Existence of the Solution of a Discrete Optimal
Control Problem for Linear Concentrated Systems
with Lions Special Quality Test
Vuqar Salmanov1, Nureli Mahmudov2
Nakhchivan State University, Nakhchivan, Azerbaijan
[email protected], [email protected]
1
Abstract— The paper deals with the existence of the solution of a
discrete optimal control problem for linear concentrated systems
with special quality test of Lions functional type.
Keywords— optimal control; discrete problem, quality test, liner
concentrated systems
I.
STATEMENT AND DISCRETIZATION OF THE
PROBLEM
Consider an optimal control problem on minimization of
the functional

 J u  
Bt   bik t 
x0 , u   xT , u  L n  0, T  
2

2
on the set
a
matrix
of
nm ,
order
f t    f1 t , f 2 t , ..., f n t  is a vector-column. We’ll
T
assume that
f  L2n  0, T  , aij  L 0, T  , i, j  1, n ,
_____
_____
bik  L 0, T  , i  1, n , k  1, m .
At first, consider difference approximation of problem (1)(3). To this end, partition the segment 0, T into N parts by

the points

_______


tk , k  0, N  :


0  t0  t1  ...  t N 1  t N  T , having accepted these points
as nodes, we replace equation (2) by difference equations with
the help of the Euler’s simplest explicit diagram. As a result,
we get the following discrete optimal control problem on
minimization of the function
N 1
 N    x0 k  xT k
 I N  u
k 0
2
tk 

on the set

U  u  u t  : u  L2m  0, T , u
L2m  0 , T 

 b0 
U N  u N : u N  u0 , u1 , ..., u N 1 , uk  R m ,
 N 1 2

k  0, N  1,   uk m tk 
 k 0

under the conditions:


is
_____
INTRODUCTION
We investigate the convergence of the difference method
for the solution of an optimal control problem for linear
concentrated systems with a special quality test of Lions
functional type. Note that a lot of papers have been devoted to
the difference method for the solution of optimal control
problems with concentrated systems. However, the problem
under consideration considerably differs from the ones studied
earlier. Therefore, study of convergence of the difference
method in the present case both is of theoretical and practical
interest.
II.
T  0, b0  0 are the given numbers, x0 , xT are the
given vectors, At   aij t  are the matrices of order n  n ,
where
__________

x p t   At x p t   Bt u t   f t 
p  0, T , 0  t  T ,
 x0 0 
x0 , xT T   xT 


2
 b0 
under the conditions
x0

1
k 1


 x0  tk Ak x0  Bk uk  f k 
k
k

m 
__________

 k  0, N  1 

xT
k 1




 xT  tk Ak xT  Bk uk  f k 
k
k
The space L2 N is a difference analogy of the space
_______


L2m  0, T  corresponding to the partition tk , k  0, N 


of the segment 0, T  .
Thus, problem (1)-(3) considered in the space
__________
 k  N  1, 0  


 x0
0

 x0 , xT  xT  
N


for each entire
For each
aijk 
1
tk
t k 1
k
 aij t dt , bir 
tk
1
tk
t k 1

0  k  N 1
t k 1
 bir t dt , 
M  const  0 .
tk
Then the inequality holds for:


max x p
0 k  N
where
__________ 
______
 x0
______
difference diagram (5)-(7) for
uN  u0 , u1, ..., uN 1  , vN  v0 , v1, ..., vN 1 

k 1
k


 x0   t j Aj x0  B j u j  f j 

xT  xT 
k
N 1
2N

j 0
j
 k  0, N  1 


with
the scalar product
m 
p  0, T 
__________
m 
 N , vN  L
u N  n  c0 ,
uN U N .
Introduce a space L2 N of discrete control functions
 u
u N U N
c0  0 is a constant independent of k, N .
tk
k
k
T
M,
N
Proof. Using (5) and (6) we have:
f i t dt , i  1, N , k  0, N  1
x p  x p u N , p  0, T , k  0, N is the solution of
k
consider difference diagram (5)-(7).
f  L2n  0, T  . Let furthermore, d N  max tk 
f k   f 1k, f 2 k, ..., f nk ,
1
f ik 
tk
uN U N
Theorem 1. Let the elements of the matrices
k
j  1, n , r  1, n
m 
At , Bt  be
bounded and measurable functions on the segment 0, T  ,
 
tk  tk 1  tk  Ak  aij  Bk  bir 
k
corresponds to discrete optimal control
problem (4)-(7) considered in the space L2 N .
where
 
