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
Email: ima [email protected]
Abstract
We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear systems. 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 analysis 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 consideration [1,2]. Variational calculation offers methods
that are difficult to use in order to investigate the existence 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 condition), 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 controllable systems express the minimal energy through
the controllability operator [11–13]. Also, stability conditions for systems whose energy tends to zero in infinite 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 propose new methods for studying the optimal control for
exactly controllable linear system. The minimization of
time and energy in transfer problem is considered. The
Copyright © 2010 SciRes
minimization of the energy can be seen as being a particular case of the linear regulator problem in automatics. 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  A, B :


dx
 A x(t )  B u (t ), x(t 0 )  x0  H ,
(1)
dt
where H is a Hilbert space with inner product
  ,   and norm  .
A : D( A)  H  H is an unbounded operator on H
which generates a strong semi group of operators on
H , S (t ) t 0  e t A t 0 .
 
B : U  H is a bounded linear operator on another
Hilbert space U , for example B  L U , H  .
u : [0, ]  H is a square integrable function representing the system control (1).
For any control function u , there exists a solution
of (1) given by
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M. POPESCU
135
B  : X U ,
t
x(t )  e
tA

x0  e
(t  s ) A
B u ( s ) d s, t  0
(2)
0
The control problem is the following one:
Given t 0  I , x(t0 ) , z  X and the constant M  0 ,
let us determine u  U such that
So,
B  S  (t n   ) x  U .
u  M and
x(t1 )  z
(i )
(ii ) t1  inf t x(t )  x 
(3)
t1
F
(4)
(13)
 S (t   ) B u ( ) d  , x
1
1
t1
By writing
t1
 S (t
k
 t0 ) B d   Lk , k  1, n,
F  un , L 1 x  u1 , L 1 x  un , L 1 x 
(5)
tn
  ) B un ( ) d 
un , L n x .
Therefore,
t1
S (t ) being a continuous linear operator, it is, therefore, bounded. It follows that we have:
S (tn ) x(t0 )  S (t1 ) x(t0 )
(15)
or

x(tn )  S (tn ) x(t0 )  S (t n   ) B un ( ) d  
t0
(14)
the Expression (13) becomes
L n un , x  L1 u1, x  un , L n x  u1 , L 1 x
t1
(6)
tn
 S (t
  ) B un ( ) d  , x 
tn
which gives
n
n
t1
t0
 S (t
 S (t
t1
tn

(12)
For every x  X , let us evaluate the difference
Here, X and U are Hilbert spaces.
Theorem 1. The optimal time control exists for the
above formulated problem.
Proof
Let t n  be a decreasing monotonous sequence such
that t n  t1 . Also, let us consider the sequence
un ( )  , with u n  u (t n )  U . We have
x(t n )  S (t n ) x(t 0 )  S (t n   ) B u n ( ) d  ,
(11)
S  (t n   ) x  U .

(16)

F  un  u1 , L 1 x  un , L n  L 1 x 
t1

un ( )  u1 ( ), B S (t1   ) x d  
(17)
t0
n
  ) B un ( ) d   0
(7)
t1
If we consider the set of all the admissible controls

  u U
u M

(8)
then,  is a weakly compact closed convex set.
As   U is weakly compact, any sequence
(u )   possesses a weakly convergent subsequence to
an element u1   . Thus, for u1   and any
u    (the dual of  ), we have
lim ui , u   u1 , u 
i 
(9)
Also, we have
Copyright © 2010 SciRes

un ( ), B  S (tn   ) x  B  S (t1   ) x d 
t0
By using the properties of the operator S (t ) , one obtains
t1
F

un ( )  u1 ( ), B S (t1   ) x d  
t0
t1



(18)
un ( ), B S (t1   ) S (tn  t1 ) x  x d  .
t0
Because the sequence u n un converges weakly to
u1 , the first term in (18) tends to zero for n  
Since u1   , it follows that
u1  M
t1
(10)
(see(9)). On the other hand, S (t1   ) is a strongly
continuous semigroup and, hence, from its boundedness,
it follows that there exists a constant K  0 such that
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M. POPESCU
136

B  S (t1   ) S (tn  t1 ) x  x



K S (tn  t1 ) x  x
1) Hilbert spaces.
(19)
So, it follows that the second term in (18) tends to
zero as n   . The sequence  x(tn )  X is weakly
convergent to x(t1 )  z  X if, for any x  X   X ,
we have
lim x(tn ), x  x(t1 ), x ,
n
(20)
which gives
2) Spaces l p , Lp , 1  p   1.
3) If the Banach spaces U1 ,U 2 ,, U n are rotund,
then the product space U1  U 2  U n is rotund,
too.
4) Uniform convex spaces are rotund, but the converse implication is not true.
2.2.2. Uniqueness
We assume that u1 and u 2 are optimal,
ui  U , i  1, 2 . Therefore,
t1

z , x  S (t1 ) x(t0 )  S (t1   ) B u1 ( ) d  , x
t1
(21)

