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Necessary Optimality Conditions of First and Second
Orders in the Problem of Control of Difference
Analogy of Barbashin Type Integro-differential
Equation
Zhale Ahmadova1, Sevinj Qadirova2, Kamil Mansimov3
1,2
Cybernetics Institute of ANAS, Baku, Azerbaijan
1,2,3
Baku State University, Baku, Azerbaijan
[email protected]
Abstract –An optimal control problem described by difference
analogy of Barbashin type integro-differential equation is
considered. Necessary optimality conditions of first and second
orders are proved.
Key words –
Barbashin type, difference analogy, necessary
optimality conditions, conjugated variables.
I.
INTRODUCTION
In our prevous papers [6] we have studied optimal
control problems described by a system of Barbashin type
integro-differential equations.
In this paper, we study an optimal control problem
described by difference analogy of Barbashin type integrodifferential equations. Necessary optimality conditions of
first and second orders are obtained.
II. STATEMENT OF PROBLEM
Assume that the admissible control in the “discrete
rectangle”
D  t , x  : t  t 0 , t 0  1,..., t1 ; x  x0 , x0  1,..., x1 
z t  1, x  

 K t , x, s, zt , s   f t , x, zt , x , ut 
,
(1)
s  x0
with the initial condition
z t 0 , x   ax , x  x0 , x0  1,..., x1.

 

(2)
Here K t , x, s, z , f t , x, z, u are the given n dimensional vector-functions continuous in totality of
variables together with partial derivatives with respect to z ,
z, u respectively, t 0 , t1 , x0 , x1 are the given numbers,


t1  t0 , x1  x0 are natural numbers, ax  is a
given n -dimensional discrete vector-function, ut  is an
moreover
ut   U  R r , t  t 0 , t 0  1,..., t1  1.
(3)
Such control functions are called admissible
controls.
On the solutions of problem (1)-(2) generated by all
possible admissible controls we determine the functional
S u  

x1
 z t , x  .
x  x0
1
(4)
Here
 z is the given twice continuously
differentiable scalar function.
The admissible control u t giving minimum to
functional (4) under constrains (1)-(3) is called an optimal
control, the appropriate process u t , z t , x an optimal
process.

    
III. NECESSARY OPTIMALITY
CONDITIONS
    
Assuming u t , z t , x
a
process, we introduce the denotation
is described by the system of difference equations:
x1
r -dimensional discrete vector of control actions with values
from the given non-empty, bounded and open set U , i.е.
fixed
admissible
H t , s, z, u,   f t, x, z, u ,
H u t   H u t , x, zt , x , ut , t ,
H z t   H z t , x, zt , x, ut , t ,
H uu t   H uu t , x, zt , x , ut , t ,
H zz t   H zz t , x, zt , x, ut , t ,
H uz t   H uz t , x, zt , x , ut , t .
Here    t, x  is an n -dimensional vector-
function of conjugated variables being a solution of the
conjugated problem
 t  1, x  
K t , s, x, z t , x 
 t , s  
z
s  x0
  t0 , t0  1,..., t1  1;   x0 , x0  1,..., x1.
x1

(5)
f t , x, z t , x , u t 
 t , x ,
z
 z t1 , x 
(6)
 t1  1, x   
.
z
r
Let u t  R , t  t 0 , t 0  1,..., t1  1 be an
r -dimensional bounded vector-function
arbitrary
(admissible variation control). Denote by z t , x  the

Relation (10) is an analogy of the Euler equation
for the considered problem.
Each admissible control being a solution of Euler
equation is called classical extremals.
Equation (7) is a Barbashin type linear difference
equation. It’s solution allows the representation[6]:
t1 1
z t , x    F t , , x  f u  , x u   
 t 0
t1 1 x1
 t 1

(11)
     Rt , x,  , s F  , , s  
 t 0 s  x0   1

 f u  , s u  .
Here F t , , x  and Rt , x;  , s  are n n
solution of the following linearized problem
δz t  1, x  

K t , x, s, z t , s 
δz t , s  
z
s  x0
x1

f t , x, z t , x , u t 
δz t , x  
z
f t , x, z t , x , u t 

δu t ,
u
z t 0 , x   0.
(7)
Rt  1, x; , s  
(8)
 
The solution z t , x of problem (7)-(8) is called a
state variation, equation (7) is called, an equation in
variation for problem (1)-(4).
Applying the development scheme in 5 , it is
proved that the first and second variations of the quality
functional (4) have the forms

t1 1 x1
 S u;u     H u t , x u t ,
1
matrix functions being the solutions of the following matrix
difference equations:

