvesÆp(1) and Æ as unknowns, and an be expressed as
(3.14) A
11
Æp(1)+A
12 Æ =
^
F
1 +
^
F
2 +
^
F
3 +F
4 =:r
1 :
Eliminating Æu(1) from the last equation in (3.11) by means of the third
equation implies
(3.15) A
21
Æp(1)= F
5 +F
3 (1)
T
(B T
p(1)+"u(1))+p(1) T
AF
4 =:r
2 :
Combining(3.14)and(3.15)weobtainthefollowinglinearsystemfor(Æp(1); Æ):
(3.16) A
Æp(1)
Æ !
= r
1
r
2 !
:
By (H1),(H2), and (H4)its unique solutionis given by
(3.17) Æ =(A
21 A
1
11 A
12 )
1
(A
21 A
1
11 r
1 r
2
); Æp(1)=A 1
11 (r
1 A
12 Æ):
MoreoverthereexistsaonstantC =C(; jxj
C(0;1) ; jpj
C(0;1) ; juj
L 2
(0;1)
; ju(1)j);
suh that
jÆp(1)j+jÆjCjFj
L 2
L 2
UR :
From (3.5) and (3.11), C an alsobe hosen suh that
j(Æx; Æp; Æu; Æ)j
X
CjFj
L 2
L 2
Proof of Theorem 3.2. We apply Theorem 3.1 with x =(x
" ;p
" ; u
" ;
" ).
Lemma 3.1 implies the required uniform bound of the generalized inverses
DF in the neighborhoodU D
F of (x
" ; p
" ; u
" ;
"
). Therefore itsuÆes to
argueNewton-dierentiabilityofF inD
F
. Thisisobviousforalloordinates
of F exept for F
3
, and speially for the mapping
F :p!
" (B
T
p)
from U
p"
W
1;2
(0;1; R n
) ! U. Utilizing the denitions of U
p"
and
" it
suÆes to onsider the restrition of F from W 1;2
(0;
t; R n
) to L 2
(0;
t; R 1
)
whih weagain denote by F. Note that F an be deomposed as
F =F
3 ÆF
2 ÆF
1 ;
where
F
1 :W
1;2
(0;
t ; R n
)!W 1;2
(0;
t; R); F
2 :W
1;2
(0;
t; R) !L 4
(0;
t; R);
F
3 :L
4
(0;
t; R) !L 2
(0;
t; R);
are given by
F
1
(u)=B T
u; F
2
(v)=max( 1; v
" ); F
3
(v)=min(1;v):
In [HIK , IK2℄ it was shown that v ! max(0;v) is Newton dierentiable
from L p
() to L q
() if 1 p > q 1, if is a bounded domain. Sine
min(1;v)=1+min(0;v 1) this implies that F
3
and similarlythat F
2 are
Newtondierentiable. FromthehainruleforNewtondierentiablemapping
in [HK ℄ it follows that F
3 ÆF
2
is Newton dierentiable. The hain rule for
a linear mapping, here F
1
, followed by the Newton dierentiable mappings
F
3 ÆF
2
; [IK1℄, implies that F is Newton dierentiable in D
F
:
4 A numerial example
The semi-smooth Newton method is used to solve a lassial time optimal
problem related to the harmoni osillator with three swithing points. We
(4.1)
>
>
>
<
>
>
>
: min
0 R
0 dt
subjet to
d
dt
x(t)=Ax(t)+Bu(t); ju(t)j1;x(0) =x
0
;x()=x
1 ;
where
A=
0 1
1 0
; B =
0
1
; x
0 =
5
5
; x
1 =
0
0
:
The optimal minimal time for the ontinuous problem is known tobe
=
10:5871. Tosolve(4.1)numeriallyatimedisretizationbasedonthe Crank
Niolson method with equidistant grid points was applied to (3.5). The
initialisation for the state was hosen as a semiirleonneting x
0
and x
1 .
Thenu(1)washosentobeative,andpwashosensothatthetransversality
ondition and the adjoint equation hold. With respet to the hoie of the
parameter = 1
"
weutilized a ontinuation proedure, startingwith a small
value and inreasing it, using the solution from the smaller value of as
initialization for the next larger -value. Certainly this proedure an be
automated as has been done elsewhere, but this was not the fous of this
paper. In Table 1 we show the number of iterates of the Netwon iteration
(outer loop) that was required for this ontinuation proedure with respet
to . The Newton iterationwasstopped when the residualof the optimality
systemintheL 2
-normwasbelow10 8
. AlsoinTable1wedepittheoptimal
minimal times
(). These results are obtained for meshsize h = 1
32
. Then
1 5 10 20
No. ofiterations 8 8 4 7
FinalTime 11:26515 10:84455 10:82977 10:81781
Table 1
the results for = 1 are interpolated to the ner grid h = 1
128
and the
ontinuationproedurewithrespettoisrepeated. Theresultsaredepited
inTable 2. Thegraphs fortheorrespondingontrols aregiveninFigure1.
