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# Pontryagin&#39;s principle of maximum for linear optimal control problems with phase constraints in infinite dimensional spaces.

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```Bulletin of PFUR
Series Mathematics. Information Sciences. Physics.
No 4. 2008. Pp. 5–19
Математика и теоретическая
механика
UDC 517.95
Pontryagin’s Principle of Maximum for Linear
Optimal Control Problems with Phase Constraints
in Infinite Dimensional Spaces
M. Longla
Department of Differential Equations and Mathematical Physics
Peoples’ Friendship University of Russia
6, Miklukho-Maklaya str., Moscow, Russia, 117198
This paper presents the conditions of optimality for a problem with linear phase constraints
in an infinite dimensional normal space with separated locally convex topology demonstrated
using the works of M.F. Sukhinin in infinite dimensional normal spaces, his theory of differential equations in these spaces when functions are not Bochner-integrable and have no derivative of Gateaux. Problems with phase constraints were analyzed in finite spaces by many
authors like L.S. Pontryagin, L. Graves, V.G. Boltyanskiy, R.V. Gamkrelidze, A.A. Milyutin,
A.V. Dmitruk, N.P. Osmolovskij and others. Using the theory of differential equations of
Prof. M.F. Sukhinin published in his monograph , applying the Gamkrelidze and Pontryagin’s method illustrated in book , we enounced and proved theorems for linear mixed
constraint in the separated locally convex space .
Key words and phrases: nonlinear optimization, topology, differential equations, constraint problems.
1.
Integral, Differentiability and Properties
When functions of the type  (, ) = sin(), were  = (),  ∈ [0, 1] or
(, (), ) =  sin(()/) are used in a problem, one should ask what we mean
talking of derivatives. The first function is nowhere differentiable by Freshet as a
function from  to  , but is –differentiable as a function  :  → ( , ). Here
is a system of bounded subsets of  ,  is the weak topology. The second has no derivative of Gateaux at no point, but is –differentiable as a function  : ∞ → (∞ , ),
were  is the weak* topology. To use properly these functions and others with the
same particularities, we need the following theory in infinite dimensional spaces.
1.1.
Integral and Properties
Here  is a Banach space with an additional locally convex topology,
– () is the unite ball of ,
– () is the set of all bounded subsets of ,
– () is the set of all sequently compact subsets of ,
– ℘() — is the set of seminorms, defining the topology ,
–  — a separated locally convex topology in , satisfying the conditions:
1. () — is closed in  ,
2. (()) — is sequently complete,
3. () ⊂ ( ).
Example. Let ,  — be Banach spaces,  = ℓ(, ) with the strong operator’s
topology . Then the mentioned properties are satisfied.
Received 16th May, 2008.
6
Longla M.
Here the set ℓ ( ,  ) is the space of linear sequently continue operators of
into  , with convergence by virtue of . ℓ( ,  ) is the same set with strong
convergence topology.
Let  = [, ] ⊂ R,  ⊂  (),  () — is the set of measurable subsets of . Let
:  →  be uniformly continuous, () ∈ (),  = 1 < 2 < . . . <  = ,
∈  =  ∩ . Let
{︂
( )( ),  ̸= 0,
=
0,  = 0,

∑︁
∫︁
and
()d =
lim
max Δ →0

=1
with respect to the topology . This limit exist and doesn’t depend on the parameters.
The defined integral∫︀ satisfies the
∫︀ next properties:
1. ∀ ∈ ℓ( , R) :  ()d = ()d.
⃒
⃒
⃒
⃒ ∫︀
∫︀
⃒
⃒
2. ∀ ∈ ℓ( , R) : ⃒ ()d⃒ 6 |||| ||()||d.
⃒
⃒

∫︀
∫︀
3. ∀1 , 2 , 1 , 2 ∈ R, (1 1 () + 2 2 ())d = 1

1 ()d + 2

∫︀
2 ()d.

4. Λ (,  ) = { :  →  | is a class of equivalent measurable functions and
)︃1/
(︃
∫︀

||()|| d < ∞} for 1 6  < ∞, ||||Λ = |||||| =
∫︀

||()|| d
.

∫︀
∫︀

5. For  ∈ Λ1 (,  ), ()d = lim→∞
()d,  — is compact in , | :
→  — is continuous, | :  →  — is bounded, and ( r  ) → 0.
The space of Bochner-integrable functions is a closed in Λ1 (,  ).
6. For  ∈ Λ1 (,  ), the function Φ :  ↦→ Φ() =
′
∫︀
()d ∈  — is differentiable

almost everywhere and its derivative is Φ () = ().
7. Φ() — is absolutely
continuous as function of  into .
{︂
8. 11 (,  ) =  | ∃ ∈ , ∃ ∈ , ∃ ∈ Λ1 (,  ) :  () =  +
∫︀
}︂
()d .