K t , x, s, z t , s 
,
z
f t , x, z  , x , u  
F t ,  1, x   F t , , x 
,
z
F t , t 1, x  E , ( E - is n n unique matrix)
Rt  1, x; t , s   
We can write formula (11) in the form
t 1
δ 2 S u; δu  

 δz t , x 
x  x0
1
2
z t1 , x 
z 2
z t , x    F t , , x  f u  , x  
 t0
δz t1 , x  

   Rt , x;  , s F  , , s  f u  , s u  .
s  x0   1

x1
(9)
t1 1 x1
   δz t , x  H zz t , x δz t , x  
t 1
 2δu t H zz t , x δz t , x   δu t H uu t , x δu t .
From the classic theory of variational calculus it is
clear that along the optimal process u t , z t , x the first
variation of the quality functional equals zero, the second is non-negative.
From the equality to zero of the first variation
quality functional it follows:
Theorem 1. If the set U is open, then for
optimality of the admissible control u t in the considered
problem, it is necessary that the relation
H u θ , ξ   0
(10)
    

(12)
Let by definition
Lt ,   F t , , x  f u  , x  
t  t 0 x  x0
be fulfilled for all


f t , x, z t , x , u t 
Rt , x;  , s ,
z
t t0 x  x0
x1
K t , x,  , z t ,  
Rt ,  ;  , s  
z
 x0
x1

x1
t 1
  Rt , x; , s F  , , s  f  , s .
u
s  x0   1
Then formula (11) is written by the formula
t1 1
z t , x    Lt , u t .
(13)
 t 0
By means of representation (12) it is proved that
 2 z t1 , x 
z t1 , x  
z 2
t1 1 t1 1
 2 z t1 , x 
  u  L t , 
Lt1 , s ,
z 2
 t 0 s t 0
z t1 , x 
t1 1 x1
  δz t , x H zz t , x δz t , x  
t  t 0 x  x0

 t 1

 t 1

    Lt , τ δu τ  H zz t , x     Lt , s δu s  
x  x0 t  t 0  τ  t 0

 s t0

Theorem 2. For optimality in the classical
extremals u t in the considered problem, it is necessary
that the inequality

t1 1 t1 1
 δu τ K τ , s δus  
x1 t1 1
 x1 1 t1 1

   δu τ    L t , τ H zz t , x Lt , s  δu s ,
τ t0 s t0
 x  x0 t  max  τ , s 1

t1 1 t1 1
τ t 0 s t 0
t1 1 x1
 t 1

 2    δu t H uz t , x Lt , τ δu τ  
t  t 0 x  x0  τ  t 0

t1 1 x1
   δu t H uu t , x δu t   0
t1 1 x1
  u t H t , x zt , x  
uz
t  t 0 x  x0
 t 1

     u t H uz t , x Lt , u  .
t t0 x  x0  t 0

t1 1 x1
(14)
t  t 0 x  x0
be fulfilled for all u t   R , t  t 0 , t 0  1,..., t1  1.
r
Inequality (14) is a rather general necessary
optimality condition of second order.
Introduce the matrix function
K  , s  
x1
   L t1 , 
x  x0

x1
   z t1 , x 
Lt1 , s  
z 2
IV.
t1 1
  Lt , H Lt , s  .
x  x0 t  max  , s 1
zz
An optimal control problem described by the
difference analogy of Barbashin type integro-differential
equation is studied. First and second order necessary
optimality conditions are obtained.
Then the second variation (9) of the quality case (4)
is represented in the form
t1 1 t1 1
REFERENCES
δ S u; δu     δu τ K τ , s δu s  
[1]
 t 1

 2    δu t H uz t , x Lt , τ δu τ  
t  t 0 x  x0  τ  t 0

[3]
2
τ t 0 s t 0
t1 1 x1
t1 1 x1
   δu t H uu t , x δu t .
t  t 0 x  x0
CONCLUSIONS
2
[2]
[4]
[5]
[6]
R. Qabasov, F.M.Kirillova. Quality theory of optimal processes.
M. Nauka, 1972. (in Russian)
I.V.Gayshun. Multiparametric control systems. Minsk. IM NAS
Byelorussia, 1996, 159 p.
K.M.Mansimov. Optimization of a class of discrete twoparametric systems.// Differen. equations. 1991, № 2, pp. 259262.
P.M.Dymkov. Extremum problems in multiparametric control
systems. Minsk. BSEU, 2005, 363 p.
R. Qabasov, F.M.Kirillova. Singular optimal controls. M.
Nauka, 1973, 256 p. (in Russian)
K.B.Mansimov,
ZH.B.Ahmadova,
S.Sh.Qadirova.
On
representations of solutions of a class of linear difference
equations of Barbashin type. // Proc. of Repub. Conference on
Contemporary problems of mathematics informatics and
economics. Baku, 2009, p. 125-128.