View of Construction of a solution for optimal control problem with phase and integral constraints

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IRSTI 27.31.15

1,*

S. Aisagaliev,

¹

Zh. Zhunussova,

²

H. Akca

1Al-Farabi Kazakh National University, Almaty, Kazakhstan

*e-mail: [email protected],

2Professor of Applied Mathematics, Abu-Dhabi University, Abu-Dhabi, UAE e-mail: [email protected]

Construction of a solution for optimal control problem with phase and integral constraints

Abstract. A method for solving the Lagrange problem with phase restrictions for processes described by ordinary differential equations without involvement of the Lagrange principle is supposed. Necessary and sufficient conditions for existence of a solution of the variation problem are obtained, feasible control is found and optimal solution is constructed by narrowing the field of feasible controls. The basis of the proposed method for solving the variation problem is an immersion principle. The essence of the immersion principle is that the original variation problem with the boundary conditions with phase and integral constraints is replaced by equivalent optimal control problem with a free right end of the trajectory. This approach is made possible by finding the general solution of a class of Fredholm integral equations of the first order. The scientific novelty of the results is that: there is no need to introduce additional variables in the form of Lagrange multipliers; proof of the existence of a saddle point of the Lagrange functional; the existence and construction of a solution to the Lagrange problem are solved together.

Key words: immersion principle, feasible control, integral equations, optimal control, optimal solution, minimizing sequence.

Introduction

One of the methods for solving the variational calculus problem is the Lagrange principle. The Lagrange principle makes it possible to reduce the solution of the original problem to the search for the extremum of the Lagrange functional obtained by introducing auxiliary variables (Lagrange multipliers).

The Lagrange principle is the statement about the existence of Lagrange multipliers, satisfying a set of conditions when the original problem has a weak local minimum. The Lagrange principle gives the necessary condition for a weak local minimum and it does not exclude the existence of other methods for solving variational calculus problems unrelated to the Lagrange functional.

The Lagrange principle is devoted to the works [1-3]. A unified approach to different extremum

problems based on the Lagrange principle is described in [4].

In the classical variational calculus, it is assumed that the solution of the differential equation belongs to the space С¹(I,Rⁿ), and the control u(t), tI is from the space С¹(I,R^m), in optimal control problems [5] the solution x

(

t

)

KC¹

(

I

,

Rⁿ

),

and the control

).

, ( )

(

t KC I R^m

u  In this work, the control u(t), I

t is chosen from L2(I,R^m) and the solution x(t), I

t is an absolutely continuous function on the interval I

= [

t₀

,

t₁

].

For this case solvability and uniqueness of the initial problem for differential equation are given in [4, 6-8].

The purpose of this work is to create a method for solving the variational calculus problem for the processes described by ordinary differential equations with phase and integral constraints that

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differ from the known methods based on the Lagrange principle. It is a continuation of the research outlined [9-10].

Problem statement

We consider the following problem: minimize the functional

0 1 1

0 0 1

0

( ( ), , ) = ( ( ), ( ), , , )

t

J u x x

F x t u t x x t dt inf







 (1.1)

at conditions

] , [

= ),

, , ( ) ( ) (

= A t x B t f x u t t I t₀ t₁

x   _(1.2)

with boundary conditions

R n

S S S x t x x t

x(₀))= ₀, (₁)= ₁) ₀ ₁= ²

(    (1.3)

in the presence of phase constraints ( ) ( ) : ( ) =

{ ⁿ/ ( ) ( , ) ( ), },

x t G t G t

x R  t Q x t  t t I



    

and integral constraints

0 1

1 0 1

1 2

( ( ), , ) 0,

= 1, ; ( ( ), , ) = 0,

= 1, ,

j

g u x x j m g u x x

j m m

 





(1.4)

0 1 1

0 0 1

0

2

( ( ), , ) = ( ( ), ( ), , , ) ,

= 1, .

j t

g u x x

f x t u t x x t dt

j m







(1.5)

where the control

).

