Variable structure extremum problem with constraints for delay discrete inclusions

(1)

VARIABLE STRUCTURE EXTREMUM PROBLEM WITH

CONSTRAINTS FOR DELAY DISCRETE INCLUSIONS

GECİKMELİ AYRIK İÇERMELER İÇİN KISITLAMALI DEĞİŞKEN YAPILI EKSTREMUM PROBLEMİ

Hijran G. MIRZAYEVA

Baku State University Institute of Applied Mathematics

hijan_2002@rambler.ru

ABSTRACT: The paper considers a variable structure extremum problem with

constraints for delay discrete inclusions. The necessary extremum conditions are obtained for the considered problem.

Keywords: Discrete Inclusion; Necessary Extremum Condition; Subdifferential;

Lipschitz Condition; Tangent Cone; Hypertangent cone

JEL Classifications: C61

ÖZET: Makalede gecikmeli diskret içermeler için kısıtlamalı değişken yapılı

ekstremum problemi araştırılmaktadır. Ele aldığımız problem için gerekli ekstremum koşulları bulunmuştur.

Anahtar Kelimeler: Diskret İçerme; Gerek Ekstremum Koşulu, Altdiferansiyel,

Lipschitz Koşulu, Teğet Koni, Hiperteğet Koni

JEL Sınıflaması: C61

1. Introduction

The extremum problems for discrete inclusions with delay are a generalization of the discrete problems of optimal control with delay. Application of nonsmooth analysis theory is of importance for investigation of such a problem. Since the definition of the subdifferential given by T. Rockafellar and F.Clarke is entirely a generalization of the smooth and convex problems, the application of the theory of nonsmooth analysis in this paper is advisable. As the necessary extremum conditions are formulated more naturally with the help of the subdifferential, the definition of subdifferential is of importance in the theory of extremum problems.

There exist a lot of papers devoted to the qualitative investigation of different problems of optimal control of discrete systems. The control problems of discrete systems, described by the different difference equations are of importance among various optimal control problems. In spite of this fact, there are many unstudied problems in the field of extremum problems for discrete inclusions. Recently, multivalued mappings became the subjectof intensivestudy.Different properties of multivalued mappings and their connection with the theory of optimization were considered in (Aubin, Ekeland, 1984: 510; Borisovich, Gelman, Mishkins, Obukhovskiy, 1986: 103).

Note that main models of mathematical economics are reduced to extremum problems for discrete inclusions.

(2)

If the state of the control system is characterized by a system of more than two equations and an operator connecting the equations of the system (connection operator) exists, then such a control system is called a variable structure system. This work generalizes some of the results, obtained in (Mirzayeva, Sadygov, 2005: 89-94; Mirzayeva, Sadygov, 2006: 106-112). An arbitrary order necessary extremum conditions are obtained for the nonconvex optimal control problem for discrete inclusions with delay in (Mirzayeva, Sadygov, 2006: 106-112). In this paper the extremum problem for delay discrete embedding is reduced to the mathematical programming problem using the method given in ( Boltyanskiy, 1975: 3-55). Further, using Clarke’s theory, necessary extremum conditions are obtained. Differently from the considered works (Mirzayeva, Sadygov, 2005: 89-94; Mirzayeva, Sadygov, 2006: 106-112; Mirzayeva, Sadygov, 2007: 67-72), in the paper the discrete extremum problem with constraints is considered. Note that the extremum problem for delay discrete inclusions, is a particular case of variable structure extremum problem for delay discrete inclusions. The known optimal control problems of a variable structure are obtained from variable structure extremum problems for delay discrete inclusions (Mirzayeva, 2007: 44-50). The optimal control problems of variable structure with discrete time delay are considered in (Mirzayeva, 2007: 44-50). Such problems are considered by many authors. Differently from the known works, in this paper a nonsmooth case is considered and the problem is reduced to the extremum problem for delay discrete inclusions with variable structure. Further, in the paper the necessary extremum condition is obtained for discrete systems with variable structure.

Variable structure extremum problem for delay discrete inclusions without constraints is considered in (Mirzayeva, Sadygov, 2007: 67-72). The necessary extremum conditions are obtained for one discrete systems class in (Mirzayeva, Sadygov, 2007: 67-72 ).

Note that optimal control problems of variable structure arise while investigating some chemical-technological processes, applied problems of economics and physics.

2. The formulation of the problem

Let X , be Banach spaces, Y : 2 2X,

t X a  t0,1,...,k1, , 2 : 2 Y t Y

b  tk,k1,...,m1 be the multivalued mappings, where 2V denotes the set of all subsets of

V

. We denote grF 



(z,v)ZV:vF(z)



.