_______


N  1 and partition tk , k  0, N  of the


0, T 
segment
L2m  0, T 
  tk  uk , vk  R m 
 t A
N
j  k 1
j 1

x  B j 1u j 1  f j 1 
j
j 1 T
k 0
__________

and with the norm
 N
 u
L2mN
1
 k  N  1, 0  


From equality (10) it is easy to get validity of the inequality

2 
   tk uk m  
 k 0

N 1

2
x0
k
 n Amax d N  x0
k 1
n
j
n
j 0

k
Theorem 2. Let the conditions of theorem 1 fulfilled.
Furthermore, the elements of the matrices Ak , Bk and vector-


k
 n Bmax  u j t j   f j t j  x0 n
m
j 0
n
j 0
__________
Proof. At first prove the continuity of function (4) on the set
U N . To this end take any control u N U N and give it the
increment hN such that u N  hN  U N .
 k  0, N  1 
where
Amax  vrai max At  ,
t0, T 
 Bmax
Let
 vrai max Bt  
t 0 , T 
k
k 1
n
 n Amax d N  x0
j 0
 nT  f N
L2mN
j
n
 nT Bmax u N
L2mN


 x0 n ,
k  0, N  1  



 
where f N   f 0 , f1 , ..., f N 1  . Applying the discrete
analogy of the Gronwall lemma [1, p.110], we get validity of
the estimation:

 x0
k
n

 c1 x0 n  u N
for k  0, 1, ..., N , where
of
L2mN
  f N
L2mN

c1  0 is a constant independent
N
p
N
, p  0, T be the solution just
u N  hN U N . Then
x   x u  h   x u 
p N
N
p
N
N
p
N
N

will be the solution of the following system:
x0
k 1


 x0  tk Ak x0  Bk hk 
k
k
 k  0, N  1 
xT
k 1



 xT  tk Ak xT
k
k 1

 Bk hk 
__________
 k  N  1, 0 
 x0
0


 0, xT  0  

N
k.
Using inequality (11), similar to obtaining estimation (14),
we can prove validity of the estimation:
 xT
k
n

 c2 xT n  u N
for k  0, 1, ..., N , where
of
p N
__________
__________

x   x u 
of this system for
Using the Cauchy-Bunyakovsky inequality, from the last
inequality we get:
x0
column f k be defined by formulae (8). Then discrete optimal
control problem (4)-(7) has at least one solution.
L m 
2N
  f N
L m 
2N
We can write this system in the form
  
x0
c2  0 is a constant independent
k 1
j 0
Now we’ll study the existence of the solution of discrete
problem (4)-(7).
j
__________
Thus, from estimations (14), (15) we get validity of
theorem’s statement. Theorem 1 is proved.
EXISTENCE OF THE SOLUTION OF DISCRETE
OPTIMAL CONTROL PROBLEM

  t j Aj x0  B j h j 
k  0, N  1  
k.
III.

k
xT
k 1

 t A
N
j  k 1
j 1



xT  B j 1h j 1 
j
j 1
__________
k  0, N  1  


From these equalities we can easily get validity of the
inequalities:
N 1
  t k x0

x0
k
 n Amax d N  x0
k 1
n
j
n
j 0
k 0
N 1
where
m
__________
 n Amax d N
 n Bmax
N

j  k 1

k


k
______
x p , k  0, N , p  0, T is the solution of system
k
uN U N , and x p k , k  0, N , p  0, T
______
 k  0, N  1  
n
n
k 0
(1.5)-(1.7), for
k 1
k 0
k 2

k
xT
n
N 1
  t k xT
 2  x0 , xT  Rn t k

 n Bmax  h j t j ,
j 0
k 2

N

j  k 1
xT

j
n

is the solution of system (16)-(18).
By estimations (9), (23) and the Cauchy-Bunyakovsky
inequality, from (24) we get validity of the inequality:

I N u N   c4 hN


 hN
L2mN
2
L2mN
 
c4  0 is a constant independent of hn . This
inequality implies the continuity of the function I n u n  on
any element u N U N i.e.
h j 1 t j 1 ,
where
m
__________
k  0, N  1  


 N   0  hN
 I N  u
Applying in these inequalities the Cauchy-Bunyakovsky
inequality and the discrete analogy of the Gronwall lemma, we
get validity of the estimations
By arbitrariness of the element
follows the continuity of
 x p
where
k
n
 c3 hN
______
L m 
 k  0, N  p  0, T  
2N
c8  0 is a constant independent of k and hn .
Now consider an increment of function (1.4) on the element
u N U N . By means of the formula of this function we
have:

I N u N   I N u N  hN   I N u N  

N 1
 2  x0  xT , x0  xT  R n tk 
k 0
k
k
k
k
L2mN
 0  
uN U N
I n u n  on the set U N .
from (26) it
From the structure of the set U N it is clear that it is a
convex, closed and bounded set in a finite-dimensional space
L2mN . It is known that such a set is compact in L2mN . By



I n  u n  proved on this set, I n  u n   0 ,  u N U N is
also continuous. Thus, all the conditions of the Weierstrass
theorem are fulfilled [2,3]. By the same theorem, the function
I n  u n  achieves its lower bound on the compact set U N ,
i.e. the discrete optimal control problem (4)-(7) has at least one
solution. Theorem 2 is proved.


REFERENCES
[1]
[2]
[3]
F.P. Vasil’ev. Solution methods for extremal problems. Moscow, Nauka,
1981 (Russian)
F.P. Vasil’ev. Numerical solution methods for extremal problems.
Moscow, Nauka, 1979 (in Russian)
A.D. Iskandarov, R.K. Tagiyev, G.Ya. Yagubov.
Optimization
methods. Teaching book, Baku-Chashioglu, 2002, 400 p. (in
Azerbaijani)