The defined integral also satisfies other properties of the Lebesgue integral necessary in this paper (see [1, § 6]).
1.2.
Differentiability and its Properties
Def 1. The function  :  →  is said to be -small at 0 ∈  , if
∀ ∈ ℘() ∀ ∈ ∃ > 0 ∀ℎ ∈
∀|| < ,
0 + ℎ ∈  :
((0 + ℎ)) 6 ||.
Def 2. The function  :  →  is said to be -equivalent to the operator  ∈
ℓ(, ) at 0 ∈  , if (ℎ) =  (0 + ℎ) −  (0 ) − ℎ is -small at 0. Moreover, if  is
defined for all ℎ ∈ , then  is -differentiable and its -derivative at 0 is .
Def 3.  is sequently (, 1 )-Lipshitzian at 0 ∈  , if ∀ ⊂  ∀{ℎ } ⊂
∀{ } ∈ 0 (R),  ̸= 0, 0 +  ℎ ∈  : {−1
[ (0 +  ℎ ) −  (0 )]} ∈ 1 .
Def 4. The mapping  :  →  is said to be open at 0 , if ∀Ω ⊂ , 0 ∈ Ω :
(0 ) ∈  (Ω). If the contrary yields, we said that the mapping  is critical at 0
(see [3, p. 781–839]).
Def 5. If ,  are seminormal spaces, then the mapping  :  →  is said to be
correct (see [4, p. 223–228]) or have the covering property ( see [5, p. 39–44], [6, p. 11–
46]) at 0 , if ∃ > 0 ∀ ∈]0, [:  (0 ) + ( ) ⊂  (0 + ()). If the contrary
yields, we said that the mapping  is quasicritical at 0 .
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
7
Def 6. Let ,  — be seminormal spaces. The mapping  :  →  is lipschitzian
at 0 ∈ , if ∃ > 0, ∃ > 0 ∀ ∈]0, [:  (0 + ()) ⊂  (0 ) + ( ).
Remark 1. Let  be a locally convex space. 1 :  →  — is a linear operator,
2 ∈ (, ) , and  :  →  and  :  ( ) →  are respectively —equivalent to 1
at 0 and 1 —equivalent to 2 at  (0 ). If  is sequently (, 1 )–Lipshitzian at 0 ,
then  ∘  :  →  is -equivalent to 2 ∘ 1 at 0 .
Remark 2. On the critical and quasicritical properties.
1. From the covering property of the mapping at a point follows its openness at this
point, or in other words, from the criticity of the mapping at a given point follows its
quasicriticity at this point.
2. If  ,  ,  — are topological spaces, the mapping  :  →  is continuous at
0 ∈ , and  :  () →  is critical at  (0 ), then  ∘  :  →  is critical at 0 .
3. If , ,  — are seminormal spaces, the mapping  :  →  is lipschitzian at
0 ∈ , and the mapping  :  () →  is quasicritical at  (0 ), then  ∘  :  →
is quasicritical at 0 .
2.
Formulation of the Problem
2.1.
General Settings
Let  be a convex functional,
()
∈
Λ1 ([0 , 1 ],  ( ,  )),

−() ∈ Λ1 ([0 , 1 ], (R ,  )), ℎ () ∈ (, ( , R)),  () ∈ (, R ).
¯ 0 , 1 , 0 , 1 ) ∈ R . Let be
¯ : (0 , 1 , 0 , 1 ) ↦→ (
Let be defined the mapping
given the equations:
()
˙
= ()() + ()(),  ∈ ,  ∈ ,
we put (0 ) = 0 , (1 ) = 1 ,
(1)
(2)
[ℎ (), ()] +  ()() 6 0,  = 1, ,
[ℎ̄ (), ()] + ¯ () 6 0,  = 1, ,
¯ — is quasicritical at (*0 , *1 , *0 , *1 ).

(3)
(4)
(5)
Question: Find de necessary conditions of existence of the solution of the system
(1)–(4), for which at the given point  is quasicritical.
In order to answer to this question, we check separately the subsystem (1)–(3), with
(5) and the subsystem (1)–(2), (4)–(5). Having the results from the two subsystems,
we combine them to find the answer to the question for the system (1)–(5).
Simultaneously, we use the answers to get the necessary conditions of existence of
the optimal control for problems with linear constraints in the form of Pontryagin’s
principle of maximum in infinite dimensional spaces. To get this result, it suffices to
¯ = (, ), if we want to investigate in infinite dimensional
take in the above problem
banach space the case of the optimal control problem formed by (1)–(4) and the next
equations:
(0 , 1 , 0 , 1 ) = 0,
(0 , 1 , 0 , 1 ) → min .
(6)
(7)
Let set the next conditions on ,  and , that allow us work with the defined
integral and to differentiate by virtue of the topology the functions that we use:
(, (), ) = sup lim
sup
∈ →0 ∈[−1 (−)∩()]
{||−1 || ( + ) −  ()||},
(,  , ) = { :  →  |(, ||.||, (), ) < ∞},
|| ||1 = || || + (, (), ).
8
Longla M.
In the space  (,  , ) is defined a topology with the next basis at 0:
Ξ(, , ) = { |( (˜
)) + (, , , ) < 1},
where  ∈ ℘( ),  ∈ , and  ∈ ( ).
1.
2.
3.
4.
5.
() — is closed in  , (()) — is sequently complete,
() ⊂  ⊂ ( ), ∀ ∈  ∀ ∈ ℓ( ,  ) ∀{ } ⊂ , { } ⊂ ,
′
′
( ∈ ,  ⊂ ) =⇒  ∈ ,
( ∈ ,  > 0) =⇒ [−,  ] ∈ ,
′
′
( ∈ ,  ∈ ) =⇒  U ∈ .
2.2.
Problem with Regular Constraints
As announced, let check the case of the subsystem (1)–(3), (5).
Let
ℎ() = (ℎ1 (), . . . , ℎ ()) ∈ [ℓ( , R)] ,
¯ = (˜
, ), () = (1 (), . . . ,  ()) ,

˜˙ () = [ℎ(), ()] + ()(). The solution {* , * , *0 , *1 , *0 , *1 } of the given system
is also solution of ():
∫︁
() =
∫︁
()()d +
0
()()d + 0 ,
0
∫︁

˜=
(8)
∫︁
[ℎ(), ()]d +
0
()()d.
(9)
0
The optimal solution of our system also satisfies the necessary conditions of optimality for the system (8)–(9) with the mapping (5), for which holds the proposition:
2.2.1.
Existence of the Admissible Solutions
Proposition 1 (Existence of the admissible solutions). The system (8)–
¯  × R ) for each  ∈ 1 (R ),
(9) has a solution
¯ = (˜
, ) ∈ Λ1 ([0 , 1 ],
∈ Λ1 ([0 , 1 ], ℓ ( ,  )) that we can express by the formulas:
() = ℜ(, 0 )0 +
∫︁
ℜ(, ) ∘ ()()d,
(10)
0
∫︁