Let us consider the delay discrete inclusions with variable structure

, ,..., 2 , 1 , ) ( 1 ,..., 1 , ), , ( 0 , 1 ,..., 1 , ) ( 1 ,..., 1 , 0 ), , ( 1 1 C y k h k h k h k t at x G y m k k t y y b y t at t c x k t x x a x m t t t h t t t t t t t t                             (2.1)

(3)

where c(t)X at t,1,...,1,0, CY,G:X Y is mapping, h

m

k, , are fixed natural numbers. As a trajectory (solution) , ({x_t}, {y_v}) of the discrete inclusion (2.1) we understand the process

, ,..., 1 , , , 1 ,..., 1 ,t k k y v k m

xt   v   for which (2.1) is satisfied.

Suppose that

 

t X R t k i n g k m k h k1, min{ 1,  1}, _i , :  , 1,..., , 1,...,  

 

t Y R t k m i n f_i , :  ,  1,..., , 1,..., . We denote x(x₁,...,x_k), y(y_k₁,...,y_m).

Consider the minimization of the function



    k t m k t t t t f y t x g y x F 1 1 0 0 0( , ) ( , ) ( , ) (2.2)

on the trajectories of discrete inclusion (2.1) and with the following constraints

 

_

 

_

 

     k t m k t t t t f y t x g y x F 1 1 1 1 1 , , , 0,



 

_

 

_

 

     k t m k t t j t j j x y g x t f y t F 1 1 , 0 , , ,

 

_

 

_

 

        k t m k t t j t j j x y g x t f y t F 1 1 1 1 1 , , , 0, (2.3)



 

_

 

_

 

     k t m k t t n t n n x y g x t f y t F 1 1 . 0 , , ,

We note that as a trajectory of (2.1) we take the pairs ( yx, ), for which (2.1) is satisfied. Denote by M the set of solutions of problem (2.1). To reduce the formulated problem to the mathematical programming problem we use the following notation. We denote smk and define the sets in XkYsas



( 1,..., , 1,..., ) : 1 0( ( ), (0))



, 0 x x y y X Y x a c c M k s m k k      _



( 1,..., , 1,..., ) : 2 1( ( 1), 1)



, 1 x x y y X Y x a c x M k s m k k      _





( ₁,..., , _₁,..., ) : __₁ _( (0), _)



,   x x y y X Y x a c x M _k _k _m k s



( 1,..., , 1,..., ) : 2 1( 1, 1)



, 1        x x y y X Y x a x x M k k m k s





( 1,..., , 1,..., ) : 1( 1 , 1)



, 1       k k m  k s k k k k k x x y y X Y x a x x M

(4)



( ₁,..., _k, _k ₁,..., _m) k s: _k ₁ _k( ( _k _h), ( _k))



, k x x y y X Y y b G x G x M  _   _  _



( ₁,..., , ₁,..., ) : ₂ ₁( ( ₁ ), ₁)



, 1        k k m  k s k  k k h k k x x y y X Y y b G x y M





( ₁,..., _k, _k ₁,..., _m) k s: _k _h ₁ _k _h( ( _k), _k _h)



, h k x x y y X Y y b G x y M          



( ₁,..., , ₁,..., ) : ₂ ₁( ₁, ₁)



, 1          h  k k m  k s k h  k h k k h k x x y y X Y y b y y M





( ₁,..., , ₁,..., ) : ₁( ₁ , ₁)



, 1       k k m  k s m m m h m m x x y y X Y y b y y M



(x₁,...,x , y ₁,...,y ) X Y :y C



. M_m  _k _k_ _m  k s _m

It is clear that the formulated problem will be reduced to the minimization of the function )F₀(x,y on the set

_

m t t M M 0 

 with the constraints

 

, 0, ,

 

, 0,

1 x y  F x y 

F _ _j F_j_₁

 

x,y 0,_,F_n

 

x,y 0.

3. The solution of the problem

Let Z be a Banach space, E be a nonempty subset of Z . Clarke’s subdifferential of the function  at the point z is denoted as ₀ (z₀) .

Suppose z₀E. The set T_E(z₀)



zZ:d_E0(z₀;z)0



is called tangent cone to E in z . ₀

The set N_E(z₀)



zZ:z,z0at zT_E(z₀)



is called normal cone to E in z . 0

Let D . The set of all hypertangents to D at the point Z z denote by D I_D(z). By the definition (see (Clarke, 1988: 279) )

  

z  vZ:0,thatytE atally



zB



E, vB,t

 

0,

I_E _ .

