Linearized and Quadratic Necessary Optimality
Conditions in one 2-D Discrete Optimal Control
Problem Involving Functional Constraints of
Inequality Type
Мehriban Nasiyati
Cybernetics Institute of ANAS, Baku, Azerbaijan
[email protected]
Abstract— The paper is devoted to derivation of linearized and
quadratic necessary optimality conditions in one 2-D discrete
two-parametric stepwise control problem.
Key words— 2-D discrete systems; functional constraints;
admissible process; two-parameter systems
I.
Assume that an admissible process is described by
the following discrete two-parametric system of equations
z i t  1, x  1 
the
given
n-
dimensional functions continuous in totality of variables
together with partial derivatives with respect to
zi , ai , bi , ui  ,
1 x  ,  i t  ,
discrete
i  1, 3 to second order inclusively,
i  1, 3 are the given n -dimensional
vector-functions,
ui t , x  , i  1, 3 are r -
dimensional vector-functions of control actions with values
from the given non-empty, bounded and convex sets
t, x  Di , i  1, 3 . (3)

ut , x   u1 t , x , u2 t , x , u3 t , x 
ui t , x   U i  R r ,
The triple
(2)
problem (1)-(2) an admissible state of the process.
Therewith the pair u t , x , z t , x is called an admissible
process.
On the solutions of boundary value problem (1)-(2)
generated by all possible admissible controls determine the
functionals
    
z1 t , x0   1 t , t  t0 , t 0  1, ... , t1 ,
z 2 t1 , x   z1 t1 , x , x  x0 , x0  1, ... , X ,
1 x0   1 t0  , z1 t1 , x0    2 t1  ,
z 2 t 2 , x0    3 t 2  .
are
given,

zt , x   z1 t , x , z2 t , x , z3 t , x  of boundary value
z1 t0 , x   1 x , x  x0 , x0  1, ... , X ,
z3 t , x0    3 t , t  t 2 , t 2  1, ... , t3 ,
i  1, 3
are
(1)
with boundary conditions
z3 t 2 , x   z 2 t 2 , x , x  x0 , x0  1, ... , X ,
x0 , X , ti , i  1, 3
where
with the above - mentioned properties will be called an
admissible
control,
its
appropriate
solution
 f i t , x, z i t , x , z i t  1, x , z i t , x  1, u i t , x ,
z 2 t , x0    2 t , t  t1 , t1  1, ... , t 2 ,
t i  1; x  x0 , x0  1, ... , X  1, i  1, 3,
U i  R r , i  1, 3 , i.e.
STATEMENT OF PROBLEM
t, x  Di , i  1,3 ,
Di  t , x  : t  t i 1 , t i 1  1, ... ,
f i t , x, zi , ai , bi , ui  ,
INTRODUCTION
A step wise problem of optimal control of discrete
two-parameter systems is considered. Under the assumption
of openness of the control domain, necessary optimality
conditions are obtained.
Note that different necessary and sufficient
optimality conditions for discrete 2-D control systems were
obtained in the papers 1-6 and others.
II.
Here
3
S j u     i j  z i t i , X  ,
j  1, p .
i 1
Here
 i j   z i  , i  1, 3 , j  1, p
are the given
twice continuously differentiable scalar functions.
The problem consists of minimization of the
functional
 2 j  t 2  1, x  1   3 j  t 2  1, x  1 
3
S 0 u     i0  z i t i , X  ,
(4)
i 1

under the constraints
S j u   0 ,
j  1, p .
(5)
ψ 2 j  t  1, X  1 
The accessible process satisfying the constraints (5)
is called an admissible process.
III.
H 2 j  t 2  1, x H 3 j  t 2  1, x

,
a 2
a3
 3 j  t 3  1, X  1  
NECESSARY OPTIMALITY CONDITIONS
    
ψ 3 j  t 3  1, x  1 
The admissible process u t , x , z t , x
being
solution of problem (1)-(4) is called an optimal process.
Let u t , x , z t , x be a fixed admissible process.
Further we’ll use the following denotation:
    
ψ 3 j  t  1, X  1 
H i j  t , x, z i , ai , bi , u i , i    i j  f i t , x, z i , ai , bi , u i ,

 H i t , x

zi2
 j
 j
Theorem
vector-functions of conjugated variables being the solutions
of the problem
 i j  t  1, x  1 
H i j  t , x  H i j  t  1, x 


z i
ai
H i j  t , x  1

, i  1, 3,
bi
(5)
 1 t1  1, x  1   2 t1  1, x  1 
 j

 j
H 1 t1  1, x H 2 t1  1, x

,
a1
a 2
 j
 j
 2 j  t 2  1, X  1   3 j  t 2  1, X  1 

 2 j  z 2 t 2 , X 
,
z 2
Along
the
optimal
process
H i j  t , x
vi t , x   ui t , x   0
min  
jJ u 
u i
t ti 1 x  x0
for all
vi t , x U i , t , x   Di , i  1, 3 .
(7)
Relation (7), being a necessary optimality condition
of first order, is an analogy of the linearized maximum
condition for problem (1)-(2).
Definition
1.
The
admissible
control

vt , x   v1 t , x , v2 t , x , v3 t , x  is called critical
with
the
ut , x   u1 t , x , u2 t , x , u3 t , x  , if
respect
to
extremum
H i j  t , x 
vi t , x   u i t , x   0 ,
min  
jJ u 
u i
t ti 1 x  x0
ti 1 X 1
1 j  z1 t1 , X 

,
z1
H 1 j  t  1, X  1
,
ψ1 t  1, X  1 
b1
1.
ti 1 X 1
 1 j  t1  1, X  1   2 j  t1  1, X  1 
 j
(6)

J u   0 I u  .
ut, x, zt, x
 ψ i j  t , x  , i  1, 3 are n - dimensional
H 3 j  t , X  1
.
b3

 j
where ψ i
H 3 j  t 3  1, x
,
a3
I u   j ; S j u   0, j  1, p ,
  H i t , x, zi t , x , zi t  1, x , zi t , x  1, ui t , x , i t , x 
,
zi2
2
 3 j  z 3 t 3 , X 
,
z 3
Let by definition
f i t , x f i t , x, z i t , x , z i t  1, x , z i t , x  1, ui t , x 

,
ui
ui
2
H 2 j  t , X  1
,
b2
i  1, 3 .
(8)
The extremum ut , x  is said to be quasi-singular
in problem (1)-(5) if there exists the appropriate critical
admissible control v t , x  u t , x .
Now, by
from
   
J1 u  denote a maximal subset of indices
J u  such that
ti 1 X 1
H i j  t , x 
  u  0 ,
t ti 1 x  x0
i
for all admissible controls
ut , x .
j  J1 u  , i  1, 3
vt , x  critical with respect to
Assume that in system (2.1)
f i t , x, zi , ai , bi , ui   Bi t , x  bi  F t , x, zi , ai , ui  .
REFERENCES
(9)
Further, the necessary optimality condition of
quasi-singular controls is proved in problem (1)-(5), (9).
[1]
[2]
IV.
CONCLUSION
[3]
[4]
Linearized and quadratic necessary optimality
conditions are established for a stepwise problem of optimal
control of discrete two-parameter systems involving
functional constraints.
[5]
[6]
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