Intelligent Information Management, 2010, 2, 134-142
doi:10.4236/iim.2010.22016 Published Online February 2010 (http://www.scirp.org/journal/iim)
Existence and Uniqueness of the Optimal Control
in Hilbert Spaces for a Class of Linear Systems
Mihai Popescu
Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy
Bucharest, Romania
Abstract
We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear sys-
tems. We are interested in the minimization of time, energy and final manifold in transfer problems. The state
variables space X and, respectively, the control variables space U, are considered to be Hilbert spaces. The
linear operator T(t) which defines the solution of the linear control system is a strong semigroup. Our analy-
sis is based on some results from the theory of linear operators and functional analysis. The results obtained
in this paper are based on the properties of linear operators and on some theorems from functional analysis.
Keywords: Existence and Uniqueness, Optimal Control, Controllable Linear Systems, Linear Operator
1. Introduction
A particular importance should be assigned to the
analysis of the control of linear systems, since they
represent the mathematical model for various dynamic
phenomena. One of the fundamental problems is the
functional optimization that defines the performance
index of the dynamic product. Thus, under differential
and algebraic restrictions, one determines the control
corresponding to functional extremisation under con-
sideration [1,2]. Variational calculation offers methods
that are difficult to use in order to investigate the exis-
tence and uniqueness of optimal control. The method of
determining the field of extremals (sweep method), that
analyzes the existence of conjugated points across the
optimal transfer trajectory (a sufficient optimum condi-
tion), proves to be a very efficient one in this context
[3–5]. Through their resulting applications, time and
energy minimization problems represent an important
goal in system dynamics [6–11]. Recent results for con-
trollable systems express the minimal energy through
the controllability operator [11–13]. Also, stability con-
ditions for systems whose energy tends to zero in infi-
nite time are obtained in the literature [13,14]. By using
linear operators in Hilbert spaces, in this study we shall
analyze the existence and uniqueness of optimal control
in transfer problems. The goal of this paper is to pro-
pose new methods for studying the optimal control for
exactly controllable linear system. The minimization of
time and energy in transfer problem is considered. The
minimization of the energy can be seen as being a par-
ticular case of the linear regulator problem in automat-
ics. Using the adjoint system, a necessary and sufficient
condition for exact controllability is established, with
application to the optimization of a broad class of
Mayer-type functionals.
2. Minimum Time Control
2.1. Existence
2.1.1. Problem Formulation
We consider the linear system


BA,
:
,)(),()( 00 HxtxtuBtxA
td
xd  (1)
where
H
is a Hilbert space with inner product




,and norm.
HHADA )(: is an unbounded operator on
H
which generates a strong semi group of operators on
H
,




0
0
)( 
t
At
tetS .
HUB :is a bounded linear operator on another
Hilbert space , for example .
U

HUB ,L
Hu 

],0[: is a square integrable function repre-
senting the system control (1).
For any control function , there exists a solution u
of (1) given by

M. POPESCU 135
0,)()(
0
)(
0 tsdsuBexetx
t
AstAt (2)
The control problem is the following one:
Given , ,
It 
0)( 0
tx
X
z

U
and the constant ,
let us determine such that
0M
uMu  and
 xtxttii
ztxi


)(inf)(
)()(
1
1

(3)
Here,
X
andUare Hilbert spaces.
Theorem 1. The optimal time control exists for the
above formulated problem.
Proof
Let
 be a decreasing monotonous sequence such
that . Also, let us consider the sequence

n
t
ttn
 )
1
 (

n
u, with . We have Utuu nn  )(
,)()()()()(
0
0
n
t
t
nnnn duBtStxtStx
 (4)
which gives




n
t
t
nn
t
t
nnnn
duBtS
duBtStxtStx
1
1
0
)()(
)()()()()( 0


(5)
)(tS being a continuous linear operator, it is, there-
fore, bounded. It follows that we have:
)()()()( 010 txtStxtS n (6)
0)()(
1


n
t
t
nnduBtS
 (7)
If we consider the set of all the admissible controls


MuUu  (8)
then, is a weakly compact closed convex set. 
As is weakly compact, any sequence
U