We note if g_i

 

,t ,i0,...,n, satisfy the Lipschitz condition in the neighbourhood of x where t, t1,...,k and fi

 

,t,i0,...,n, satisfy the Lipschitz condition in the

neighbourhood of y_t, where tk1,...,m, then F_i

 

x,y,i0,...,n satisfy the Lipschitz condition in the neighbourhood of

 

x,y . Denoted by  the set of solutions of problem (2.1), satisfying condition (2.3). The pairs

 

x,y  are called optimal , if F0

 

x,y F0

 

x,y at

 

x,y .

(5)

We use Lagrange’s generalized function for the nonsmooth problem of mathematical programming. Let



 





 

   n i M i iF x y r d x y r y x L 0 , , , , ,    , where



  n



 ₀, ₁,, , _i,rR ,i0,...,n. The following corollary follows from Theorem 6.1.1 ( Clarke, 1988: 279).

Corollary 1. Let

 

x,y  minimize the functional F₀ on the set , the functions

n

F F

F₀, ₁,_ satisfy the Lipschitz condition in the neighbourhood of

 

x,y, M be a closed set. Then for sufficiently large r the numbers ₀ 0,₁0,_,_j 0 and

n

j 

 _₁,_, may be found, not all equal to zero simultaneously, such that

 



  j i i iF x y 1 0 ,  and 0L



 

x,y,,r



.

In the paper, the necessary extremum conditions are obtained, considering the additional conditions A, B of R.T. Rockafellar (Rockafellar, 1998: 733), the conditions C, D (Sadygov, 2007: 59). Similarly to subdifferential calculation the certain additional conditions are required for calculating normal cones. Such problems (the problems of calculation of normal cones) were studied by some authors as F.H. Clarke, R.T.Rockafellar, Dubovitsky-Milutin and others. These conditions are not equivalent, though they are considered for the same purpose.

Condition A. _m i IM z z M T i 1 ( )) . ( ) ( 0  _  Condition B.

_X

_

_R

n1

_,

_Y

_

_R

n2_{, from} *

_N

₍

_z

₎

i M i





and

0

* * 1 * 0













m





it follows that



_i*



0

at i0,...,m. Condition C. _X _Rn1_,_Y _Rn2_, ₍ ₎ ₍ ₎ 1 2 _, 1 0 s n k n l i M M z T z R R T _l   _i   



l1,...,m,

and Mi are closed sets at i0,...,m.

Condition D. X Rn1,Y Rn2, ( ) ( ) 1 2 , 1 s n k n m l i M M z T z R R T _l  _i    



l0,...,m1, and Mi are closed sets at i0,...,m.

Proposition 1. If one of the conditions A, B, C or D holds, then .) ( ) ( 0



  m i M M z N z N _i

Further we suppose that b_t_:Y2 C

 

Y _{, where}_C₍_Y₎_{denote the families of all}

nonempty closed subsets of .Y We denote

 



 

  n i i iF x y y x F 0 . , ,  

Theorem 1. Let z

  

x,y  x1,,x_k,y_k_1,y_m



 minimize the functional

0

F on the set , gra_t at t0,...,k1, grb_t at tk,...,m1 and C be closed sets, G:X Y continuous operator, the functions gi

 

,t :X R, i0,...,n

(6)

satisfy the Lipschitz condition in the neighbourhood of x_t at t1,...,k, the functions f_i

 

,t , i0,...,n satisfy the Lipschitz condition in the neighbourhood of

t

y at tk1,...,m. If in addition one of the conditions A, B, C or D holds, then there exist vectors _tN_M_t

 

z,t0,1,,m and the numbers

0 , , 0 , 0 ₁ 0    j 

 _ и _j_₁,_,_n may be found, not all equal to zero simultaneously, where



 

  j i i iF x y 1 0 ,

 and F_

 

x,y such that



  __ m i i 0   .

Proof. Using Theorem 1.3.12 (Borisovich, Gelman, Mishkins, Obukhovskiy, 1986: 103) we have that M is closed at _s s ,...,k kh. So the intersection of finitely many closed sets is a closed set, therefore according to this condition we obtain that

M is closed. It is straightforward to check that the conditions of corollary 1 are satisfied. Then for enough large r the numbers ₀ 0,₁ 0,_,_j 0 and

n

j 

 1,, may be found, not all equal to zero simultaneously such that

 



  j i i iF x y 1 0 ,  and

 

_

 



             _   n i M i i n i M i iF r d x y F x y r d x y 0 0 , , , 0    

 



    n i M i iF x y N x y 0 , ,  . Since

 



 

  m t M M x y N x y N _t 0 , ,

, then we have that 0

 

,

 

, .

