ON EXACT NULL CONTROLLABILITY OF DISTRIBUTED
SYSTEMS
B. Shklyar
Holon Institute of Technology, Holon, Israel
INTRODUCTION
The linear moment problem is defined as follows:
Given sequences cn , n  1, 2 ,
and x n  X , n  1, 2



,

find necessary and
sufficient conditions for the existence of linear functional g  X  such that


cn  x n , g , n  1, 2
(1)
, .
The moment problem (1) has many important applications. In this paper we present
applications of linear moment problem (1) for the investigation of exact nullcontrollability for linear evolution control equations.
PROBLEM STATEMENT
Let X ,U
be Hilbert spaces, and let

continuous C 0 -semigroups S t
A
be infinitesimal generator of strongly
in X
[1]. Consider the abstract evolution
control equation
x t  Ax t  Bu t , x 0  x 0, 0  t  ,

(2)

  
where x t  , x  X , u t  , u U , B : U  X is a linear possibly unbounded
operator, W  D A   X  V are Hilbert spaces with continuous dense injections
0
0
(see [2] for the description of spaces W and V ).
Let x t , x 0 , u 
be a mild solution of equation (2) with initial condition


 
x 0  x 0.
Definition Equation (2) is said to be exact null-controllable on 0, t 1  by controls


vanishing after time moment t 2 if for each x 0  X there exists a control

 


   0.
u   L 2 0, t 2  ,U , u t  0 a.e. on [t 2, ) such that x t 1, x 0, u 


THE ASSUMPTIONS
The assumptions on A are listed below.
1) The operators A has purely point spectrums  A with no finite limit points.
Eigenvalues of A have finite multiplicities.
2)
There exists T  0 such that
all mild solutions of the equation
x t  A x t are expanded in a series of generalized eigenvectors of the operator A


converging uniformly for any t  T 1,T 2  ,T  T 1  T 2.


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MAIN RESULTS
As a preliminary we consider the following:
1) The operator A has all the eigenvalues with multiplicity 1. .
2) U 
(one input case). It means that the possibly unbounded operator
is defined by an element b V , i.e. equation (2) can be written in
B :U 
the form
3) x t  Ax t  bu t , x 0  x 0, b V , 0  t  .


  
The operator defined by b V
is bounded if and only if b  X .
Let the eigenvalues j   A , j  1, 2,
of the operator A
be enumerated in the
j ,  j , j  1, 2,
order of non-decreasing of absolute values, and let
,
be the
eigenvectors of the operator A and the adjoint operator A  respectively.
Denote:



 




x j t  x t , x 0, u  , j , x 0 j  x 0, j , bj  b, j , j  1, 2, ..., .
Theorem 1. For equation (2) to be exact null-controllable on 0, t 1  , t 1  T ,
by


controls vanishing after time moment t 1  T , , it is necessary and sufficient that the
following infinite moment problem
x 0 j  
t1 T   
j
e
0

u   L
bj u  d , j  1, 2, ...,
with respect to
2
(3)
0, t  T  is solvable for any x 0  X
 1

.
SOLUTION OF MOMENT PROBLEM (3)
The solvability of moment problem (4) for each x 0  X essentially depends on the
properties of eigenvalues j , j  1, 2, ..., .

Definition. The sequence x j  X , j  1, 2, ...


x

is said to be minimal, if
x j  span x k  X , k  1, 2, ..., k  j .
It is well-known that the sequence
y
there exists a sequence
x
j
 X , j  1, 2, ...,
Let Gn 
n
i
 X , j  1, 2, ...,

 X , j  1, 2, ... ,
x , x  , i, j  1, 2, ..., n 
x , ..., x 
1

j
j
of above sequence. Denote by  nmin
hence any first
n
x
j

 X , j  1, 2, ..., n , ...
elements
is minimal if and only if
biorthogonal to the sequence
be the Gram matrix of
j
Each minimal sequence

n
first elements
the minimal eigenvalue of G n .
is a linear independent sequence,
x , ..., x , n  1, 2, ..., are linear independent, so
1
n
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 nmin  0, n  1, 2, ..., .
It is easily to show that the sequence
decreases, so there exists lim  nmin  0.
n 
x
Definition. The sequence
j

 X , j  1, 2, ..., n , ...

min
n

, n  1, 2, ..., n , ...
is said to be strongly minimal, if
min lim
min 0.
n n
Theorem 2. For equation () to be exact null-controllable on 0, t 1  , t 1  T ,


t
T
controls vanishing after time moment 1
, it is necessary, that the sequence
by
0, t 1 
, j 1, 2, . . . , n, . . .  #
e j b j, t  
is minimal, and sufficient , that:



1) the sequence e j bj , t  0, t 1  T  , j  1, 2, ..., is strongly minimal,



2)
 x , 
0
2
j
e
2 Re j t1
 , x 0  X .
j 1
SOLUTION OF MOMENT PROBLEM (1) IN THE CASE OF THE NORMAL
SEQUENCE OF THE EIGENVECTORS OF THE OPERATOR A
Denote by
max
n

the maximal eigenvalue of G n . It is easily to show that the

x
increases , so there exists lim  nmax  . .
sequence  nmax , n  1, 2, ...,
Definition The sequence
and lim 
n 
max
n
j

 X , j  1, 2, ...
n 
is said to be normal, if lim  nmin  0
n 
 .
Theorem 3. Let the sequence
 , j  1, 2, ..., of eigenvectors of the operator A
j
be
a normal sequence .
0, t  ,
t 1  T , by controls
 1
, it is necessary, that the sequence
For equation () to be exact null-controllable on
vanishing after time moment
e
j t

t1  T
bj , j  1, 2, ... is minimal,
and sufficient , that it is strongly minimal .
It is well-known [2] that both functional differential linear control systems and linear
partial differential control equations of parabolic and hyperbolic type can be written in
the form (1), so Theorems 1-3 can be applied for the establishing of exact null
controllability conditions for above classes of control distributed systems.
REFERENCES
1. E. Hille, R. Philips, Functional Analysis and Semi-Groups, AMS, 1957.
2. D. Salamon, Infinite dimensional linear systems with unbounded control and
observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987),
383–431.
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