S (t1 ) x(t0 )  S (t1   ) B u1 ( ) d   z
t0
From (21), one gets
t1

S (t1 ) x(t0 )  S (t1   ) B u1 ( ) d   z

S (t1 ) x(t0 )  S (t1   ) B u2 ( ) d   z
t1
(22)
t0
t  [t0 , t1 ] .
t0
S (t1 ) x(t0 ) 
t1
1
 S (t   ) B 2 u ( )  u ( )d   z
1
1
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 U is said to be rotund if the following
equivalent conditions are satisfied:
a) if x1  x2  x1  x2 , it follows that there ex-
ists a scalar   0 , such that x2   x1 ;
b) each convex subset K  U has at least one element that satisfies
uK,
zK;
c) for any bounded linear functional f on U
there exists at least an element x   such that
x, f  f ( x )  f ;
d) each x    is a point of extreme of  .
Examples of rotund spaces:
(25)
2
t0
It follows that u1 ( ) , u2 ( ) , 1 2 u1 ( )  u2 ( ) 
are optimal. By using Theorem 2, we have
u1 ( ), u2 ( ),
Copyright © 2010 SciRes
(24)
By adding Equations (23) and (24), one obtains
and, hence, u1 is the optimal control.
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 u (t ) is
the control of optimal time in the problem formulated in
(2.1.1), ([2, 10]) it follows that u (t )   , for almost all
u  z ,
(23)
t0
uk  1,
1
u1 ( )  u2 ( )     
2
1
u1  u2  1.
2
Since the condition (i) is satisfied, U is a rotund
space and u1   u 2 . Thus,
u1  u2  1  

u2  2
(26)
and, therefore
  1  u1  u2
(27)
which ends the proof of the uniqueness.
3. Minimum Energy Problem
3.1. Problem Formulation
Let  A, B (see(1)) be a controllable system with finite
dimensional state space X [6,10,12, 14].
Let I  [0, t1 ] , x(0)  a  X , b  X be given. Let U be
the Hilbert space L 2 ( I ,U ) . The minimum norm control
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M. POPESCU
problem can be formulated as follows: determine
u (t )  U such that for some t1  I ,
1) x(t1 )  b ,
2)
is minimized on [0, t1 ] , where  represents
u
the norm on L 2 ( I ,U ) .
Let us fix t1  0 . Consider the linear operator
Ln : L 2 (0, t1 ; U )  H1
137
ei
Since
f i , u  0, i  1,, n
are independent,
and, so, M  ker L t . L t maps M bijectively to the
range of L t and hence L tM is an injective mapping
into H . Let
u  u1  u2 , u1  M , u2  M  , u2  0 .
(28)
(36)
Because L t u2  0 ,
L t u  L tM u1  x  H
defined by
t1

Ln u : S (t1  s ) B u ( s ) d s
(29)
We have
u
x(t1 )  S (t1 ) a  Lt1 u  b.
(30)
Theorem 3. Let L t : U  H be a linear mapping
between a Hilbert space U and a finite dimensional
space H [11]. Then, there exists a finite dimensional
space M  U such that the restriction L tM of L t
to M is an injective mapping.
Proof
Let  ei  i  1,, n , be a basis for the range of L t
in H . Given any u  U , L t u can be written as
Lt u 
u1 , u2  0 , we have
Since
0
n
 e ,
(31)
i i
 u1  u 2 , u1  u 2  u1
2
2
 u2
 
1
(38)
x exists and
u1  L tM x is the minimum norm element satisfying
Lu  x.
Remark 1. The unique solution u (t ) of the equa-
tion L u  x , with minimum norm control, is the projection of
u (t ) onto the closed subspace
M   f1 , , f n  .
Hence, from (30), there exists a control
u (  ) L 2 (0, t1 ; U ) transferring a to b in time t1 if
and only if b  S (t1 ) a   Im L t1 . The control which
achieves
this
t1
fi  U   U .
 i  fi , u
2
Then, a unique minimum norm L tM
i 1
Et1 
where each  i can be expressed by
(37)

u (s)
2
and
minimizes
the
functional
d s , called the energy functional, is
0
u  L t1 b  S (t1 ) a 
(32)
(39)
1
Define the linear operator
Then,
Lt u 
t
n

f i , u ei .