)(u
 u
possesses a weakly convergent subsequence to
an element . Thus, for and any
(the dual of ), we have

1
u
1
u


 uuuui
i,,lim1 (9)
Since , it follows that 
1
u
Mu
1 (10)
Also, we have
.)(
,:
UxtS
UXB
n




(11)
So,
.)(UxtSB n   (12)
For every Xx

, let us evaluate the difference
xduBtS
xduBtSF
n
t
t
t
t
nn
,)()(
,)()(
1
1
1
11






(13)
By writing
,,1,)(
1
1
0nkdBttS k
t
t
k
L
 (14)
the Expression (13) becomes
xuxuxuxu nnn
  1n LLLL ,,,, 111 (15)
or
.,
,,, 1
xu
xuxuxuF
n
nn

 
n
111
L
LLL
(16)
Therefore,



dxtSBxtSBu
dxtSBuu
xuxuuF
t
t
nn
t
t
n
nn








1
0
1
0
)()(),(
)(),()(
,,
1
11
11n1 LLL
(17)
By using the properties of the operator, one ob-
tains
)(tS

.)()(),(
)(),()(
1
0
1
0
11
11


dxxttStSBu
dxtSBuuF
t
t
nn
t
t
n






(18)
Because the sequence un converges weakly to
, the first term in (18) tends to zero for
n
u
1
u

n
(see(9)). On the other hand, is a strongly
continuous semigroup and, hence, from its boundedness,
it follows that there exists a constant such that

)( 1  tS
K0

M. POPESCU 136

xxttSK
xxttStSB
n
n




)(
)()(
1
11

(19)
So, it follows that the second term in (18) tends to
zero as . The sequence is weakly
convergent to if, for any ,
we have
n

Xtx n)(
xXztx )(1XX 
,),(),(lim1xtxxtx n
n

 (20)
which gives
xduBtStxtSxz
t
t
,)()()()(,
1
0
1101

 (21)
From (21), one gets
zduBtStxtS
t
t
  
1
0
)()()()( 1101 (22)
and, hence, is the optimal control.
1
u
An important result which is going to be used for
proving the uniqueness belongs to A. Friedman:
Theorem 2 (bang-bang). Assuming that the set

is convex in a neighborhood of the origin and is
the control of optimal time in the problem formulated in
(2.1.1), ([2, 10]) it follows that , for almost all
.
)(tu
)(tu
],[ 10ttt
2.2. Uniqueness of Time-Optimal Control
2.2.1. Rotund Space
Let

be the unity sphere in the Banach space U
and let

 be its boundary [11].
The space is said to be rotund if the following
equivalent conditions are satisfied:
U
a) if 2121 xxxx  , it follows that there ex-
ists a scalar 0

, such that 12 xx

;
b) each convex subset has at least one ele-
ment that satisfies
UK 
;,, KzKuzu 
c) for any bounded linear functional on
there exists at least an element
f U

x such that
;)(,fxffx 
d) each

x is a point of extreme of

.
Examples of rotund spaces:
1) Hilbert spaces.
2) Spaces , ,
p
lp L

p11.
3) If the Banach spaces are rotund,
then the product space U is rotund,
too.
n
UUU ,,, 21 
n
UU  
21
4) Uniform convex spaces are rotund, but the con-
verse implication is not true.
2.2.2. Uniqueness
We assume that and are optimal,
1
u2 u
2,1,


iUui. Therefore,
zduBtStxtS
t
t
  
1
0
)()()()( 1101 (23)
zduBtStxtS
t
t
 
1
0
)()()()( 2101 (24)
By adding Equations (23) and (24), one obtains
 zduuBtS
txtS
t
t