0



    m t M x y N y x F _t

Therefore there exist _tN_M_t

 

z,t 0,1,,m, and F_

 

x,y _{such that}



  __ m t t 0 

 . The theorem is proved.

Theorem 2. If z(x,y)



x₁,,x_k,y_k_₁,,y_m



 minimizes functional F on ₀ the set  , gra at _t t0,...,k1, grb at _t tk,...,m1 and C are closed sets,

Y X

G:  is continuous operator, the functions g_i

 

,t :X R,i0,...,n, satisfy the Lipschitz condition in the neighbourhood of x at _t t1,...,k, the functions

 

t i n

f_i , , 0,..., , satisfy the Lipschitz condition in the neighbourhood of y at _t m

k

t 1,..., and in addition one of the conditions A, B, C or D holds, then the numbers 0₀ 0,₁ 0,_,_j  and _j_₁,_,_n may be found, not all equal to zero simultaneously, where



 

  j i i iF x y 1 0 ,

(7)

k t t x g x i t n i i t ( , ) at 1,..., 0 * 0



   , ( , )at 0 * 0 f y t x i t n i i t



    tk1,...,m,

 

    , 0

 

1 1 0 N 0 x x  _a _c__ _c ,



(), * ()



₍ ₍ _),₎( , ₁), 1,..., , 1 * _ _ _       t N x x t x t x_t _t _gra_t _c _t _t _t















x_t t,x___t t,x___t_₁ t



N_gra___t



x_t ,x___t,x___t_₁



att1,...,k1, , ) , , ( )) ( ), ( ), ( ( ( (), ()) 1 * 1 * *      h k k  grb G G k h k k k k x k y k N x x y x _k





x_k _t _h(k t),y_k _t(k t),y_k* _t (k t) N_grb_k _t₍_G₍_),₎(x_k _t _h,y_k _t,y_k _t ₁),t 1,...,h 1 * * _ _ _ _ _              ), , , ( )) ( ), ( ), ( ( *_ *_ _ *_ __₁ _ _ _ _ __₁          _k _h _t _k _h _t _grb _k _t _k _h _t _k _h _t t k k h t y k h t y k h t N y y y y t h k , 1 , 1m k h t    *( ) ( ) m C m m N y y 

such that in the case h the relations are fulfilled:

0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  x_t _t _t _t at t1,...,k; 0 ) ( ) ( ) 1 ( * * * * 0 , y t y t y t  x_t _t _t _t at tk1,...,mh1; (3.1)

 

0 ) 1 ( * * 0 , y t y t  xt t t at tmh,...,m;

in the case h the relations are fulfilled:

0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  xt t t t at t1,...,k1; 0 ) ( ) 1 ( * * * 0 , x t x t  x_t _t _t at tk,...,kh1; 0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x th  x_t _t _t _t attkh;...,k; (3.2)

 





0 ) 1 ( * * 0 , y t y t y th  xt t t t at tk1,...,mh1; 0 ) ( ) 1 ( * * * 0 , y t y t  xt t t at tmh,..., m;

in the case h the relations are fulfilled:

; 1 ,..., 1 at 0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  t kh x_t _t _t _t ; 1 ..., , at 0 ) ( ) ( ) ( ) 1 ( * * * * * 0 , x t x t x t x th  tkh k  x_t _t _t _t _t (3.3) ; ,..., at 0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t h t k k x_t  _t   _t  _t     ; 1 ,..., 1 at 0 ) ( ) ( ) 1 ( * * * * 0 , y t y t y th  tk mh x_t _t _t _t 0 ) ( ) 1 ( * * * 0 , y t y t  x_t _t _t at tmh,..., m;

Proof. Using the corollary of Theorem 2.4.5. (Clarke, 1988: 279), from the definition of

M

₀

,

M

₁

,

_

,

M

_m we obtain that



( ,..., , ,..., ) : ( )



, ) ( ₁ ₁ ₁ ₀₍ ₍ _),₍₀₎₎ ₁ 0 z x x y y X Y x T x Т k s _a _c _c m k k М      



( ,..., , ,..., ) :( , ) ( , )



) (  ₁ k k₁ m  k s t t₁  gra ₍c₍t₎_,₎ t t₁ М z x x y y X Y x x T x x Т _t _t

(8)

  ,...,1 t

,

 





( ₁,..., , _₁,..., ) k s:( _t__, _t, _t_₁) _gr_a ( _t__, _t, _t_₁)



m k k M z x x y y X Y x x x T x x x T _t _t at t 1,...,k1,



    ( ,..., , _ ,..., ) :( _ , , _ ) ) ( ₁ _k _k ₁ _m k s _k _h _k _k ₁ М z x x y y X Y x x y Т _k      , 