, ( )

( L₂ I R^m

u   (1.6) Here

A (t ), B (t )

are matrices with piecewise- continuous elements of orders nn, nr,

respectively, a vector function

))

, , ( , ), , , ( (

= ) , ,

( x u t f

₁

x u t f x u t

f

 _r _is

continuous with respect to the variables ,

) , ,

(x u t RⁿR^mI satisfies the Lipschitz condition by x, i.e.

I R R t u y t u x

y x t l t u y f t u x f

m

n 











) , , ( ), , , (

|,

| ) (

| ) , , ( ) , , (

|

(1.7)

and the condition

0 2 1

| ( , , ) | (| | | | ) ( ), ( , , ),

f x u t c x u c t x u t

  

 (1.8)

where

l ( t )



0,

l(t)L₁(I,R¹), c₀=const>0,

0,

)

1

( t



c

c₁(t)L₁(I,R¹).

The vector function ))

, ( , ), , ( (

= ) ,

(x t F₁ x t F x t

F  _s is continuous with

respect to the variables (x,t)RⁿI. Function ),

, , , , ( (

= ) , , , ,

( ₀ ₁ ₀₁ ₀ ₁

0 x u x x t f x u x x t

f

)) , , , , (

,f₀_m₂ x u x₀ x₁ t

 satisfies the condition

0 0 1 2 2 0 1

3

| ( , , , , ) | (| | | | | | | |) ( ),

f x u x x t c x u x x c t

    



0 1 0 1

( , , , , ),( , , , , ) ,

n

m n n

x u x x t y u x x t R

R R R I

  

   

).

, ( ) ( 0, ) ( 0,

= ₃ ₃ ₁ ¹

2 const c t c t L I R

c   

Scalar function F₀(x,u,x₀,x₁,t) is defined and continuous with respect to the variables together with partial derivatives by variables (x,u,x₀,x₁),

(t ),

  (t ),

tI– are given s– dimensional functions. S is given bounded convex closed set of

,

R

2n the time moments t₀,t₁ are fixed.

In particular, the set 0,

) , ( / ) , {(

= x₀ x₁ R² H x₀ x₁ 

S ⁿ _j j=1,p₁;

0,

>=

) , ( ,

<aj x₀ x₁ j= p₁1,p₂}, where ),

, (x₀ x₁

H_j j=1,p₁ are convex functions, ,

j R2n

a  _j₌ _p₁_1,_p₂ are given vectors.

Note, that if the conditions (1.7), (1.8) are satisfied for any controlu()L₂(I,R^m)_{and the} initial condition x(t₀)=x₀ of the differential

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equation (1.2) has a unique solution.

x (t ),

tI. This solution has derivative

x



L

₂

( I , R

ⁿ

)

_and satisfies equation (1.2) for almost all tI.

It should be noted that integral constraints

1

0 1 0 0 1

0

1

( ( ), , ) = ( ( ), ( ), , , ) 0,

=1, ,

t

j j

t

g u x x f x t u t x x t dt

j m





 _(1.9)

by introducing additional variables d_j 0, ,

1,

= m₁

j can be written in the form . 1,

= ,

= ) , ), (

(u x₀ x₁ d j m₁

g_j   _j

Let the vector be

,

,0) ,0,0, ,

, (

= d

₁

d

_m₁

R

^m²

c

    whered_j 0,

. 1,

= m₁

j Let a set be

}, 1,

= 0, / {

= c R ² d j m₁

Q  ^m _j  _whered_j 0,

1, 1

= m

j are unknown numbers.

Definition 1.1. The triple

1

* 0

* 1 0

*( ), , )

(u t x x U S S is called by admissible control for the problem (1.1) – (1.6), if the boundary problem (1.2) – (1.6) has a solution. A set of all admissible controls is denote by

 ,

1

.

0

S

S U

 





From this definition it follows that for each element of the set Σ the following properties are satisfied: 1) the solutions

x

_*

( t ),

tI of the differential equation (1.2), issuing from the point

0,

0* S

x  satisfy the condition

x

_*

( t

₁

) = x

₁^*

S

₁

,

and also (x₀^*,x₁^*)S₀S₁ =S; 2) the inclusion

), ( )

*

( t G t

x 

_t_I holds; 3) for each element of the set Σ we have the equalityg(u(),x0,x1)=c, where

* * * *

0 1 1 0 1

* *

* * 0 1 2 *

( ( ), , ) = ( ( ( ), , ), , ( ( ), , )).

m

g u x x g u x x g u x x

 





The following problems are set:

Problem 1.2. Find the necessary and sufficient conditions for the existence of a solution of the boundary value problem (1.2) – (1.6).