1
0
)()(
2
1
)(
)()(
211
01
(25)
It follows that )(
1

u,)(
2

u,

)()(2121


uu

are optimal. By using Theorem 2, we have

.1
2
1
,1
)()(
2
1
),(),(
21
2121


uuu
uuuu
k

Since the condition (i) is satisfied, is a rotund
space and
U
21 uu


. Thus,


21 221  uuu  (26)
and, therefore
21
1uu   (27)
which ends the proof of the uniqueness.
3. Minimum Energy Problem
3.1. Problem Formulation
Let (see(1)) be a controllable system with finite
dimensional state space
BA,

X
[6,10,12, 14].
Let ],0[ 1 tI

,Xax


)0(
),(
2UIL
,be given. Let be
the Hilbert space. The minimum norm control
Xb U

M. POPESCU 137
problem can be formulated as follows: determine
such that for some , Utu )(
tx(1
It 
1
1) , b)
2) u is minimized on , where],0[ 1 trepresents
the norm on .
),(
2UIL
Let us fix . Consider the linear operator
0
1t
1
20(: HL
nL1);,Ut

1)Bst
1
at L
H
(28)
defined by


1
0
)((:
t
nsdsuSuL (29)
We have
.)()( 11 butStx  (30)
Theorem 3. Let be a linear mapping
between a Hilbert space and a finite dimensional
space
U
t:L
U
H
[11]. Then, there exists a finite dimensional
space such that the restriction of
to
UMM
t
Lt
L
M
is an injective mapping.
Proof
Let , be a basis for the range of
in

niei,,1 t L
H
. Given any , can be written as Uu u
t
L
,
ii e

1



n
i
tuL (31)
where each i

can be expressed by
., UUfiufii    (32)
Then,
.
1



n
i
tuL,ii euf (33)
i
f are linearly independent and generate a dimen-
sional subspace .
n
UM
From the properties of Hilbert spaces (the projection
theorem), it follows that .

MMU
Let . Then,

Mu 0,  ii uf  . Thus,


.ker0 tt uUuM LL
 (34)
Let . We get
t
uLker
.,0
1



n
i
iit eufuL (35)
Since are independent,
i
eniufi,,1,0,
and, so, . maps
t
LkerMt L
M
bijectively to the
range of and hence is an injective mapping
into
t
LM
t
L
H
. Let
.0,,, 22121  uMuMuuuu (36)
Because , 0
2u t
L
Hxuu M
tt  1
LL (37)
Since 0, 21 uu , we have
2
2
2
12121
2,uuuuuuu  (38)
Then, a unique minimum norm exists and
is the minimum norm element satisfying

x
M
t
1
L
xu M
t
L
1
xu

L.
Remark 1. The unique solution of the equa-
tion
)(tu
xu

L, with minimum norm control, is the pro-
jection of onto the closed subspace )(tu


n
ff,,
1M

.
Hence, from (30), there exists a control
transferring to in time if
and only if
);,0()( 1
2UtLu  a b1
t


Im)( 1atS 1
t
Lb


. The control which
achieves this and minimizes the functional


1
1
0
2
)(
t
tsdsuE, called the energy functional, is
 atSbu t)( 1
1
1
L  (39)
Define the linear operator
0,)()(
0
 trdrSBBrSQ
t
t (40)
We have the following results (see [10–12]):
Proposition 1.
1) The function , is the unique solution of
the Equation [1,13]
0, tQt
IQADx
xBxxQxxQ
td
d
tt




0
2
,)(
,2,
(41)
where

 .)(,
,,)(
ADxxC
yxARCHyAD
H
H

 
(42)