 , , 1

 

T_gr_b_k_G _G x_k _h x_k y_k



( ,..., , ,..., ) :( , , ) ( , , )



) (  ₁ _₁  k s _t__h _t _t_₁  _gr_b₍_G₍__),_₎ _t__h _t _t_₁ m k k М z x x y y X Y x y y T x y y Т _t _t h k k t 1,...,  ,



( ,..., , ,..., ) :( , , ) ( , , )



) (  1 k k1 m  k s th t t1  grb th t t1 М z x x y y X Y y y y T y y y Т _t _t 1 ,..., 1    k h m t ,

 



( ₁,..., _k, _k ₁,..., _m) k s: _m _C ( _m)



. М z x x y y X Y y T y Т _m  _    Therefore we have



( ,0,...,0) : ( )



, ) ( 1 1 0(( ),(0)) 1 0 z x X Y x N x N a c c k М s       _ _ _ 



( , , , ,0,...,0) :( , ) ( , ) ) ( ( ( ),) 1 * 1 * 2 1             _k k _t _t _gr_a _c_t _t _t М z x x x X Y x x N x x N _t  s _t ,



1 , at 0     _i _t _t xi at t ,...,1 ,











1 1 * 1,..., ,0, ,0) : , , , , ( ) (       t t t  gra t t t k k М z x x X Y x x x N x x x N _t  s _t ,



1 , , at 0      _i _t _t _t xi at t1,...,k1,





  _ _ _           1 * 1 * 1,..., , ,0,...,0) : , , ( ) ( _k _k k _k _h _k _k М z x x y X Y x x y N s k NgrbkG   ,G 



xk h,xk,yk 1



, xi 0at ikh,k



     (3.4)





, ,..., 1 at 1 , at 0 , at 0 ), , , ( ) , , (: ) ,..., , ,..., ( ) ( 1 ) ), ( ( * 1 * 1 1 h k k t t t i y h t i x y y x N y y x Y X y y x x z N i i t t h t G b gr t t h t k m k k М t s t                              





1 , , at 0 ), , , ( ) , , (: ) ,..., , ,..., ( ) ( 1 * 1 * 1 1                       t t h t i y y y y N y y y Y X y y x x z N i t t h t b gr t t h t k m k k М t s t , 1 ,..., 1 attkh m     _ _ _  ₍₀_,...,₀_,₀_,...,₀_, ₎   _:  ₍ ₎ ) ( *_m k _m _C _m М z y X Y y N y N s m .

According to Theorem 1 the numbers ₀ 0,₁0,_,_j 0and_j_₁,_,_n may be found, not all equal to zero simultaneously, where 

 



j

(9)

 

_

 

_

 

_

 

_

 



        _ _ _            m t k t m k t k t t t t M n i i iF x y N t x y g x t f y t g x t 0 1 1 1 1 1 0 0 0 0 0 , , , , , 0    

 

_

 

_

 



__



_



__ _  _   _           m t n i i i M m k t k t m k t t n n t n n t t g x t f y t N x y g x y f _t 0 0 1 1 1 1 1 1 ,  ,  , ,  ,1  













_

 



                m t M n i n i n i m i i k i i k i ig x k f y k f y m N t x y 0 0 0 0 1, 1 , , ,      (3.5)

 









                      k t m k t m t M n i t i i n i t i ig x t f y t N t x y 1 0 1 0 0 . , , ,  

Using the relations (3.4) and (3.5) it is straightforward to check the correctness of Theorem 2.

Theorem 3. Let z

 

x,y  minimize the functional F on the set 0 ,grat at

, 1 ,..., 0

t k grb_tat tk,...,m1and С be closed sets, G:X Y be continuous operator, the functions g_i

 

,t :X R,i0,...,n satisfy the Lipschitz condition in the neighbourhood of x_t att1,...,k, the functions f_i

 

,t ,i0,...n, satisfy the Lipschitz condition in the neighbourhood of yt attk1,...,m,

    ,



x,x1



att1,...,, Igratc t t t Igrat



xt,xt,xt1



at t1,...,k1, G(),G()( kh, k, k1) grb x x y I _k , Igrbi(G(),)(xth,yt,yt1) at tk1,...,kh, ) , , ( th t t1 grb y y y

I _t at tkh1,...,m1 and I_C

 

y_m be nonempty sets. Then the numbers 00 0,1 0,,j  and j1,,n may be found, not all equal to

zero simultaneously, where 

 



j

i 1 iFi x,y 0,

 and there vectors exist , 0 0   _ t t z x  where 0,1, ( , )at 1,... , 0 * 0 g x t t k z i t n i i t 



