Control of Loaded Parabolic System with Minimal
Energy
Rashad Mammadov1, Gasim Gasimov2
1,2
Azerbaijan State Oil Academy, Baku, Azerbaijan
[email protected]
Abstract— The problem of control of loaded parabolic system
with minimal energy is considered in the work. The statement of
the general solution to the considered problem and the statement
of the problem in infinite-dimensional phase space are
investigated. The solution to the problem on control with
minimal energy is given in the work.
II.
Each concrete admissible control determines the unique
solution. This solution can be formulate in the form of Fourier
series on nonsingular functions of the boundary problem
X ( x)   2 X ( x)  0, 0  x  1;
X (0)  0, X (1)  X (1)  0.
Keywords— minimal energy; loaded parabolic system; general
solution; infinite-dimensional phase space; control.
I.
INTRODUCTION
Let the state of the controlled object is described by
the function u (t , x) , which satisfies the equation
1
(1)
k 1
(2)
and
formula

G ( x,  , t )   e  a  t X n ( x) X n ( ),
0
u ( x) are the given
p(t , x) is a control parameter.
We will assume that all the functions in L2 (Q) are the
functions in L2 (0,1), but
admissible controls.
The problem of control with minimal energy consists of
determining of such control from the class of admissible
controls that, the corresponding solution to the problem
satisfies the condition
u(T , x)   ( x)
(7)
m

   G ( x,  , t   )  bk ( )u ( , x k )  p( ,  ) dd ,
 k 1

0 0
where G( x,  , t ) is the Grin function and is determined by the
t 1
u x (t ,0)  0, u x (t ,1)   u x (t ,1)  0,   const  0 (3)
on the border of this oblast. Here 0  x1  x 2  ...  x m  1 are
the known phase points, bk (x)
u ( x, t )   G ( x,  , t )u 0 ( )d 
0
in the oblast Q  0  x  1, 0  t  T  and satisfies the
initial and boundary conditions
u(0, x)  u 0 ( x),
(6)
The system of functions X n (x )generates the complete
system of orthonormal functions. So using the known method
we can find the following expression for the solution of the
problem (1)-(3):
m
u t  a 2 u xk   bk ( x)u (t , x k )  p(t , x)
STATEMENT OF THE GENERAL SOLUTION TO THE
PROBLEM (1)-(3)
(4)
2 t
n
n 1
Here  n are the nonsingular values of the boundary problem
(6).
We’ll obtain the system of integral equations
corresponding to the functions u i (t )  u (t , x i ) , if consider the
values x  x i , i  1,2,..., m in the equalities (7):
t
m
u i (t )    Rik (t ,  )u k ( )d   i (t ), i  1,2,..., m, (8)
0 k 1
where
t
Rik (t ,  )   G ( x i ,  , t   )bk ( )d ,
and the functional
I [ p]  p
2
L2 ( Q )
0
(5)
takes on it’s minimal admissible value, where  ( x)  L2 (0,1)
is a given function.
t
t 1
0
0 0
 i (t )   G ( x i ,  , t )u 0 ( )d    G ( x i ,  , t   ) f ( ,  )dd .
If we denote
R(t , )  Rik (t , )
The class of admissible controls is the class of infinite
dimensional vector functions p(t )  ( p1 (t ), p 2 (t ),...) , which
____
j 1, m
____
,
1, m
T
u (t )  u i (t )i 1, m ,  (t )   i (t )i  1, m,
____
satisfy the following condition
then we can write the system (8) in the vector form:
0
and which solution we can organize in the following form:

u (t )   v n (t ); v 0 (t )   (t ),
IV.
THE SOLUTION TO THE PROBLEM ON CONTROL WITH
MINIMAL ENERGY
n 0
If we’ll denote
t
v n (t )   R(t ,  )v n 1 ( )d , n  1,2,...
y (t )  (u1 (t ), u 2 (t ),...), p(t )  ( p1 (t ), p 2 (t ),...),
0
Consequently, the solution to the problem (1)-(3) is
determined by the formula (7).
STATEMENT OF THE PROBLEM IN INFINITEDIMENSIONAL PHASE SPACE
To solve the problem with minimal energy let’s formulate
it in infinite dimensional phase space. So we’ll find the
solution to the problem (1)-(3) in the following form:

y 0  (u10 , u 20 ,...), y 1  ( 1 , 2 ,...),
  a 2 12  b1
b2
... 


A  b2  a 2 12  b1
... ,
 ...
...
... 

then we can write the problem (9)-(10) in the vector form:
u (t , x)   u n (t ) X n ( x),

n 1
1
u n (t )   u (t , x) X n ( x)dx.
Then to determine the coefficients u n (t ) we’ll obtain the
infinite system of ordinary differential equations

y (t )  Ay (t )  P(t ),
(13)
y(0)  y
(14)
y(T )  y 1 .
(15)
We can represent the solution to the problem (13)-(14) by
Cauchy formula in the following form:
__
t
(9)
y (t )  e At y 0   e A( t  ) p ( )d .
(10)
We’ll obtain that the control function which satisfies the
torque condition
n 1
0
with initial conditions
u n (0)  u n0
m
here
0
But the final condition takes on form
0

(t )dt   .
So, it is required to find such admissible control that,
corresponding to this control the solution to the problem (9)(10) satisfies the final condition (11) and in addition to this the
functional (12) gets minimal value.
t
u n (t )  a 2  2n u n (t )   b n u n (t )  p n (t )
2
n
0
u (t )   R(t , )u ( )d   (t )
III.
p
 bkn X n ( x k ) a u n0 , bkn , Pn (t ) are the coefficients
k 1
T
e
A (T  t )
p (t )dt  c
(16),
0
of Fourier functions u 0 ( x), bk ( x), p(t , x) .
The final condition (4) takes on the following form:
u n (T )   n
(11)
where  n are the coefficients of the Fourier functions  (x) .
In addition to the functions X n ( x), n  1,2,... are generate
the complete system of orthonormal functions, the functional
takes on the form :
T

I [ p ]    p n2 (t ) dt
0 n 1
(12)
if we consider condition (13), where c  y 1  e AT y 0 .
We can write the expression (16) in the following form
(hi , p)  ci ,
i  1,2....
(17)
A (T  t )
if we’ll denote i-th row of the matrix e
by hi (t ) .
Consequently, it is required to find such control
among the admissible controls that, this control will be
satisfied the condition (17) and in addition to this its norm will
be minimal.
The control p(t ) satisfying this condition we’ll find
in the following form:

p(t )    i hi (t ).
(18)
i 1
As well as this function satisfies conditions (17), then the
vector   ( 1 ,  2 ,...) is the solution to the infinite system of
equations
M  c
(19)
where the elements of the matrix M is determined in the
following form:
M  hi , h j i, j  1 .


REFERENCES
[1]
[2]
[3]
Egorov A.I. The bases of control theory, M.:phys.math.lit., 2004, 504
pp.(in Russian)
Aida-zade K.R., Abdillayev V.M. Numerical solution to optimal control
problems for loaded concentrated systems //The journal of
computational mathematics and mathematical physics, Moscow, Russia,
2006, Vol 46, No9.
Mamedov R.S. Construction of the general solution to loaded equation
of parabolic type / The 4th Congress of the Turkic World Mathematical
Society, Baku, Azerbaijan, 2011.