Note, that the optimal control problem (1.1) – (1.6) has a solution if and only if the boundary value problem (1.2) – (1.6) has a solution.

Problem 1.3. Find an admissible control

* *

* 0 1 0 1

( ( ), , ) u t x x    U S   S .

If problem 1.2. has a solution, then there exists an admissible control.

Problem 1.4. Find the optimal control

**

( ) ( ),

u t  U t

the point (x ^*0, x ) S0 S1=S

*

1  

and the optimal trajectory

x t t x

*

( ; , ),

0 ^*0

t  I ,

where

x t

_*

( )  G t ( ), t  I , x t

* 1

( ) = x

1^*

 S

1

,

* *

0 1

( (), , ) 0,

*

g u x x

j

 

j=1,m₁,

g u x x

_j

( (), , ) =0,

_*



^*⁰ ¹^*

, 1,

=m₁ m₂

j 

J u ( ( ), , )

*

 x x

^*0 1^* =inf J(u(),x0,x1), .

) , ( ) , ), (

(u  x⁰ x¹ L₂ I R^m S₀S₁



One of the methods for solving the problem of variation calculus is the Lagrange principle. The Lagrange principle allows to reduce the solution of the original problem to the search for an extremum of the Lagrange functional obtained by introducing auxiliary variables (Lagrange multipliers).

In the classical variation calculus, it is assumed that the solution of the differential equation (1.2) belongs to the space �^�� ^�) and the control u(t), t

∈ I of the space �� ^�) in the optimal control problems [5], the solution x ∈^KC¹^{(I, R}ⁿ) and control u(t) ∈^KC¹^{(I, R}^m). In this paper, the control u(t), t ∈^I is chosen from L2(I, �^�), and the solution x(t), t ∈^I is an absolutely continuous function on the interval I

= [t0, t1]. For this case, the existence and uniqueness of the solutions of the initial problem for equation (1.2) are presented in the references [4, 6, 7, 8].

The purpose of this paper is to create a method for solving the problem of the variation calculus for processes described by ordinary differential equations with phase and integral constraints that differ from the known methods based on the Lagrange principle. It is a continuation of the scientific research presented in [9-16].

Existence of a solution

We consider the following optimal control problem: minimize the functional

1 1 2 0 1

1 1 0

( ( ), ( ), ( ), ( ), , , ) = ( ( ), ) inf

t

I u p v v x x d

F q t t

   





 (2.1)

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1 1 1 2 2 0

= ( ) ( ) ( ) ( ), ( ) = 0, z A t z B t v t B v t z t

t I

 



 (2.2)

), , ( ) ( ), , ( )

( ₂ ₂ ₂ ²

1

r v L I Rm

R I L

v     (2.3)

2

0 1 0 1

( ) ( ), ( ) ( , ),

( , ) = , .

p t V t u L I Rm

x x S S S d

  

    (2.4) We introduce the following notations:

2 2 2

2 1

2

= ( , ) ( , ) ( , )

( , )

m s r

m n n m

H L I R L I R L I R

L I R R R R

  

    ,

2 2

2 2 0 1

= ( , ) ( , )

( , )

m r

m

X L I R V L I R

L I R S S H

  

      , vector function

H X d x x t v t v t p t u

t ) = ( ( ), ( ), ( ), ( ), , , )

 

(

₁ ₂ ₀ ₁



,

)) ( ), ( ), ( (

= )

( t z t z t

₁

t

q 

.

The optimization problem (2.3) – (2.6) can be represented in the form:

. )

( , inf ) ), ( (

= )) (

( ¹ ₁

0

1 F q t t X H

I

t













^



Let the set be

))}.