˜() =
∫︁
[ℎ, ()]d +
0
()()d.
(11)
0
This is a direct consequence of the next result from :
Proposition 2 (Existence and uniqueness of the integral equation’s solution). Let
∈ Λ∞ (,  ), 0 ∈ , () = ( ),  — is an infinite dimensional normal space,
— is a separated locally convex topology in , and  ⊂ (), () : ([0 , 1 ] →
ℓ ( ,  )) is -integrable. Then the equation
() −
∫︁
∫︁
()()d = ()
has
() = () +
0
ℜ(, )()()d
0
as unique solution in Λ∞ (,  ). Here 11 (, ℓ ( ,  )) ∋ ℜ :  ×  → ℓ ( ,  )
is the resolvent kernel of ′ = (). We have
′
ℜ (, ) = −ℜ(, ) ∘ (),
′
ℜ (, ) = () ∘ ℜ(, ).
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
Proof. ∀ ∈ N, ∀ ∈ , ∃Π () ∈ [0 , 1 ],  ∈ ,
∈ Λ1 (, ℓ (R ,  )),  ∈ 1 (, ℓ(R , R )) :
1. ([0 , 1 ] r Π ()) < 1 , Π () — is compact;
2.  — is continuous on Π () and lim  ()=();
˙
9
∈ Λ1 (, ℓ ( ,  )),
→∞
3.  — is continuous on Π () and lim  ()=();
˙
→∞
4.  — is continuous on Π () and lim  ()=();
˙
→∞
5.  — is continuous on Π () and lim  ()=();
˙
Π () ⊂ Π+1 ().
→∞
Using the defined approximations for the given systems on Π(), we obtain a
continuous solution  by the enounced proposition with ℜ (0 , ). Taking into account
the properties
of , it easily comes out that ℜ (0 , ) → ℜ(0 , ) and  →  on
⋃︀
Π() = ∞
=1 Π ().
Let now study the problem on Π () with the defined sequences. In order to
reduce the quantity of indexes in this paper, we will denote the sequences just as their
limits, as they can’t be misunderstood in this case, keeping in mind that after all our
operations the results should be turned to the limits of the used functions.
Let set 0 () = 0 +0 , 1 () = 1 +1 ,
¯0 () =
¯*0 +
¯0 , (, ) = * ()+(),

¯1 () =
¯*1 +
¯1 + 0 (
¯*0 , ).
(12)
Equations (10)–(11) define  (¯
0 , , 0 , 1 ) =
¯(1 ), and (12) a continuous operator
: R+ → (¯
0 ) × (¯
1 ) × (0 ) × (1 ).
For the case of the optimal control problem, let  = (*0 , *1 , *0 , *1 ).
Then for each solution of (8)–(9) we have (¯
0 ,  (¯
0 , , 0 , 1 ), 0 , 1 ) = 0 and
(¯
0 ,  (¯
0 , , 0 , 1 ), 0 , 1 ) > . And, for the quasicritical mapping we have
¯ ∘ ()(

¯0 ,
¯1 , 0 , 1 ) − is critical at  = 0.
From here we find some , for which  > 0. (see )
Therefore, the optimal solution of the initial problem should satisfy the necessary
conditions of optimality given by the result of the next problem:
¯
()
=  ∘ ()(
¯0 ,
¯1 , 0 , 1 ) = 0,
¯
() =  ∘ ()(
¯0 ,
¯1 , 0 , 1 ) → min .
(13)
(14)
For (13)–(14) and vectors from the above solution, as the critical value of  is 0,
we obtain for some  ∈ R ,  ∈ R r {0}, () : [*0 , *1 ] → R the following:
1 +
0 + ¯1 Δ¯
1 + 0 0 + 1  + ¯0 Δ¯
0 + ¯1 Δ¯
¯0 Δ¯
*
∫︁1
˙
[(), ()
− ()() − ()()]d +
+ 0 0 + 1  +
*
0
*
∫︁1
+
[(), [ℎ, ()] + ()()]d 6 0, (15)
*
0
Δ() = ℜ(, *0 )0 + ℜ(, *1 )(()* () + ()* ()) −
−
ℜ(, *0 )((*0 )*0
+
(*0 )* (*0 ))0
∫︁
+
*
0
ℜ(, ) ∘ ()()d, (16)
10
Longla M.

˜˙ () = ([ℎ, * ()]+()* ())−([ℎ, 0 ]+(*0 )* (0 ))0 +[ℎ, ()]+()(). (17)
From (15)–(16), we obtain the following:
− [(1 + 1 + (*1 )) ∘ ℜ(*1 , *0 ), (*0 )*0 + (*0 )* (*0 )] +
+ [0 + 0 , (*0 )*0 + (*0 )* (*0 )] + 0 + 0 = 0, (18)
[1 + 1 , (*1 )*1 + (*1 )* (*1 )] + 1 + 1 = 0,
(1 + 1 + (*1 )) ∘ ℜ(*1 , *0 ) + 0 + 0 − (*0 ) = 0,
*
⎡
⎢
*
⎣1 + 1 + (1 ),
∫︁1
(19)
(20)
⎤
ℜ(*1 , )()()d⎦ 6 0.
⎥
(21)
*
0
Knowing from the above inequalities that () = max (, ) = (, * ),
∈
(, ) = [(), ()], using the regularity of the set  , we find functions
(), 1 (), · · · ,  (), for which
*
() ∘ () = ∇ (,  ) = () ∘ () +