Q t  S (r ) B B S (r ) d r , t  0
(33)
i 1
0
f i are linearly independent and generate a n dimensional subspace M  U .
From the properties of Hilbert spaces (the projection
theorem), it follows that U  M  M  .

Let u  M . Then,


M   u U L t u  0  ker L t .
the Equation [1,13]
d
Q t x, x  2 Q t x, x  B x
dt
(34)
Let u  ker L t . We get
n

f i , u ei .
(35)
2
(41)
x  D( A ) , Q 0  I
where
i 1
Copyright © 2010 SciRes
We have the following results (see [10–12]):
Proposition 1.
1) The function Q t , t  0 , is the unique solution of
f i , u   i  0 . Thus,
Lt u  0 
(40)

D ( A )  y  H  C  R  , A x, y
C x

,  x  D( A) .
H
H

(42)
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M. POPESCU
138
2) If A generates a stable semigroup, then
lim Q t  Q
(43)
t 
In order to make Equation (48) true, u (t ) must be
chosen on the interval [t0 , t1 ] . Consider the i-th component ei of the vector e :
exists and is the unique solution of the equation
2
2 Q A x, x  B  x
 0,
t1
x  D ( A ) ,
(44)
The proof of Proposition 1 is given in [10,11].
The following theorem gives general results for the
functionals E (a, b) , the minimal energy for transfer-
ring a to b in time t1 , and E (0, b) , where
a, b  H , t1  0 (see [12]).
Theorem 4.
1) For arbitrary t1  0 and a, b  H
 
1 2 1
E (a, b)  Q
 
1
2
b
where i is the row of the matrix  and f is the
unique functional corresponding to the inner product
(Rieszrepresentation theorem).
Let i be an arbitrary scalar. Then,
i ei  i f ( i )  f (i  i ) ,
n

2
(45)
.
  A, B 
is null
controllable in time t0  0 , then
E (0, b)  Q 1 2
, bH
(46)
Q 
2
b
 
C t0  Q
 E t1 (0, b) 
1 2 1
2
b
n
 f (  ) 
i
n

n
i  i , u
 u
u
p


f



n
i
i
i
,
i
i 1
q
(53)
i
 
i
i 1

e  b  S (t1   ) B u ( ) d  
(48)
 
i

i
q
(54)
.
n
t1
i 1
n
i 1
q
, e
is minimized for p  [1, )
Now,

n
 
 e
i i

i 1
i 1
Copyright © 2010 SciRes
 
n
   
i
1) x(t1 )  b ,
1
p
From Equations (51) and (53),
u(t )  U such that
 (t   ) u( ) d 
(52)
.
i
1 1

   1 .
p q

I  R [9,10,13]. Given x0  0 , t0 , t1 , b , determine
t0
i
i 1
i 1
1
t0
  ,u
By Hölder’s inequality,
, b  H , t1  t0
Let  A, B be a dynamic system with U  R m , X  Rn ,
t1
(51)
n

(47)
, e
i
i 1
 n

f  i i  


 i 1

4.1. Presentation of the Numerical Methods
up
(50)
 being a vector with arbitrary components 1,, n .
If Equation (51) is true for at least n different linearly
independent, i , i  1,, n then
Equation (48) is also true.
We have
4. Numerical Methods for Minimal Norm
Control
2)
i ei 
i 1
3) Moreover, there exists C t 0  0 such that
1 2 1
(49)
t0
n
Since R is a Hilbert space, the inner product
S (t1 ) a  b 
2) If S (t ) is stable and the system

ei  i (t1   ) u ( ) d   i , u  f (i )
i
q
Every control driving the system to the point b 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
a n vectors  such that the right-hand side of Equa-
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M. POPESCU
139
tion (54) takes its maximum.
b  , u
The relation (63) becomes
4.2. A Simple Example
u (t )  k sign ( (t ))  (t ) .
We consider a single output linear dynamic system
 A, B , where
U  R 1,
(64)
Then,
t1

.
(66)
t0
Because t1  I fixed, then S (t1   ) B   ( )
Thus,
t1

2
b  k  2 ( ) d   k 
X  R 1, I  R 1
e  b  ( ) u ( ) d  .
(65)
b
k
(55)
t0
(67)
2

and the optimal control is given by
Therefore,
t1
e b

t1
 ( ) u ( ) d  
t0

 ( ) u ( ) d 
b  (t )
u (t ) 
(56)
2

(68)
.
t0
and from Hölder’s inequality,
5. Exact Controllability
t1
 ( ) u( ) d  

(57)
u ,
5.1. Adjoint System
t0
where we have assumed that  (t )  L 2 ( I ) . The minimum norm control, if exists, will satisfy
e
u 