   ( , ) 0 * 0 f y t z i t n i i t



    attk1,...,m, x1

 

0 Na0_c_{   },c0_

 

x1  _ _,





_  () ), ( *₁ * _t _x _t x_t _t ) , ( ₁ ) ), ( (    N_gra_t _c _t x_t x_t at t1,...,,



x_t



t



,x___t



t



,x___t_₁



t



 Ngrat(xt,xt,xt1)att1,...,h1,





( ), ( ), * ( ) ₍ ₍_),₎( , , ₁), 1 * *            t h  k t  k t   grb _ G k t h k t k t k k t y k t y k t N x y y x t k        1,...,h,(y*_ (k h t),y*_ _ (k h t),y*_ __₁(k h t)) t _k _t _k _h _t _k _h _t ) ( ) ( , 1 , 1 ), , , ( _k _t _k _h _t _k _h_t ₁ _m* _C _m grb y y y t m k h y m N y N t h k       _ __ ___  

such that in the case h the relations (3.1) are fulfilled, in the case h the relations (3.2) are fulfilled, in the case h the relations (3.3) are fulfilled.

Proof. It is straightforward to check that



( 1,..., , 1,..., ) :( , 1) ( ( ),)( , 1



) ( 1     t t  gra ct  t t s k m k k M z x x y y X Y x x I x x I _t   ,...,1 at t ;

(10)



    ( ,..., , _ ,..., ) :( __, , _ ) ) ( ₁ ₁ k s _t _t _t ₁ m k k M z x x y y X Y x x x I _t



, , 1





I_gra_t x_t x_t x_t att1,...,k1

;



    ( ,..., , _ ,..., ) :( _ , , _ ) ) ( ₁ _k _k ₁ _m k s _k _h _k _k ₁ M z x x y y X Y x x y I _k



1





)) ( ), ( (    , ,  I_grb_k _G _G x_k _h x_k y_k ;



( 1,..., , 1,..., ) :( , , 1) ( (),)



, , 1





) (  _  k s _t__h _t _t_  _grb _G__ _t__h _t _t_ m k k M z x x y y X Y x y y I x y y I _t _t ; ,..., 1 attk kh



( 1,..., , 1,..., ) :( , , 1)



, , 1





) (  _  k s _t__h _t _t_  _grb__₁ _t__h _t _t_ m k k M z x x y y X Y y y y I y y y I _t _k _h ; 1 ,..., 1 attkh m



( ,..., , ,..., ) : ( )



. ) ( ₁ _k _k ₁ _m k s _m _C _m M z x x y y X Y y I y I _m  _    If _m t M M z I z T _t 1 ( )) , ( ) ( 0  

 then according to lemma 5.11 ( Girsanov, 1970: 118) we can find the linear functionals * _N (_z),_t 0,1,...,_m,

t

M

t  

 not all equal to zero,

such that *₀₁*..._m* 0. Then we obtain that at 0 the statement of Theorem 3 is satisfied.

According to Theorem 3 we have that I_M_t

 

z intT_M_t

 

z att1,...,m. Therefore, if

_

m

t Mt

M

z

I

z

T

1

0

(

)



(

_

(

))





, then the conditions of Theorem 2 with

the condition A are satisfied. Then it follows from Theorem 2, that at





1

the statement of Theorem 3 is satisfied. The theorem is proved.

Theorem 4. If z(x,y)



x1,,x_k,y_k_1,,y_m



, gra at t t0,...,k1, t

grb at t k,...,m1 and C are convex sets, G:X Y is a linear operator, the functions gi

 

,t :X R,i0,...,n are convex at t1,...,k, the functions

 

t Y R i n

f_i , :  , 0,..., are convex at t k1,...,m, besides j and n 0 , , 0 , 1 ₁ 0    j 

 _ may be found, where 

 



 j i 1iFi x,y 0 , , ,... 1 at ) , ( 0 * 0 g x t t k x n _i _t i i t     0 ( , )at * 0 f y t x n _i _t i i t   tk1,...,m,

and there the vectors exist

 

    , 0

 

1 1 0 N 0 x x  _a _c__ _c ,



xt*(t),x*t1(t)



Ngrat(c(t),)(xt ,xt₁), t1,...,,















xt t,xt t,xt1 t



Ngrat



xt ,xt,xt1



att1,...,k1, , ) , , ( )) ( ), ( ), ( (xk*h k xk* k y*k1 k Ngrbk(G(),G()) xkh xk yk1





x_k*__t__h(kt),y_k*__t(kt),y_k*__t_₁(kt)N_grb_k__t₍_G₍__),_₎(x_k__t__h,y_k__t,y_k__t_₁),