( inf (

= )) ( (

| ) ( {

=

_* ₁ _* ₁

*    





 X I



I

X

Lemma 2.1. Let the matrix be positive definite

0 >

) , ( t

₀

t

₁

T

. In order to the boundary value problem (1.2) – (1.6) have a solution, it is necessary and sufficient that

lim

1

(  ) =

1*

= inf

1

(  ) = 0



I

n X

n  ,

where

{ 

_n

(



)}



X

is a minimizing sequence in the problem (2.1) – (2.4).

Proof of the lemma follows from Theorem 2.3.

and Lemmas 2.4. and 2.5. [9].

Theorem 2.2. Let the matrix be

T ( t

₀

, t

₁

) > 0

, the function

F

₁

( q , t )

be defined and continuous in the set of variables

( q , t )

together with the partial derivatives with respect to q and satisfies the Lipschitz conditions

,

|,

|

| ) , ( ) , (

|F₁q qq t F₁q q t l q tI (2.5) where

1 1 1 ( )1 1 1

11 12 10 11 1

( , ) = ( ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )),

q z z t u p

v v x x d

F q t F q t F q t F q t F q t F q t F q t F q t F q t F q t

2 2

1 1 2 0 1

2 1

= ( , ( ), , , , , , , ) ^{n m} ^{n m}

m m

m s r n n

q z z t u p v v x x d R R

R R R R R R R

 

  

       ,

) , , , , , , ), ( , (

= z z t

₁

u p v

₁

v

₂

x

₀

x

₁

d

q

        

 _,

0 >

= const

l

.

Then the functional (2.1) under the conditions (2.2) – (2.4) is continuously Frechet differentiable,

the gradient ¹ ¹ ¹ ¹¹ ¹²

10 11 1

( ) = ( ( ), ( ), ( ), ( ), ( ), ( ), ( ))

u p v v

x x d

I I I I I

I I I H

    

  

    

    at

any point

  X

is calculated by the formula

1 1 1 1 1 1

11 1*

* 1

12 12 2 10 10

0

1 1

11 11 1 1

0 0

( ) = ( ( ), ), ( ) = ( ( ), ), ( ) = ( ( ), ) ( ) ( ),

( ) = ( ( ), ) ( ), ( ) = ( ( ), ) , ( ) = ( ( ), ) , ( ) = ( ( ), ) ,

u u p p v

v

t

v v x x

t

t t

x x d d

t t

I F q t t I F q t t I

F q t t B t t

I F q t t B t I F q t t dt

I F q t t dt I F q t t dt

  



  

 

  

 

  

 



 

(2.6) where

z (t )

, tI is the solution of the differential equation (2.2), and the function

 (t )

, tI is the solution of the conjugate system

1 1* 1

1 1 ( )1 0

= ( ( ), ) ( ) , ( ) = ( ( ), ) .

z t t z t

F q t t A t t

F q t t dt

   

 





(2.7)

In addition, the gradient

I

₁

 (  )

_,

  X

satisfies the Lipschitz condition

1 1 1 2 1 2 1

2

| ( ) ( ) | | , ,

,

I I K

X

    



     



| | | ||

(2.8) where

K = const > 0

.

Proof. Let

 ( t ),  ( t )



 ( t )



X

_,

z ( t , v

₁

, v

₂

)

,

) ,

,

( t v

₁

v

₁

v

₂

v

₂

z    

_,_t_I be a solution of the system (2.2), (2.3). Let

) ( ) , , (

= ) ,

,

( t v

₁

v

₁

v

₂

v

₂

z t v

₁

v

₂

z t

z      

_,_t_I_. Then

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.|

|

| ) (

|  z t  C

₁

|  v

₁

|  C

₂

|  v

₂

|

_(2.9)

The increment of the functional (see (2.5))

1 1 1

1

1 1

0

1 * *

1 1

0

1* 1 1

* *

2 12 0 10

* *

1 11 1

*

= ( ) ( ) =

[ ( ( ) ( ), ) ( ( ), )] =

= [ ( ) ( ( ), ) ( ) ( ( ), ) ( ) ( ( ), )