∑︁
∇ ˜ (* ) ⇒
=1
⇒ ()()() = [(), ()()].
On the other hand, taking into account rang(())=,
˙ we find a measurable Λ(),
satisfying [() + Λ()(), ()] = 0. Then [(), Λ()] = −(), and
˙
()
= −() ∘ () + ()ℎ,
*
(*1 ) = −1 − 1 ,
(*0 ) = (*1 )ℜ(*1 , *0 ) −
∫︁1
()ℎ ∘ ℜ(, *0 )d.
*
0
Taking into account this fact, (21) vanishes and (18)–(20) become:
[(*0 ), (*0 )*0 + (*0 )* (*0 )] = −0 − 0 ,
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 + 1 ,
(*0 ) = 0 + 0 .
(18′ )
(19′ )
(20′ )
Let define 1 , · · · ,  ,  : 0 < 1 ,  6 +1 ,  = 1 ,  ∈ Π (). Here Π () ⊂  ()
with measure (Π ()) = 1/, and the used functions are uniformly continuous on
Π (), for all  ∈ N. Such a subset exist according to [1, § 6.9.10].
Let 1 > 0, . . . ,  > 0,  ∈ R and {1 , . . . ,  } ⊂  , ( =  is possible). Let
= [ +  ,  + ( +  )],  = 1, , for  defined as follows:
=
⎧
⎪
⎨  − ( + · · · +  ), if  =  ;
⎪
⎩
− ( + · · · +  ), if  =  <  ;
− ( + · · · +  ), if  = +1 = · · · =  < +1
( < ).
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
11
We choose  so that  ∩  = ∅, and  ⊂ [0 , 1 + ], and define
{︃
() =
* (), ∀ ∈
/ Π () ∩ ∪1  ,
,
∀ ∈  ∩ Π ()
{︃
or
() =
0,
∀ ∈
/ Π () ∩ ∪1  ,
− * (), ∀ ∈  ∩ Π ().
Considering in (15) that the integrals vanish, 0 = 0, 0 = 0,  =  = 0, we
obtain [1 + 1 , (*1 )(*1 )] > 0, what leads to
((*1 ), (*1 )) 6 ((*1 ), * (*1 )),
[() ∘ ℎ, * ] + () ∘ () = 0,
˙ + () ∘ ℎ.
() ∘ () = −()
(21′ )
(22)
(23)
Using () = −[(), Λ()], we obtain
˙
()
= −()(() + Λ() ∘ ℎ) and
∫︁
() =
(() + Λ() ∘ ℎ)()d.
(23* )
*
0
Therefore, (16) becomes
Δ() =

∑︁
ℜ̃(*1 ,  ) ∘ ( )( − * ( )) ,
(16′ )
=0
where ℜ̃(, *0 ) is the resolvent kernel of (23* ). Coming back to (15) with the previous
changes and (16′ ), we obtain
( ) = (*1 )ℜ̃(*1 ,  ), [( ), ( )( )] 6 0 or
(( ), ( )) 6 (( ), * ( )) for all .
(21′′ )
We can conclude that (21′′ ) holds for all  ∈ Π (). And taking the limit in the proved
expressions,⋃︀we obtain the necessary conditions for the optimal control and (21′′ ) holds
on Π() = ∞
=1 Π ().
The result makes sense only if (*1 ) = 1 + 1 ̸= 0. This condition is guaranteed if for some 0 , {01 , 1 } is linearly independent. The satisfaction of the above
condition and those of the problem formulation leads to the following theorems.
2.3.
Theorems
Here we enounce theorems for different cases, taking into account the above transformation.
Theorem
1
(Analog
of
the
maximum
principle).
Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable optimal solution of
(1)–(3), (6)–(7).
Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ).
Let
∈ Λ1 (,  ).  is (2 ) × (R2 ) — differentiable at (*0 , *1 , *0 , *1 ). If
: (*0 ) × (*1 ) × (*0 ) × (*1 ) → R is continuous in ((*0 , *1 )), rang (())=
˙
and for some 0 , {01 (*0 , *1 , *0 , *1 ), 1 (*0 , *1 , *0 , *1 )} is linearly independent,
then ∃() ∈ 11 ([*0 , *1 ], ℓ ( , R)), and () ∈ 1 (, R ), for which holds:
∀ ∈  ∃Π() :
¯(Π()) = *1 − *0 and ∀ ∈ Π(),
˙ * () − ()* () =  ((), * ()) ≡ ()* (),
(24)
12
Longla M.
˙
()
= −()() + () ∘ ℎ(),
*
((), ()) 6 ((),  ()), [() ∘ ℎ(), * ()] + () ∘ ()* () = 0,
(25)
(26)
where () comes from the maximum’s condition;
[(*0 ), (*0 )*0 + (*0 )* (*0 )] = −0 − 0 , (*0 ) = 0 + 0 ,
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 + 1 , (*1 ) = −1 − 1 .
(27)
(28)
From this theorem follows:
Corollary
1
(Integral
form
of
the
maximum
principle). Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable optimal solution of
the given system.
Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ).
Let
If
∈ Λ1 (,  ).  is (2 ) × (R2 ) — differentiable at (*0 , *1 , *0 , *1 ).
*
*
*
*