(11)

, ... 1 h t ( _ (   ), _ _ (   ), *_ __ (   )) 1 * * _k _h _t _y _k _h _t _y _k _h _t y_k _t _k _h _t _k _h _t h k m t y y y N_grb _k _t _k _h _t _k _h _t t h k      _ _ _ _ __   ( , , 1), 1,..., 1 ,

(

)

(

)

* m C m

m

N

y



such that in the case h the relations are fulfilled:

0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  x_t _t _t _t at t1,...,k; 0 ) ( ) ( ) 1 ( * * * * 0 , y t y t y t  x_t _t _t _t at tk1,...,mh1; (3.6)

 

0 ) 1 ( * * 0 , y t y t  x_t _t _t at tmh,..., m;

in the case h the relations are fulfilled:

0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  x_t _t _t _t at t1,...,k1; 0 ) ( ) 1 ( * * * 0 , x t x t  xt t t at tk,...,kh1; 0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x th  x_t _t _t _t at tkh,...,k; (3.7)

 





0 ) 1 ( * * 0 , y t y t y th  x_t _t _t _t at tk1,...,mh1; 0 ) ( ) 1 ( * * * 0 , y t y t  x_t _t _t at tmh, m... ;

in the case h the relations are fulfilled:

; 1 ,..., 1 at 0 ) ( ) ( ) 1 ( * * * * 0 , x t x t x t  t kh xt t t t ; 1 ,..., at 0 ) ( ) ( ) ( ) 1 ( * * * * * 0 , x t x t x t x th  tkh k  xt t t t t k k t h t x t x t x xt*,0 t*( 1) t*() t*(  )0at  ,..., ; ; 1 ,..., 1 at 0 ) ( ) ( ) 1 ( * * * * 0 , y t y t y th  tk mh x_t _t _t _t (3.8) 0 ) ( ) 1 ( * * * 0 , y t y t  x_t _t _t at tmh,...,m

Then z

  

x,y  x₁,_,x_k,y_k_₁,_y_m



 minimizes the functional F on the ₀ set  .

Proof. Consider the case h. Since ( , )

0 * 0 g x t x n _i _t i i t    at t1,...,k and ) , ( 0 * 0 f y t x n _i _t i i t 

  at tk1,...,m, then from the relation (3.6) we have

) , ( ) ( ) ( ) 1 ( 0 * * * _t _x _t _x _h _t _g _x _t x n _i _t i i t t t        at t1,...,k; ) , ( ) ( ) ( ) 1 ( 0 * * * _t _y _t _y _t _h _f _y _t y n _i _t i i t t t         at tk1,...,mh1;

 

( , ) ) 1 ( 0 * _t _y _t _f _y _t y n _i _t i i t t       _ _at_t__m__h_{,..., m}_; where

(12)

), ( ) 0 ( ₍ ₍ _), ₍₀₎₎ ₁ * 1 N 0 x x  _a _c _c



xt*(t),x*t1(t)



Ngrat(c(t),)(xt ,xt1), t1,...,h,







xt ht,x*t(ht),xt₁



ht



Ngrat (xt,xt,xt₁), ,t1,...k1h , ) , , ( )) ( ), ( ), ( (xk*h k x*k k yk*₁ k Ngrbk₍G₍_),G₍₎₎ xkh xk yk₁



( ), ( ), * ( )



₍ ₍_),₎( , , ₁), 1 * *            t h  k t  k t   grb G k t h k t k t k k t y k t y k t N x y y x _k _t h t1,..., ,(y_k*__t(kht),y_k*__h__t(kht),y_k*__h__t_₁(kht)) , 1 ,..., 1 ), , , (y y y ₁ t m k h N_grb _k _t _k _h _t _k _h _t t h k      _ _ _ _ __   ( ) ( ) * m C m m N y y  . Therefore t t t t t t i n i i t i n i i x x t h x t x t x t x g t x g             ( , ) ( , ) ( 1) () ( ), * * * 0 0   at t 1,...,k; t t t t t t i n i i t i n i if y t f y t y t y t y t h y y             ( , ) ( , ) ( 1) () ( ), * * * 0 0   at ;tk1,...,mh1 t t t t t i n i i t i n i if y t f y t y t y t y y           ( , ) ( , )) ( 1) (), * * 0 0   at tmh,..., m. Hence follows