( ) ( ( ), ) ( ( ), ) ( ( ), ) ( ( ), ) ( )

t

t t

u p

t

v

v x

x d

I I I

F q t q t t F q t t dt

u t F q t t p t F q t t v t F q t t

v t F q t t x F q t t x F q t t d F q t t z t F

  

   

   

   

 

   





1 * 1 1 ( )1

9

=1

( ( ), ) ( ) ( ( ), )]

,

z z t

i i

q t t z t F q t t dt

R



  



(2.10)

where | | ¹| ( )|| ( )| ,

0 1

1 l u t q t dt

R

t







,

| ) (

||

) (

|

| ¹

0 2

2 l p t q t dt

R

t







,

| ) (

||

) (

|

| ¹ ₁

0 3

3 l v t q t dt

R

t







,

| ) (

||

) (

|

| ¹ ₂

0 4

4 l v t q t dt

R

t







,

| ) (

||

|

| ¹ ₀

0 5

5 l x q t dt

R

t







,

| ) (

||

|

| ¹ ₁

0 6

6 l x q t dt

R

t







,

| ) (

||

|

| ¹

0 7

7 l d q t dt

R

t







,

| ) (

||

) (

|

| ¹

0 8

8 l z t q t dt

R

t







dt t q t z l R

t

| ) (

||

) (

|

| ¹ ₁

0 9

9 



  by the Lipschitz

condition (2.5). We note that (see (2.7), (2.9))

1 *

1 1 ( )1 0

1 * * * *

1 1 2 2

0

1 *

1 0

( ) ( ( ), ) =

[ ( ) ( ) ( ) ] ( )

( ) ( ( ), ) .

t t z t t

t t t z

z t F q t t dt

v t B t v t B t dt z t F q t t dt





     

 



(2.11)

From (2.10) and (2.11) we get

1 * *

1 1 1

0

* *

1 11 1

= { ( ) ( ( ), ) ( ) ( ( ), ) ( )[ ( ( ), ) ( ) ( )]

t

u p

t

v

I u t F q t t p t F q t t v t F q t t B t t

    

  



* *

2 12 2

* *

0 10 1 11

( )[ ( ( ), ) ( )]

( ( ), ) ( ( ), )

v

x x

v t F q t t B t x F q t t x F q t t



  

   

,

>

), (

=<

)}

), (

(

⁹ ₁

1 1 =

*

F q t t dt R I R

d

_i _H

d  i   





 ^ ^

where _i

i

R R 

⁹

1

=

, |R |C₃ || ||²,

0 ||

||

|







R

,

at

||



 ||



0

.

This implies the relation (2.6). Let

1 1 1 2

2 0 0 1 1

= ( , , ,

, , , )

u u p p v v v

v x x x x d d

       

       ,

X d x x v v p

u , , , , , , )



(

=

₁ ₂ ₀ ₁



2 . Since

,

|

| ) (

|

| ) (

|

| ) ( ) (

|I₁₁  I₁ ₂ ²l₁₀ q t ² l₁₁  t ² l₁₂  ²

,||

||

| ) (

|,

||

| ) (

|



q t



l

₁₃ 





 t



l

₁₄ 



that

1 1 1 2 2

1 2 2

1 1 1 2 15

0

|| ( ) ( ) || =

| ( ) ( ) | || || ,

t

I I

I I dt l

 

  

  

 





  

(6)

where

l

_i

= const > 0

,

i = 10,15

. This implies the estimation (2.8), whereK = l₁₅ . The theorem is proved.

Lemma 2.3. Let the matrix be

T ( t

₀

, t

₁

) > 0

, the function

F

₁

( q , t )

be convex, with respect to the variable

q



R

^N,

N = 4 n  m  s  r  m

₁_{, i.e.}

1 1 2 1 1 1 2

1 2

( (1 ) ) ( , ) (1 ) ( , ),

, ^N, , [0,1].

F q q F q t F q t

q q R

   

 

    

   

(2.12) Then the functional (2.1) under the conditions (2.2) – (2.4) is convex.

Proof. Let



₁

, 

₂

 X

,





[0,1]

. It can be shown, that

1 2

1 2 1 2

( , (1 ) , (1 ) ) =

( , , ) (1 ) ( , , ),

z t v v v v

z t v v z t v v

   

 

   

  

).