*
*
: (0 ) × (1 ) × (0 ) × (1 ) → R is continuous in ((0 , 1 )), rang (())=
˙ and
for some 0 , {01 , 1 } is linearly independent, then ∃() ∈ 11 ([*0 , *1 ], ℓ ( , R)),
and () ∈ 1 (, R ), for which holds : ∀ ∈  ,
*
() =
*0
∫︁
+
()* ()d,
˙ = −()() + () ∘ ℎ(),
(29)
*
0
*
*
∫︁1
∫︁1
((), ())d 6
*
0
((), * ())d,
(30)
*
0
where () comes from the maximum’s condition;
[(*0 ), (*0 )*0 + (*0 )* (*0 )] = −0 − 0 , (*0 ) = 0 + 0 ,
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 + 1 , (*1 ) = −1 − 1 ,
[() ∘ ℎ(), * ()] + () ∘ ()* () = 0.
(31)
(32)
(33)
If  = (1 , 0 , 1 ) and  = (1 , 0 , 1 ), and 0 is a known vector, then the
variation vanishes at this point, and we easily get from the proof of the theorem 1 and
corollary 1:
Corollary 2 (Case of fixed initial point). Let {* (), * (), *1 , *0 6  6 *1 } —
be the measurable optimal solution of the problem. Let * () ∈ 11 (,  ) and
* () ∈ ∞ (, R ). Let  ∈ Λ1 (,  ).  is ( )×(R2 )− differentiable at (*1 , *0 , *1 ).
If  : (*1 ) × (*0 ) × (*1 ) → R is continuous in (*1 ) for fixed values *0 , *1 ,
rang (())=
˙ and for some 0 {01 (*1 , *0 , *1 ), 1 (*1 , *0 , *1 )} is linearly independent,
then there exist () ∈ 11 ([*0 , *1 ], ℓ ( , R)), and () ∈ 1 (, R ), for which holds:
∀ ∈  ∃Π() :
¯(Π()) = *1 − *0 ,
˙
()
= −()() + () ∘ ℎ(),
*
((), ()) 6 ((),  ()), [() ∘ ℎ, * ()] + () ∘ ()* () = 0,
(34)
(35)
where () comes from the maximum’s condition;
[(*0 ), (*0 )*0 + (*0 )* (*0 )] = −0 − 0 ,
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 + 1 , (*1 ) = −1 − 1 .
(36)
(37)
If 0 , 0 are known parameters, then 0 = 0, 0 = 0 and the next corollary hold:
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
13
Corollary 3 (Case of fixed initial point and initial time). Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable optimal solution of the problem.
Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ). Let  ∈ Λ1 (,  ).  is ( ) × (R)−
differentiable at (*1 , *1 ). If  : (*1 ) × (*1 ) → R is continuous in (*1 ) for the
fixed value *1 , rang (())=
˙ and for some 0 {01 (*1 , *1 ), 1 (*1 , *1 )} is linearly
independent, then there exist () ∈ ([0 , *1 ], ℓ ( , R)), and () ∈ 1 (, R ), for
which holds: ∀ ∈  ∃Π() :
¯(Π()) = *1 − 0 , ∀ ∈ Π(),
˙
()
= −()() + () ∘ ℎ(),
((), ()) 6 ((), * ()), [() ∘ ℎ(), * ()] + () ∘ ()* () = 0,
(38)
(39)
where () comes from the maximum’s condition;
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 + 1 ,
(*1 ) = −1 − 1 .
(40)
Corollary 4 (Integral form for fixed initial point and time). Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable solution of the problem for which
¯ is quasicritical. Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ). Let  ∈ Λ1 (,  ).

¯ is ( ) × (R)− differentiable at (*1 , *1 ). If
¯ : (*1 ) × (*1 ) → R is continuous

¯ 0 (*1 , *1 ) ̸= 0,
in (*1 ) for the fixed value *1 , rang (())=
˙ and for some 0 ,
1
1
*

then there exist () ∈ 1 ([0 , 1 ], ℓ ( , R)), and () ∈ 1 (, R ), for which holds
∀ ∈  :
*
() =
*0
∫︁
+
˙
(()* () + ()* ())d, ()
= −()() + () ∘ ℎ(),
(41)
0
*
*
∫︁1
∫︁1
((), ())d 6
((), * ())d,
(42)
*
0
0
[() ∘ ℎ(), * ()] + () ∘ ()* () = 0,
where () comes from the maximum’s condition;
[(*1 ), (*1 )*1 + (*1 )* (*1 )] = 1 ,
2.4.
(*1 ) = −1 .
(43)
Case of Irregular Phase Constraints
Now let be given the system ()* that consist of (1)–(2), (4)–(5). For this system, we suppose that for each  the constraints’ mapping have almost everywhere
derivatives. We set  = max{ }, where  is the least number for which
]︀
)︀
d (︀[︀
ℎ̄ (), () + ¯ () |(1) = [ℎ (), ()] +  ()() + ¯ (),

d
.
() ̸= 0, () = (1 , . . . ,  ), rang (()) = .
Let define the next variables:
, () = −
]︀
)︀
d − (︀[︀
ℎ̄ (), () + ¯ () |(1) ,  = 1,  − 1,

−

d
, () = 0,  =  ,  − 1,
(︀[︀
]︀
)︀
2
() = − ℎ̄ (), () + ¯ () ,
,
= (1, , . . . , , ) ,
˙  = (1, ˙ 1, , . . . , , ˙ , ) .
14
Longla M.
Using the properties of the defined parameters, we have
˙1 () = −[ℎ(), ()] − ()() − ¯(),
˙ () = −1 (),  = 2,  − 1,
2 ()˙ () = (1,1 −1 , . . . , , −1 ),

¯() = ((), 1 (), . . . ,  ()).
The system ()* equivalent to a system of the kind () with a new
(0 , 1 , 0 , 1 )
1
⎜,1 () +
([ℎ̄ (), ()] + ¯ ()) |(1)
⎜
⎜
..
¯ 0 ,
(¯
¯1 , 0 , 1 ) = ⎜
⎜
.
⎜

1
⎝, −1 () +
([ℎ̄ (), ()] + ¯ ()) |(1)
= 1,
⎛
⎞
−1
−1
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
Hence holds an equivalent of the expression (15) for all variations with some
¯ = (, 11,1 , . . . , 11, , . . . , 1,1 , . . . , 1, ):
1, ∈ R,  ∈ R r {0} and
¯  )1 +
¯  )0 + (
¯  )Δ1 + (
¯  )Δ0 + (
¯
¯
¯
(
¯
1
0
1
0
+
1
∫︀
∑︀
˙ ,1 + [ℎ (), ()] +  ()())d+
,1 (
=1 0
+
1
∫︀
∑︀
=1 0
, (2, ˙ , − , −1 )d +
*
1
−1 ∫︀