( 1) () ( ),



( 1) (), . ), ( ) ( ) 1 ( ) , ( ) , ( ) ) , ( ) , ( ( )) , ( ) , ( ( * * * 1 1 * * 1 * * * 0 0 1 1 0 0 1 0 0



                                              _      m h m t t t t t t t t h m k t t t k t t t t t t m h m t t i n i i t i n i i h m k t t i n i i t i n i i k t t i n i i t i n i i y y t y t y y y h t y t y t y x x t h x t x t x t y f t y f t y f t y f t x g t x g      

Since according to the condition that gra_t,grb_t,C are convex sets and G:X Y is a linear operator, then we have

0 ), 0 ( ₁ ₁ * 1 x  x  x at x₁a₀(c(h),c(0)), 0 ) , ( ) , ( )), ( ), ( (x_t* t x_t*_₁ t x_t x_t_₁  x_t x_t_₁  at (x_t,x_t_₁)gra_t(c(ht),) and ,t1,...,h







 







__ ___        _, ₍ _), _,₍ _, _, ₎ 1 1 * t t t t t t h t x h t x h t x x x x 0 ) , , ( ₁   x_t x___t x___t_ at (x_t,x___t,x___t_₁)gra___t and t1,...,k1h, 0 ) , , ( ) , , ( )), ( ), ( ), ( (x_k*__h k x*_k k y*_k_₁ k x_k__h x_k y_k_₁  x_k__h x_k y_k_₁  at , (.)) (.), ( ) , , (x_k__h x_k y_k_₁ grb_k G G



x_k*__t__h(kt),y*_k__t(kt),y_k*__t_₁(kt)



, 0 ) , , ( ) , , (x_k__t__h y_k__t y_k__t_₁  x_k__t__h y_k__t y_k__t_₁  at t1,...,h,



       _ _ _ __ _ _ _ _ __  ( ), ( ), ( ),( , , ) (y_k* _t k h t y_k* _h _t k h t y*_k _h _t ₁ k h t y_k _t y_k _h _t y_k _h _t ₁

(13)

0 ) , , ( ₁   y_k__t y_k__h__t y_k__h__t_ at (y_k_t,y_k_h_t,y_k_h_t1)grb_k_h_t and , 1 ,..., 1 m k h t    *( ),( _ ) _0 m m m m y y y at y_m . C If ({x_t},{y_v})M , then hence we have

          h t xt t xt t xt xt xt xt x x x 1 1 1 * 1 * 1 1 * 1(0), ( (), ()),( , ) ( , )







x h t x



h t



x_h _t



h t



h k t t h t               1 1 1 , ,

,

(xt,xh_t,xh_t_1)(xt,xh_t,xh_t_1)     (x*_k__h(k),x_k*(k),y_k*_₁(k)),(x_k__h,x_k,y_k_₁) (x_k__h,x_k,y_k_₁)





             h t 1 xk t h k t yk t k t yk t k t xk t h yk t yk t 1 * 1 * * ₍ _), ₍ _), ₍ ₎_,₍ _, _, ₎







                  h k m t k t k h t t k t k h t k y y y k h t y k h t x 1 1 * * 1) ( ), , , , (



    _ _ _ _ __ _ _ _ _ __    ( ),( , , 1) ( , , 1) * 1 k t k h t k h t k t k h t k h t t h k k h t y y y y y y y . 0 ) ( ), ( * _ _   y_m m y_m y_m Hence it follows . 0 ), ( ) 1 ( ), ( ) ( ) 1 ( ), ( ) ( ) 1 ( * * 1 1 * * * 1 * * *                   



       m h m t t t t t t t h m k t t t t k t t t t t t y y t y t y y y h t y t y t y x x t h x t x t x Then we obtain 0 ), ( ) 1 ( ), ( ) ( ) 1 ( ), ( ) ( ) 1 ( ) , ( ) , ( ( ) ) , ( ) , ( ( )) , ( ) , ( ( * * * * 1 1 * 1 * * * 0 0 1 1 0 0 1 0 0                               



 



                    m h m t t t t t t t t t h m k t t k t t t t t t t i n i i m h m t t i n i i h m k t t i n i i t i n i i k t t i n i i t i n i i y y t y t y y y h t y t y t y x x t h x t x t x t y f t y f t y f t y f t x g t x g       i.e. 0 ) ) , ( ) , ( ( ) ) , ( ) , ( ( )) , ( ) , ( ( 0 0 1 1 0 0 1 0 0       



             m h m t t i n i i t i n i i h m k t t i n i i t i n i i k t t i n i i t i n i i t y f t y f t y f t y f t x g t x g       at ({x_t}, {y_v})M. It is clear that