, ( ) , ( ), ,

( 1 2 1 2 ₂ ²

m

Rr

I L v v v

v  ^

 Then

1

1 1 2 1 1

0

2 1 1 1 2

( (1 ) ) = ( ( )

(1 ) ( )) ( ) (1 ) ( ),

t

I F q t

q t dt I I

   

    

  

    



1 2 1 1 1 1 2 0 1

1 1 2 0 1

2 1

, , = ( , , , , , , ),

= ( , , , , , , ).

X u p v v x x d

u p v v x x d

  



 

The lemma is proved.

The initial optimal control problem (2.1) – (2.4) can be solved by numerical methods for solving extremal problems [9,10]. We introduce the following sets

U = { u (



)



L

₂

( I , R

^m

)/ || u ||



 },

},

||

/ ) , ( ) ( {

= ) ,

(

₁ ₂ ₁

1

I R v

 

L I R v





V

^r ^r

}),

||

) , ( ) ( {

= ) ,

( ² ₂ ₂ ² ₂

2 I R v  L I R v 

V ^m ^m /

},

|

| 0, / {

= ¹

1   

 d R^m d d

0  >

is a sufficiently large number. We construct sequences

1 1

0 2

1, , , , }

, , {

= }

{ u p vn vn xn xn dn X

n n

n 

 ,

0,1,2, 

=

n

by the algorithm

1 1 1 1

1 1

1 1 1 11 2 2 2 12

1 1

0 0 0 10 1 1 1 11

1 1 1

0

= [ ( )], = [ ( )],

= [ ( )], = 0,1,2, ,

0 < 2

2

n U n n u n n V n n p n

n n n n

V n v n V n v n

n n n n

S n x n S n x n

n n n d n

n

u P u I p P p I

v P v I v P v I

x P x I x P x I

d P d I n

K

   

 

 

 

 

 

 

 

 

 

 

 

 



 , > 0,

 (2.13)

where

P

_

[  ]

is the projection of the point on the set

_,

K = const > 0

from (2.8).

Theorem 2.4. Let the conditions of Theorem 2.2.

be satisfied, in addition, the function

F

₁

( q , t )

be convex with respect to the variable

q



R

^N and the sequence

{ 

_n

}



X

₁ be determined by formula (2.13). Then:

1) the lower bound of the functional (2.1) is reached under the conditions (2.2) – (2.4)

; ),

min (

= ) (

= )

inf (

1 * 1

1

* 1 1

1

X I

I

X

X 



   



2) the sequence

{ 

_n

}



X

₁ is minimizing

)

inf (

=

= )

lim (

1

* 1 1

1

 



I

n X

n  ;

3) the sequence

{ 

_n

}



X

₁ weakly converges to the point



_*

 X

₁,

u

_n

  u

_*

, p

_n

  p

_*,

1n 1*

v   v

,

v

₂ⁿ

  v

^*₂, x₀ⁿ  x₀^*,

x

₁ⁿ 

x

₁^*,

d

*

d

_n  at n, where



_*

= ( u

_*

, p

_*

, v

₁^*

, v

₂^*

,

1

* *

* 1

0,x ,d ) X

x  ;

4) in order to the problem (1.2) – (1.6) have a solution, it is necessary and sufficient that

0 =

= ) lim I

1

(

_n

I

1*

n





 ;

5) the following estimation of the rate of convergence holds

1 1* 0

0

0 ( ) , = 1,2, ,

= > 0.

n C

I I n n

C const



    (2.14)

Proof. Since the function

F

₁

( q , t )

, tI_is convex, it follows from Lemma 3.3. that the

(7)

functional

I

₁

(  )

,





X

₁ is convex on a weekly bicompact set

X

₁. Consequently, I₁()C¹(X₁) is weakly lower semicontinuous on a weakly bicompact set and reaches the lower bound on

X

₁. This implies the first statement of the theorem.