∑︀
∑︀
(15* )
˙ , − ,−1 )d+
, (
=1 =2 0
∫︀1
˙
+ [(), ()
− ()() − ()()]d > 0.
*
0
Therefore, we obtain for the optimal control problem the conditions:
(*1 ) = 1 + 1 +

∑︁
∑︁
(︃
1,
=1 =1
)︃⃒*1
⃒
⃒
⃒ ,
⃒
]︀
]︀
d [︀
d−1 [︀
*
ℎ̄
(),

()
−
ℎ̄ (), * ()

−1

d
d

∑︁
∑︁
(*0 ) = −0 − 0 −
(︃
1,
=1 =1
˙
()
= −()() +

∑︁
(1)
)︃⃒*0
⃒
]︀
d [︀
⃒
*
ℎ̄
(),

()
⃒ ,

⃒
d
(1)
(*0 ) = 0 + 0 ,
,1 ℎ (),
(44)
=1
(*1 )
= −1 − 1 −

∑︁
∑︁
⃒*1
1,
=1 =1
( (*1 )) −

∑︁
⃒
−1
⃒
*
[
ℎ̄
(),

()]
⃒

⃒
−1
,
(45)
(1)
,1 (*1 )  (*1 )  (*1 ) + ¯ (*1 ) 6
(︁
)︁
=1
6  (* (*1 )) −

∑︁
,1 (*1 )  (*1 ) * (*1 ) + ¯ (*1 ) , (46)
(︁
)︁
=1
˙ , () = −,+1 (),
, (*1 ) = −1, ,
, (*0 ) = 0,
= 1,  − 1,
(47)
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
( )
,1 () [ℎ (), * ()] +  ()* () + ¯ () = 0,
(︀
)︀
15
= 1, .
(48)
Varying now the points of contact with the boundary set
= { ∈  : ℎ̄ (), () + ¯ () = 0},
[︀
]︀
by setting the points 1 =  , 2 =  , where  — is the point of entry and  —
is the point of exit, we obtain a varied solution that differs from the optimal solution
only in the small neighborhoods of  and  . Then, we set Δ0 = Δ(−
) = 0,
+
+
−
+
Δ1 = Δ( ), , ( ) = 0, , ( ) = 0, Δ, ( ) = 0,  = 2,  ,  = 1, .
Using these considerations in (15* ), we obtain:
(+
)
=
(−
)
−

∑︁
,1 (−
)
(︁[︀ (︀ )︀
(︀ )︀ * (︀ − )︀ ¯ (︀ − )︀)︁
(︀ − )︀]︀
*
+   , (49)
+  −
ℎ −

,
=1

(︀ )︀
(︀ + )︀ ∑︁
(︀ + )︀ (︁[︀ (︀ + )︀ * (︀ + )︀]︀
(︀ + )︀ * (︀ + )︀ ¯ (︀ + )︀)︁
−
=

−

ℎ

,

+

+   . (50)
,1

=1
Therefore, we can introduce the next conditions on the used functions:
Condition 1.  :  → ℓ( ,  ) − is measurable, and
(.) ∈ 1−1 (, ℓ( ,  )),
Condition 2.  :  → ℓ(R ,  ) − is measurable, and
(.) ∈ Λ̃1 (, ℓ(R ,  )),
Condition 3. ℎ̄ :  → ℓ( , R ) − is measurable, and ℎ̄(.) ∈ 1 (, ℓ(,  , R )),
Condition 4. ¯ :  → ℓ(R , R ) − is measurable, and ¯(.) ∈   (, R ),
1
Condition 5.  and  - (2 ×R2 )-differentiable at (*0 , *1 , *0 , *1 ) and continuous
in its neighborhood, where (*0 , *1 , *0 , *1 ) — is the quasicritical (optimal parameter) of
the problem.
Theorem
2
(Irregular case). Let hold conditions 1–5.
Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable optimal solution of the
problem. Let {︀* () ∈ 11 (,  ) and * () ∈ }︀∞ (, R ). If rang (())=
˙ and
for some 0 , 01 (*0 , *1 , *0 , *1 ) , 1 (*0 , *1 , *0 , *1 ) is linearly independent, then
∃() ∈ 11 ([*0 , *1 ] , ℓ ( , R)), and , () ∈ 11 (, R), for which holds : ∀ ∈
∃Π() :
¯(Π()) = *1 − *0 ,
˙ * () = ()* + ()* (),
˙
()
= −()() +

∑︁
(51)
,1 () ∘ ℎ (),
(52)
=1
(()) −

∑︁

∑︁
,1 () ()() 6 (* ()) −
=1
,1 () ()* (),
(53)
=1
where ,1 () comes from the maximum’s condition;

∑︁
−
(+
) −   = −
(︀
)︀
,1 −

(︀
)︀ (︁[︀
* −
ℎ −

,
(︀
)︀
(︀
)︀]︀
* −
+  −
+ ¯ −

)︀)︁
,
* +
+   +
+ ¯ +

)︀)︁
,
(︀
)︀
(︀
)︀
(︀
=1
+
−
−   = −
(︀
)︀
(︀
)︀

∑︁
=1
,1 +

(︀
)︀ (︁[︀
* +
ℎ +

,
(︀
)︀
(︀
)︀]︀
(︀
)︀
(︀
)︀
(︀
16
Longla M.

