Investigation of Singular Controls in Moiseev’s
Control Problem
Ilaha Nagiyeva
Cybernetics Institute of ANAS, Baku, Azerbaijan
[email protected]
Abstract— A problem on optimal control of ordinary dynamic
systems with non-standard quality test is considered. A
number of necessary optimality conditions are established.
Key words— Moiseev’s control problem; functional; admissible
control; necessary optimality conditions; singular controls
I.
III. NECESSARY OPTIMALITY CONDITION
INTRODUCTION
In the paper [1] N.N. Moiseev considered a
problem of optimal control of ordinary dynamic systems
with non-standard quality test and proved necessary
optimality condition of Pontryagin’s maximum principle
type. In the papers [2-4] and etc. different necessary
optimality condition were obtained for Moiseev type
optimal control problems.
In the present paper new optimality conditions are
obtained for singular controls in N.N. Moiseev’s problem.
II. STATEMENT OF PROBLEM
It is required to minimize the functional
t1 t1
S 0 u    0 xt1    F0 t , s, xt , xs ds dt
(1)
t0 t0
under constraints
ut U  R r ,
t  T  t0 ,t1  ,
S i u    i  xt1  
t1 t1
   Fi t , s, xt , xs  ds dt  0, i  1, p,
t0 t0
x  f t , x, u  , t  T ,
Here
xt 0   x0 .
(2)
(3)
(4)
 i x  , i  0, p are the given continuously
differentiable in
R n scalar functions, Fi t , s, a, b ,
i  1, p are the given scalar functions continuous in
T  T  R  R together with partial derivatives with
respect to a, b to second order inclusively, f t , x, u  is
n

to x to second order inclusively, u t is an r -dimensional
piecewise-continuous (with finite number of first order
discontinuity points) vector of control actions with values
r
from the given non-empty and bounded set U  R .
n
a given n- dimensional vector-function continuous in
T  R n  R r together with partial derivatives with respect
Suppose that ut , xt 
process. Introduce the denotation
is a fixed admissible
H t , x, u, i    i  f t , x, u 
v t  H i  t   H t , xt , vt , i t  
 H t , xt , u t , i t ,
 v t  f t   f t , xt , vt   f t , xt , u t  ,
H xi  t   H x t , xt , u t , i t  ,
 vt  H xi  t    it   v f x t 
Fi t , s  Fi t , s, xt , xs 

;
a
a
Fi t , s  Fi t , s, xt , xs 

;
b
b
 2 Fi t , s   2 Fi t , s, xt , xs 
;

a 2
a 2
 2 Fi t , s   2 Fi t , s, xt , xs 
;

a b
a b

 2 Fi t , s   2 Fi t , s, xt , xs 
,

b 2
b 2

I u   i : Si u   0 , i  1, p , J u   0 I u  .
Here vt U , t  T is an arbitrary control
function,  i t  is an n-dimensional vector-function of
conjugated variables being a solution of the conjugated
problem
 Fi t , s  Fi s, t 

ds ,
a
b 
t0 
t1
 i   H xi  t    
 i t1   
 i xt1 
, i  0, p .
x
For simplicity, we’ll assume that
J u   0,1, 2,..., m
 2 i  xt1 
 t1 ,   
x 2
t1
 t1

 2 Fi t , s 



     t ,  

t
,

dt
 ds 
a 2
 max  ,  
t0 

K  ,     t1 ,  
m  p .
It holds
Theorem 1. For optimality of the admissible
control u t it is necessary that the inequality
t1 t1


 2 Fi t , s 
     t ,  
 s,   ds  dt 
a b

 


m1

 
min
iJ u
j 1
j
 v j H i   j   0
(5)
 t1

 2 Fi t , s 
     s,  
 t ,   dt  ds 
b a

 

t1
v j  U ,  j  0 ,  j  t 0 ,t1 
be fulfilled for all
t0  1   2  ...   m1  t1  .
We can show that inequality (5) is equivalent to
the classic maximum condition for problem (1)-(4). But
theorem 1 allows to investigate a special case in the
considered problem.
Definition 1. The admissible control u t is said
to be Pontryagin’s extremum in problem (1)-(4) if is
satisfies optimality condition (5).
Suppose that there exits a non-empty subset
J 0 u   J u  such that for all i  J 0 u  ,
t1
 t1

 2 Fi t , s 
     s,  
 s,   ds  dt .
2
b

t0 
 max  ,  

qrad i xt1   0 , Fi t , s  a  0 ,
Fi t , s  b  0 .
The following fact is true.
Theorem
2.
Let
IV. CONCLUSION
Necessary optimality conditions are obtained in an
optimal control problem with non-standard quality test.
 i xt1   0 ,
REFERENCES
Fi t , s  a  0 , Fi t , s  b  0 for all i  J 0 u  .
[1]
Then for optimality of Pontyagin’s extremum, it is
necessary that for any natural number  the inequality
[2]
 

min   j  s  v j f  j K i  j , s  vs f  s   0
iJ u 
 j 1 s 1

 j  0 , v j  U ,  j  t 0 ,t1 
be fulfilled for all
[3]
t
0
 1  ...     t1  , j  1,  such that
m
min
 
j 1
iJ u \ J 0 u
Here by definition
 j  v j H i   j   0 .
[4]
[5]
(6)
N.N. Moiseev. Elements of optimal systems theory. M.Nauka.
1995. (in Russian)
I.F. Nagiyeva. Quasi-singular controls in Moiseev’s control
problem.// Vestnik BSU ser. phys. math. Sei 2006, №4, pp. 5762. (in Russian)
K.B. Mansimov, I.F. Nagiyeva. Necessary optimality condition
of singular controls in control of Moiseev type.// Problemy
upravleniya i informatiki. 2006, №5, pp.57-63 (in Russian)
K.B.Mansimov, I.F. Nagiyeva. Investigation of quasi singular
controls in a discrete control problem.// Dokl. NANA Azerb.
2005, vol. LXI №2, pp.30-35 (in Russian)
K.B. Mansimov. Singular controls is delay systems. Baku,
ELM, 1999, 174p. (in Azerbaijan)