The same proedure with h=1=512 and =100 gives the optimaltime
No. ofiterations 5 46 4 4 3
FinalTime 11:1088 10:6092 10:6034 10:6033 10:6031
Table 2
0
2
4
6
8
10
12
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
11.1088
time
control
0
2
4
6
8
10
12
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
10.6092
time
control
0
2
4
6
8
10
12
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
10.6033
time
control
Figure1: N=128and=1(left),10(middle)and100(right)
the lowest values of the full Newtonstep was toolarge. Therefore we used
a one-dimensionallinesearh based onaquadratipolynomial interpolation
for the L 2
norm of the residual ombinedwith an Armijorule.
Table 3depits the quotients ju
k +1
u
()j
L 2
ju k
u
()j
L 2
, whereu
() is the solution to
the disretized version of (2.17)for =50. It shows that the algorithmisin
fat superlinearlyonvergent.
No. ofiterations 1 2 3 4
k
0:94138 0:00037 0:00001 0:00000
Table 3
In this paper we hose to regularize by the ramp funtions
" with
inreasing slops as " ! 0 +
. Certainly other alternatives are possible as for
instane ~
(s) =
2
atan(s): This family of C 1
funtions also has the
property that it onverges to as ! 0, but it appears to be less apt for
the purpose of approximating the disontinuous swithing struture of the
optimal ontrols sine has to be taken signiantly larger for ~
than for
1
toobtain omparable results.
Aknowledgement: We thank Mrs. J. Rubesa for providing us with the
[BPW℄ Ch. Buskens, H.J. Pesh, and S. Winderl: Real-time solutions for
bang-bang and singular optimalontrol problems, in: M. Grotshel,
et al. (Eds), Online Optimization of Large Sale Systems,
Springer-Verlag, Berlin,2001, 129-142.
[Fa℄ H.O. Fattorini: Innite Dimensional Linear Control Systems: The
Time Optimal and Norm Optimal Problems, Elsevier, Amsterdam,
2005.
[HL℄ H. Hermes and J. LaSalle: Funtional Analysis and Time Optimal
Control, Aademi Presse, 1969.
[HK℄ M.HintermullerandK.Kunish: PDE-onstrainedoptimization
sub-jet to pointwise ontrol and zero- and rst-order state onstraints,
preprint.
[HIK℄ M. Hintermuller, K. Ito and K. Kunish: The primal{dual ative
set strategy as a semi{smooth Newton method, SIAM Journal on
Optimization,13(2002), 865-888.
[IK1℄ K. Ito and K. Kunish: The primal-dual ative set method for
non-linear optimal ontrol problems with bilateral onstraints, SIAM J.
on Control and Optimization,43(2004), 357-376.
[IK2℄ K.ItoandK.Kunish: On theLagrangeMultiplierApproahto
Vari-ational Problems and Appliations, SIAM, Philadelphia,2008.
[KN℄ C.Y.KayaandJ.L.Noakes: Computationalmethodsfortime-optimal
swithing ontrols, JOTA, 117(2003),69-92.
[Ke℄ H.B. Keller: Numerial Mathods for Two-Point Boundary Value
Problems: Blaisdell,London, 1968.
[KLM℄ J.-H.R.Kim, G.L. Lippi, and H. Maurer: Minimizing the transition
time in lasers by optimal ontrol methods. Single-mode
semiondu-torlasers withhomogenoustransverse prole: PhysiaD, 191(2004),
238-260.
[LM℄ E.B. Lee and L. Markus: Foundations of Optimal Control Theory:
Springer Verlag, New York, 1982.
[MO℄ H. Maurer and N.P. Osmolovskii: Seond order suÆient optimality
onditions fortime-optimalbang-bang ontrol, SIAM J.Controland
Optimization,42(2004), 2239-2263.
[MZ℄ H. Maurer and J. Zowe: First and seond-order neessary and
suf-ient optimality onditions for innite-dimensional programming
problems, MathematialProgramming 16(1979), 98{110.
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spaeraft reorientation problem,J. Guidane, 13(1990), 859-866.
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