M. POPESCU 138
2) If generates a stable semigroup, then A
QQt
t 
lim (43)
exists and is the unique solution of the equation
,)(,0,22 ADxxBxxAQ (44)
The proof of Proposition 1 is given in [10,11].
The following theorem gives general results for the
functionals , the minimal energy for transfer-
ring to in time , and , where
,t(see [12]).
),(baE

b
0
a
H
1
t),0(bE
ba,1
Theorem 4.
1) For arbitrary and 0
1tHba ,
  .)(),(
2
1
1
21 batSQbaE 

(45)
2) If is stable and the system )(tS


BA,
 is null
controllable in time , then 0
0t
 HbbQbE  
,),0(
2
1
21 (46)
3) Moreover, there exists such that 0
0
t
C


01
2
1
21
2
1
21
,,
),0(
0
1
ttHbbQC
bEbQ
t
t




(47)
4. Numerical Methods for Minimal Norm
Control
4.1. Presentation of the Numerical Methods
Let be a dynamic system with ,
BA,
m RU n RX

,
1
R
I
[9,10,13]. Given 0
0

x,, , , determine
such that
0
t1 tb
U)tu(
1) , btx )( 1
2) p u is minimized for ),1[p
Now,




1
0
1
0
)()(
)()(
1
1
t
t
t
t
dut
duBtSbe


(48)
In order to make Equation (48) true, must be
chosen on the interval . Consider the i-th compo-
nent of the vector :
)(tu
],[ 10tt
e
i
e
)(,)()(
1
0
1ii
t
t
ii fudute
   (49)
where i

is the row of the matrix and is the
unique functional corresponding to the inner product
(Rieszrepresentation theorem).
f
Let i

be an arbitrary scalar. Then,
,)()( iiiiiiffe







(50)
Since n
R
is a Hilbert space, the inner product
efe
n
i
ii
n
i
ii ,)(
11
   (51)

being a vector with arbitrary components n


,,
1.
If Equation (51) is true for at least different linearly
independent,
n
ni
i,,1, 


then
Equation (48) is also true.
We have
.,
11
uf
n
i
ii
n
i
ii  









 (52)
By Hölder’s inequality,
.1
11
,,
11









qp
uu
q
n
i
ii
p
n
i
ii

(53)
From Equations (51) and (53),
.
,
1
1
1
1
1
q
n
i
ii
q
n
i
ii
n
i
ii
q
n
i
ii
n
i
ii
p
e
e
f
u

























(54)
Every control driving the system to the point must
satisfy Equation (54), while for the optimal (minimum
norm) control, Equation (54) must be satisfied with
equality (Theorem 2). Numerically, one must search for
vectors
b
a n

such that the right-hand side of Equa-

M. POPESCU 139
tion (54) takes its maximum.
4.2. A Simple Example
We consider a single output linear dynamic system
, where
BA,

111 ,, RIRXRU 
Because fixed, then It 
1)()( 1


 BtS

1
0
.)()(
t
t
dube  (55)
Therefore,
 
1
0
1
0
)()()()(
t
t
t
t
dudube  (56)
and from Hölder’s inequality,
,)()(
1
0
udu
t
t


 (57)
where we have assumed that . The mini-
mum norm control, if exists, will satisfy
)()( 2ILt 
.

e u (58)
From (56) and from (57), we have
],[,))((sign))((sign 10tttttu 

 (59)
and, respectively,
,],[
,)()(
10ttt
tuhtpq

 (60)
h arbitrary constant.
The control u satisfies the relation
pqpq
ptkthtu )()()( 1 (61)
or
.)((u(t))sign)( pq tktu  (62)
From condition (59),
.)())((sign)( pq ttktu  (63)
By substituting Equation (63) into Equation (55), the
constant can be determined. k
4.2.1. The Particular Case 2

qp
In this case, the Equation (55) represents the inner
product in ,
);,(10
2UttL
ub , (64)
The relation (63) becomes
.)())((sign)( ttktu (65)
Then,
.)( 2
2
1
0
kdkb
t
t
 (66)
Thus,
2

b k (67)
and the optimal control is given by
.
)(
)( 2


tb tu (68)
5. Exact Controllability
5.1. Adjoint System
Let ],0[),( 1 tttS

, be the fundamental solution of an
homogeneous system associated to the linear control
system (see(1)) [4,5,13] .
BA,