(58)
.
Let S (t ), t  [0, t1 ] , be the fundamental solution of an
homogeneous system associated to the linear control
system  A, B (see(1)) [4,5,13] .
Thus, we have
d S (t )
 A S (t ),
dt
From (56) and from (57), we have
sign (u (t ))  sign ( (t )) ,  t  [t0 , t1 ]
(59)
From

 (t )
 h u (t )
p
,
(60)
 t  [t0 , t1 ] ,
q p
d 1
S (t )  S 1 (t ) A .
dt
 k  (t )
q p
(61)
which implies that
S
or
u (t ) sign (u(t))  k  (t )
q p
(62)
.
From condition (59),
u (t )  k sign ( (t ))  (t )
q p
(70)
Since
1
S
1
  A S


(t )
   S (t ) 
(t )

1
(t )
1

(71)
.
(72)
.
(73)

the system becomes
.
(63)
By substituting Equation (63) into Equation (55), the
constant k can be determined.
4.2.1. The Particular Case p  q  2
In this case, the Equation (55) represents the inner
product in L 2 (t0 , t1 ;U ) ,
Copyright © 2010 SciRes
(69)
one obtains
h arbitrary constant.
The control u satisfies the relation
u (t )  h 1 p (t )

d
dI
 0.
S (t ) S 1 (t ) 
dt
dt
and, respectively,
q
S (0)  I , t  [0, t1 ]




1
d  1
S (t )   A S  (t ) , t  [0, t1 ]
dt

(74)

1
It follows that S  (t ) , t  [0, t1 ] , is the fundamental
solution for the adjoint System (69).
5.2. A Class of Linear Controllable Systems
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M. POPESCU
140
We consider the class of linear control systems in vectorial form [1,3,4,8,13,15]
dx
 A x  B u  C , x(0)  x0
dt
(76)
By this transformation, system (75) becomes
dx
 A x  B u  B u  C .
dt


(77)
Thus, one transfers the origin of the coordinates of
space Em in u  . At the same time, the origin of the
coordinates of space Em is an internal point inside the
domain U . Let us denote by x  (t ) the solution of system (75) which corresponds to the control u  0 .
This control is admissible, as the origin of the coordinates belongs to the domain U and satisfies the initial
condition x (0)  x0 . As a result,
d x (t )
 A x  (t )  C .
dt
(78)
Let u1 (t ), 0  t  t1 , be any control and x1 (t ) be the
trajectory corresponding to system (75).
We have
d x1 (t )
 A x1 (t )  B u1 (t )  C , x1 (0)  x0 .
dt
(79)
We take
x (t )  x1 (t )  x  (t )
J  F x(t1 )  
(75)
Let u0 be any internal point of the closed bounded
convex control domain U .
This domain contains the space Em of variables
u  (u1,, um ) . We take
u  u  u
that extremises the functional of final values
(80)
and satisfies the differential constraints represented by
the system (81).
The adjoint variable y  R satisfies equation
y  
H
x
(83)
where H is the Hamiltonian associated to the optimal
problem
H  y, x
(84)
Since the final conditions x(t1 ) are free and the final
time has been indicated from the condition of transversality, one obtains y (t1 )  yt1 .
It follows that the adjunct system becomes
 y   A y
 A, B : 
 y (t1 )  yt1
(85)
with solution
y (t )  S  (t1  t ) yt1 ,
yt1  H .
(86)
Proposition 2.
For the class of optimum problems under consideration, the following identity holds true:
t1
xt1 , yt1
H
 u, B


y
U
0
Proof
Assuming that u  C1  0, t1 ,U

(87)
dt
and y (t1 )  D( A )
it follows that x, y  C1  0, t1 ,U  .
Integrating by parts, since x(0)  0 , y (t1 )  yt1 ,
B : U  H , B : H U , we have
0
 x  A x  B u, y
dt 
H
0
t1
Hence, for u  u1 (t ) , the system (75) becomes
(81)
Thus, the linear control system belongs to the class
 A, B defined by (81).
5.3. Optimal Control Problems
For a given t1 , let us determine the optimal control u~
Copyright © 2010 SciRes
(82)
i i 1
t1
d x (t )
 A x (t )  B u1 (t ) .
dt
x  Rn ,
 c x (t )
i 1
Then, from (78) and (79), one obtains
 x  A x  B u
 A, B : 
 x(0)  0
n
t1

x, y

u , B y

x,  A y
0
t1
0
t1
0
t1

A x, y
H
dt 
0
 u, B
0
dt 
H

y
U
U
d t  x, y
t1
H
dt 

0
t1
0

x, A y
(88)
H
dt 
dt
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M. POPESCU
From (93) and (94), for xt1  yt1 , we have
One obtains
t1
xt1 , yt1
H