(*0 )
= −0 − 0 −

∑︁
∑︁
1,
(︂
=1 =1

(*1 )
= 1 + 1 +

∑︁
∑︁
1,
(︂
=1 =1
(54)
)︂⃒ *
]︀
]︀ ⃒1
d [︀
d−1 [︀
*
*
ℎ̄ (),  () −  ℎ̄ (),  () ⃒⃒ ,
d−1
d
(1)
(55)
(*0 ) = 0 + 0
(*1 ) = −1 − 1 −
)︂⃒ *
]︀ ⃒0
d [︀
*
ℎ̄ (),  () ⃒⃒ ,
d
(1)
and hold (47) and (48),

∑︁
∑︁
1,
=1 =1
d−1
d−1
(56)
⃒*
[︀
]︀⃒1
*
ℎ̄ (),  () ⃒⃒ .
(57)
(1)
¯
We say that a solution of the system of equations is quasicritical for the mapping ,
if it offers the quasicritical point of this mapping. As consequence of the above theorem
we have the integral form of the theorem:
Corollary 5 (Integral form of the conditions of singularity). Let hold the conditions 1–5. Let {* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable quasicritical solution of the problem. Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ). If
rang (())=
˙ and for some 0 , 01 ̸= 0, then ∃() ∈ 11 ([*0 , *1 ], ℓ ( , R)), and
, () ∈ 11 (, R), for which holds: ∀ ∈  ,
*
() =
*0
∫︁
+
(()* () + ()* ()) d,
(58)
*
0
˙ = −()() +

∑︁
,1 () ∘ ℎ ().
(59)
=1
*
*
∫︁1
( (()) −

∑︁
∫︁1
( ( ()) −
,1 () ()())d 6
=1
*
0
*

∑︁
,1 () ()* ())d, (60)
=1
*
0
where ,1 () comes from the maximum’s condition;

∑︁
(︀ − )︀ * (︀ − )︀ ¯ (︀ − )︀)︁
(︀ − )︀ (︁[︀ (︀ − )︀ * (︀ − )︀]︀
(︀ − )︀
(︀ )︀
+   ,
,

+

ℎ
=
−
+
−

,1

=1

∑︁
(︀ )︀
(︀ + )︀
(︀ )︀ (︁[︀ (︀ + )︀ * (︀ + )︀]︀
(︀ )︀ * (︀ + )︀ ¯ (︀ + )︀)︁
−
,1 +
ℎ  ,   +  +
+   ,
−   = −

=1
(*0 ) = −0 −

∑︁
∑︁
=1 =1

(*1 )
= 1 +

∑︁
∑︁
1,
=1 =1
(*0 )
(︂
(︂
)︂⃒ *
]︀ ⃒0
d [︀
*
⃒ ,
ℎ̄
(),

()

⃒
d
(1)
)︂⃒ *
]︀
]︀ ⃒1
d−1 [︀
d [︀
*
*
ℎ̄ (),  () −  ℎ̄ (),  () ⃒⃒ ,
d−1
d
(1)
= 0 ,
(*1 ) = −1 −
1,

∑︁
∑︁
=1 =1
and hold (47) and (48),
(︂
)︂⃒ *
]︀ ⃒1
d−1 [︀
*
⃒ .
ℎ̄
(),

()
1,

⃒
d−1
(1)
(61)
(62)
(63)
(64)
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
2.5.
17
General Case of Linear Mixed Constraints
It’s easy to prove, that other versions of the two theorems also hold for this case.
For the case when all the equations are taken into account, the irregular constraints
influence the maximums’ condition, and using the first two cases and the condition of
regularization of the general system, we can state the following theorem:
Theorem 3 (General case). Let hold the conditions 1–5.
Let
{* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable quasicritical solution
of the problem (1)–(5). Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ). If
rang ((), ())=
˙ + . Then ∃() ∈ 11 ([*0 , *1 ], ℓ ( , R)), and , () ∈ 11 (, R),

() ∈ 1 (, R ), for which holds: ∀ ∈  ,
*
() =
*0
∫︁
+
(()* () + ()* ()) d,
(65)
*
0
˙ = −()() +

∑︁
,1 () ∘ ℎ () + () ∘ ℎ(),
(66)
=1
*
*
∫︁1
( (()) −

∑︁
∫︁1
=1
*
0
where ,1 (),
( (* ()) −
,1 () ()())d 6

∑︁
,1 () ()* ())d, (67)
=1
*
0
() comes from the maximum’s condition;

(*0 )
¯ 0 −
= −

∑︁
∑︁
=1 =1
¯ 1 +
(*1 ) =

∑︁
∑︁
1,
=1 =1
¯ 1
(*1 ) = −
(︂
1,
(︂
)︂⃒ *
]︀ ⃒0
d [︀
*
ℎ̄ (),  () ⃒⃒ ,
d
(1)
)︂⃒ *
]︀
]︀ ⃒1
d−1 [︀
d [︀
*
*
⃒ ,
ℎ̄
(),

()
−
ℎ̄
(),

()

⃒
d−1
d
(1)
(*0 ) = 0 ,
)︂⃒ *
(︂

∑︁
∑︁
]︀ ⃒1
d−1 [︀
*
1
ℎ̄ (),  () ⃒⃒ ,
,
−
d−1
(1)
(68)
(69)
(70)
(71)
=1 =1
−
(+
) − ( ) = −

∑︁
* −
−
* −
¯ −
ℎ (−
),  ( ) +  ( ) ( ) + ( ) ,
,1 (−
)
(︁[︀
,1 (+
)
(︁[︀
]︀
)︁
=1
(−
)
−
(+
)
=−

∑︁
* +
+
* +
¯ +
ℎ (+
),  ( ) +  ( ) ( ) + ( ) ,
]︀
)︁
=1
[()ℎ(), ()()] + ()()* () = 0,
and hold (47) and (48).
As a consequence we have the Pontryagin’s principle of maximum for linear optimal
control problems:
Corollary 6 (General Case for the optimal control problem). Let hold the conditions 1–5. Let {* (), * (), *0 , *1 , *0 6  6 *1 } — be the measurable optimal
solution of the general problem. Let * () ∈ 11 (,  ) and * () ∈ ∞ (, R ).
If rang((), ())=
˙ +  and for some 0 , {01 (*0 , *1 , *0 , *1 ), 1 (*0 , *1 , *0 , *1 )} is
18
Longla M.
linearly independent, then ∃() ∈ 11 ([*0 , *1 ], ℓ ( , R)), and , () ∈ 11 (, R),
() ∈ 1 (, R ), for which holds: ∀ ∈  ∃Π() :
¯(Π()) = *1 − *0 ,
˙ * () = ()* () + ()* (),