Thus, we have
],0[,)0(),(
)(
1
ttIStSA
td
tSd  (69)
From

 .0)()( 1

td
Id
tStS
td
d (70)
one obtains
.)()( 11 AtStS
td
d  (71)
which implies that

 .)()( 11 


 tSAtS
 (72)
Since

  .)()( 1
1


tStS (73)
the system becomes

 
],0[,)()( 1
1
1tttSAtS
td
d 
 (74)
It follows that

 ],0[,)( 1
1tttS 
, is the fundamental
solution for the adjoint System (69).
5.2. A Class of Linear Controllable Systems

M. POPESCU 140
We consider the class of linear control systems in vecto-
rial form [1,3,4,8,13,15]
0
)0(, xxCuBxA
td
xd  (75)
Let be any internal point of the closed bounded
convex control domain .
0
u
U
This domain contains the space of variables
. We take
m
E
),,( 1m uuu 

uuu  (76)
By this transformation, system (75) becomes
 .CuBuBxA
td
xd 
 (77)
Thus, one transfers the origin of the coordinates of
space in . At the same time, the origin of the
coordinates of space is an internal point inside the
domain . Let us denote by the solution of sys-
tem (75) which corresponds to the control .
m
E
U

u
m
E
)(tx
0u
This control is admissible, as the origin of the coordi-
nates belongs to the domain and satisfies the initial
condition . As a result,
U
0
)0( xx 

.)(
)( CtxA
td
txd  

(78)
Let , be any control and be the
trajectory corresponding to system (75).
110),(tttu  )( 1tx
We have
.)0(,)()(
)(
0111
1xxCtuBtxA
td
txd (79)
We take
)()()( 1txtxtx   (80)
Then, from (78) and (79), one obtains
.)()(
)(
1tuBtxA
td
txd 
Hence, for , the system (75) becomes )(
1tuu 
,
0)0(
:
,
n
BARx
x
uBxAx 






(81)
Thus, the linear control system belongs to the class
defined by (81).
BA,

5.3. Optimal Control Problems
For a given , let us determine the optimal control
1
tu
~
that extremises the functional of final values




n
i
ii txctxFJ
1
11 )()( (82)
and satisfies the differential constraints represented by
the system (81).
The adjoint variable Ry

satisfies equation
x
H
y


 (83)
where
H
is the Hamiltonian associated to the optimal
problem
xyH  , (84)
Since the final conditions are free and the final
time has been indicated from the condition of transver-
sality, one obtains
)(1 tx
1
)(1t yty

.
It follows that the adjunct system becomes










1
)(
:
1
,
t
BA yty
yAy

(85)
with solution
.,)()( 11
1HyyttStytt   (86)
Proposition 2.
For the class of optimum problems under considera-
tion, the following identity holds true:


1
11
0
,,
t
U
H
tt tdyBuyx (87)
Proof
Assuming that



 UtCu ,,01
1
 and
it follows that
)()( 1

ADty



 UtCy ,,01
1
x,.
Integrating by parts, since , 0)0( x1 )( 1t yty

,
, , we have HUB :UHB 
:












1
11
1
0
1
11
1
0
00
0
00
0
,
,,
,,
,,
,0
t
U
t
H
t
H
t
U
t
H
t
H
t
H
tdyBu
tdyAxtdyAx
yxtdyBu
tdyxAtdyx
tdyuBxAx
t


(88)

M. POPESCU 141
One obtains
0,,
1
11
0
 
t
U
H
tt tdyBuyx (89)
The identity has been demonstrated.
This result can be extended for arbitrary
and .
);,0(1
2UtLu Hty )(1
An important result referring to the exact controllabil-
ity of the linear system (81) is stated in
Theorem 5. The system is exactly controllable
if only if the following condition is satisfied
BA,