141
 u, B

y
U
0
dt 0
(89)
The identity has been demonstrated.
This result can be extended for arbitrary
u  L 2 (0, t1;U ) and y (t1 )  H .
An important result referring to the exact controllability of the linear system (81) is stated in
Theorem 5. The system  A, B is exactly controllable
yt1

B S  (t ) y0
0
2
U
,
(90)
Proof
“  ” We assume that  A, B is exactly controllable.
Let u  L 2 (0, t1;U ) and y (t1 )  H .
We consider that the application
(91)
t1
restriction
1
ker L t1

Since
L t1 u  x(t1 )  u  L t11  x(t1 )    x(t1 )  it
follows that transfers 0 in x(t1 ) for the system  A, B .
We choose y (t1 )  H and x(t1 )  y (t1 ) , u    x (t1 )  . It
follows that
yt1
t1
2
 yt1 , yt1 
H
  (x
t1 ), B

y
0
U
dt
(93)
For  ( xt1 )  u  L 2 , B y  L 2 B, using Hölders’s inequality, one obtains
1



  ( xt )B y d t    ( xt ) B y  d t 
1
12
t1
  (x
t1 )
2
dt
0
Copyright © 2010 SciRes
t1

12
By
0

2
dt
(95)
12
2

2
yt1
(96)
H
y0 becomes
yt1
and
U
2
yt1
2
t1
U
0
c yt1
H
2

dt  BS(t1 t) y0 dt 
(97)
2
H

B y
2
U
d t  c yt1
For any yt1  H we take the set
2
(98)
H
 u(t )  B

y (t )


( y (t ) is solution of  A, B ).
We consider the solution x(t ) for  A, B which corresponds to the above mentioned control u (t ) . Let us
define the bounded operator


 : yt1  H  x(t1 )  L t1 B  y (  )  H
(99)
Since u (t )  B y , the identity (89) becomes
t1
H


B  y (t )
0
2
U
d t  c yt1
2
H
(100)
Therefore, there exists a constant c  0 for which
1
0

The direct implication has been demonstrated.
“  ” We assume that condition (90) is fulfilled.
Then
t1
0
0
0
 yt1 , yt1
t1
2
0
is injective.
1
dt 
U
 B S (t)y
 
(92)

12
S  (t ) is transformed into S  (t1  t ) .
Therefore, we get one equivalent relation of controllable system in which yt1 substitutes y0 .
t1
 L t M   L t  
2
B y
By changing t  t1  t 
is well defined.
Let  : H  L 2 (0, t1;U ) be the inverse of L t1 . From
Theorem 3 it follows that there exists a finite dimensional subspace M  H , M   ker L t1 such that the


2
d t 
U



d t 
U


2
or
t1
L t1 : U  x(T )
 t1
 B y

0
H
0
 y0  H
1
yt1

2
U
d t  c y0

H

if only if the following condition is satisfied
t1
 t1
  ( yt1 )  B y

0
2
(94)
(100) is satisfied. It follows that
 yt1 , yt1
H
is posi-
tively defined.
This resumes the conclusion that the system  A, B is
controllable.
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M. POPESCU
142
Thus, since  is inversable, for any xt1  H , state
yt1   1 ( xt1 )
is such that there exists a control
Sivasundaram), Cambridge Scientific Publishers 2, pp. 135–
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
u (t )  B y which transfers 0 in yt1 . The theorem has
been demonstrated.
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 controllable linear systems.
A necessary and sufficient condition of exact controllability 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 system 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, satellite dynamics, space pursuing), [3,4,8,9]. Also, this theory can be successful applied to automatics, robotics and
artificial intelligence problems [16–18] modelled by linear control systems.
7. References
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[11] M. Popescu, “Stability and stabilization dynamical systems,” Technical Education Bucharest, 2009.
[12] G. Da Prato, A. J. Pritchard, and J. Zabczyk, “On minimum energy problems,” SIAM J. Control and Optimization, Vol. 29, pp. 209–221, 1991.
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[14] F. Gozzi and P. Loreti, “Regularity of the minimum time
function and minimum energy problems: The linear
case,” SIAM Journal on Control and Optimization, Vol.
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