Using the properties of the projection of a point on a convex closed set

X

₁ and taking into account that I₁()C^1,1(X₁) it can be shown that

1 2 1

1

1( _n) I ( _n_ ) || _n  _n_ ||

I      _,

n = 0,1,2, 

,

0  >

. It follows that: 1) the numerical sequence )}

(

{I₁ _n strictly decreases; 2) ||_n _n_₁||0_at

 n .

Since the functional is convex and the set

X

₁ is bounded, the inequality holds

1 1 * 1 1

1

0 ( ) ( ) || ||,

=ⁿ > 0, = 0,1,2, .ⁿ ⁿ

I I C

C const n

    _

   

 (2.15) Hence, taking into account that

0

||

||_n _n_₁  at n,, we have: the

sequence {_n} is minimizing.

) inf (

= ) (

= )

lim (

1

* 1 1

1

  



I

n X

n  .

Since{_n} X₁,

X

₁ is weakly bicompact, that,

weakly *



n

  

at n.

As it follows from Lemma 3.1., if the value

0 = ) (

_*

1



I

, then the problem of optimal control (1.1) – (1.6) has a solution.

The estimation (2.14) follows directly from the inequalities (2.15),

1 2 1

1

1( _n)I ( _n ) || _n  _n ||

I      .

We briefly outlined above, the main steps in proof of the theorem. Detailed proof of an analogous theorem is given in [16]. The theorem is proved.

For the case when the function

F

₁

( q , t )

is not convex with respect to the variable q, the following theorem is true.

Theorem 2.5. It is supposed, that the conditions of Theorem 2.2. are satisfied, the sequence

} 1

{_n X is determined by formula (2.13). Then:

1) the value of the functional I₁(_n) strictly decreases for

n = 0,1,2, 

; 2) ||_n _n_₁||0 at

 n .

Proof of the theorem follows from Theorem 2.4.

From the results it follows that 1) if

1

* *

* 1

* 0

* 2 1

*

* =(u ,p ,v ,v ,x ,x ,d )X

 is the solution of

optimal control problem (2.1) – (2.4),

for which

I

₁

( 

_*

) = 0

, then

1

* 0

* 1 0

*

*= ( ), , )

(u u t x x US S is admissible control; 2) the function x_*(t;t₀,x^*₀), tIis the solution of differential equation (1.2), satisfies the conditions: x(t₁;t₀,x^*₀)=x₁^*, x_*(t;t₀,x₀^*)G(t),

I

t , the functionals g_j(u_*(),x₀^*,x₁^*)0, 1, 1

= m

j , ( (), *, 1^*)=0

0

* x x

u

g_j  , j=m₁1,m₂ ; 3) the necessary and sufficient condition for the existence of a solution of the boundary value problem (1.2) – (1.6) is

I

₁

( 

_*

) = 0

where



_*

X

₁ is the solution of problem (2.1) – (2.4); 4) for the admissible control, the value of the functional (1.1) equals to

* *

* 0 1

1 * *

0 * * 0 1 *

0

( ( ), , ) =

( ( ), ( ), , , ) = ,

t

J u x x

F x t u t x x t dt 



 

^(2.16)

where x_*(t)=x_*(t;t₀,x₀^*), tI. In the general case, the value

* *

* 0 1

( ( ), , ) ( , , ) = J u  x x  J u x x

0 1

inf ( ( ), , ), J u x x 

. )

, ( ) , ), (

(u  x₀ x₁ L₂ I R^m S₀S₁ Construction of an optimal solution

We consider the optimal control problem (1.1) – (1.6). We define a scalar function

 (t )

_,tI as:

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

= )

(

₀ ₀ ₁

0

I t d x x u x F t

^t

t

 ^ ^ ^ ^





Then (t)=F₀(x(t),u(t),x₀,x₁,t)_,(t₀)=0_, ,

) , ), ( (

=

= )

(t₁  I u  x₀ x₁ 

 _{ _R¹ _/

0

,

0

}, where inf ( ( ), , ) I u x x

0 1 0

,

  

   



  



the value



₀ is bounded from below, in particular



0= 0, if

F

₀



_0.

Now the problem of optimal control (1.1) – (1.6) can be written in the form (see (2.1))