Hy
yctdytSB U
t
U



0
2
0
0
2
0,)(
1
(90)
Proof
“” We assume that is exactly controllable.
Let and
BA,

ty);,0(1
2UtLu H

)(1.
We consider that the application
)(:
1TxULt (91)
is well defined.
Let be the inverse of . From
Theorem 3 it follows that there exists a finite dimen-
sional subspace
);,0(:1
2UtLH 1
t
L
H
M
, such that the
restriction
1
ker t LM 

 



1
ker
11
t
L
t
M
tLL (92)
is injective.
Since it
follows that transfers 0 in for the system .
)(1
1txuLt
 )()(11
1
1txtxLu t
)( 1 tx BA, 

We choose ynd xHt )( 1a)()( 11 tyt ,

 )(1 txu . It
follows that


1
1111
0
2),(,
t
U
ttt
H
ttdyBxyyy (93)
For ,B, using Hölders’s ine-
quality, one obtains
2
)( 1Luxt 2 LyB 

  
21
0
2
21
0
2
00
11
1
1
1
1
1
)(
)()(






tt
t
t
t
t
t
tdyBtdx
tdyBxtdyBx
(94)
From (93) and (94), for , we have
11 tt yx
21
0
2
21
0
2
2
1
1
1
11)(



























t
U
H
t
t
U
t
H
t
tdyBy
tdyByy
(95)
or
2
2
0
2
1
11
H
t
t
UytdyB 

 (96)
By changing 

ttt 1
(S
becomes and
is transformed into .
0
y
)
1t
1
t
y
)(tSt
Therefore, we get one equivalent relation of controlla-
ble system in which substitutes .
1
t
y0
y
22
2
0
2
01
0
2
0
11
11
1
)()(
H
t
H
t
t
U
t
U
ycy
tdyttSBtdytSB


 
(97)
The direct implication has been demonstrated.
“

” We assume that condition (90) is fulfilled.
Then
2
0
2
1
1
H
t
t
UyctdyB 
 (98)
For any Hyt

1 we take the set


)()(tyBtu  
( is solution of ). )(ty BA,
We consider the solution for which cor-
responds to the above mentioned control . Let us
define the bounded operator
)(tx BA,

u)(t


HyBLtxHytt  )()(: 11 1 (99)
Since , the identity (89) becomes yBtu  )(
2
0
2
1
1
11 )(, H
t
t
U
H
tt yctdtyByy   (100)
Therefore, there exists a constant for which
(100) is satisfied. It follows that
0c
H
tt yy 11, is posi-
tively defined.
This resumes the conclusion that the system is
controllable.
BA,


M. POPESCU 142
Thus, since is inversable, for any , state
is such that there exists a control
which transfers 0 in . The theorem has
been demonstrated.
Hxt
1
)( 11
1
ttxy

yBtu  )( 1
t
y
6. Conclusions
The main contribution in this paper consists in proving
existence and uniqueness results for the optimal control
in time and energy minimization problem for controlla-
ble linear systems.
A necessary and sufficient condition of exact control-
lability in optimal transfer problem is obtained. Also, a
numerical method for evaluating the minimal energy is
presented.
In Subsection 5.2 the nonhomogeneous linear control
systems are transformed into linear homogeneous ones,
with a null initial condition. For the control of such a
class of systems, one needs to consider the adjoint sys-
tem corresponding to the associated optimal transfer
problem (Subsections 5.1 and 5.3).
The above theory can be used for solving various
problems in spatial dynamics (rendez-vousspatial, satel-
lite dynamics, space pursuing), [3,4,8,9]. Also, this the-
ory can be successful applied to automatics, robotics and
artificial intelligence problems [16–18] modelled by lin-
ear control systems.
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