∑︁
˙
()
= −()() +
(72)
,1 () ∘ ℎ () +  ∘ ℎ(),
(73)
=1
(()) −

∑︁
,1 () ()() 6 (* ()) −
=1

∑︁
,1 () ()* (),
(74)
=1
where ,1 (), () come from the maximum’s condition, hold (47) and (48);
−
+
−   = −
(︀
)︀
(︀
)︀

∑︁
,1 −

)︀ (︁[︀
,1 +

)︀ (︁[︀
(︀
* −
ℎ −

,
(︀
)︀
(︀
)︀]︀
* −
+  −
+ ¯ −

)︀]︀
* +
+  +
+ ¯ +

(︀
)︀
(︀
)︀
(︀
)︀)︁
,
)︀)︁
,
=1
+
−
−   = −
(︀
)︀
(︀
)︀

∑︁
(︀
* +
ℎ +

,
(︀
)︀
(︀
(︀
)︀
(︀
)︀
(︀
=1

(*0 )
= −0 − 0 −

∑︁
∑︁
=1 =1
(*1 ) = 1 + 1 +

∑︁
∑︁
1,
=1 =1
(︂
1,
(︂
)︂⃒ *
]︀ ⃒0
d [︀
*
ℎ̄ (),  () ⃒⃒ ,
d
(1)
)︂⃒ *
]︀
]︀ ⃒1
d [︀
d−1 [︀
*
*
⃒ ,
ℎ̄
(),

()
−
ℎ̄
(),

()

⃒
d−1
d
(1)
(76)
(*0 ) = 0 + 0 ,
(*1 ) = −1 − 1
(75)
[()ℎ(), ()()] + ()()* () = 0,
)︂⃒ *
(︂

∑︁
∑︁
]︀ ⃒1
d−1 [︀
*
1
ℎ̄ (),  () ⃒⃒ .
,
−
d−1
(1)
(77)
(78)
=1 =1
In addition to this, one can obviously show that, if  is a bounded convex set, then
the optimal control * () takes values on its boundary, precisely on the intersections
of consecutive components of this boundary [2, § 17].
The results of this paper also hold if in (1)–(5) the equation of the trajectory has
the form ˙ = ()() + ()()() + ()() and the constraints are of the form
[ℎ () + ()  (), ()] + ()() > 0. The only difference between this version and the
detailed one is given by the conditions on the constraints.
In this work  * (), (* ()) denotes ((), * ()).
Remark 3. The theory seems to be a familiar one, but if one doesn’t understand
the meaning of the sequently continuity, the derivation by virtue of the system of
bounded subsets , the operators’ equivalency in topological Banach spaces, the integrability with respect to the topology, he will be unable to value this paper, as it’s
very easy to get lost thinking of the usual problems [1, § 6.1].
References
1. Сухинин М. Ф. Избранные главы нелинейного анализа. — М.: Изд-во РУДН,
1992.
2. Понтрягин Л. С. Математическая теория оптимальных процессов. — М.: Наука, 1976.
3. Гамкрелидзе Р. В., Харатишвили Г. Л. Экстремальные задачи в линейных топологических пространствах // Известия АН СССР. Сер. Мат. — Т. 33, № 4. —
1969. — С. 781–839.
4. Сухинин М. Ф. Об ослабленным варианте правила множителей Лагранжа в
банаховом пространстве // Математические заметки. — Т. 21, № 2. — 1977.
Pontryagin’s Principle of Maximum for Linear Optimal Control Problems . . .
19
5. Дмитрук А. В. К обоснованию метода скользящих режимов для задач оптимального управления со смешанными ограничениями // Функциональный
анализ и его приложения. — Т. 10, № 3. — 1976.
6. Дмитрук А. В., Милютин А. А., Осмоловский Н. П. Теорема Люстерника и
теория экстремума // Успехи математических наук. — Т. 35, № 6. — 1980.
7. Колмогоров А. Н., Фомин С. В. Элементы теории функций и функционального анализа. — М.: Наука, 1972.
8. Алексеев В. М., Тихомиров В. М., Фомин С. В. Оптимальное управление. —
М.: Наука, 1979.
9. Васильев Ф. П. Методы решений экстремальных задач. — М.: Наука, 1981.
10. Лонгла М. Условия оптимальности в бесконечномерном пространстве. — М.:
ВИНИТИ № 412–В2008, 2008.
УДК 517.95
Принцип максимума Понтрягина в линейных задачах со
смешанными ограничениями в бесконечномерном
пространстве
М. Лонгла
Кафедра дифференциальных уравнений и математической физики
Российский университет дружбы народов
ул. Миклухо-Маклая, 6, Москва, Россия, 117198
Выведены необходимые условия оптимальности в некоторых задачах с линейными
регулярными и нерегулярными ограничениями в нормированном пространстве с особой отделимой локально выпуклой топологией, основываясь на трудах М.Ф. Сухинина.
Используемые функции могут не быть интегрируемыми по Бохнеру и не быть дифференцируемыми по Гато в обычном смысле. Здесь изложена попытка обобщать результаты, полученные в конечномерных пространствах Л. Грейвзом, Л.С. Понтрягиным,
В.Г. Болтянским, Р. В. Гамкрелидзе, А.В. Дмитруком, А.А. Милютиным, Е.Ф. Мищенко,
Мак-Шейном и др. Не исследованные задачи описанного выше типа рассматриваются
в данной работе, опираясь на теории дифференцирования по системе подмножеств, эквивалентности функций и операторов в локально выпуклом банаховом пространстве, и
интегрирования по локально выпуклой топологии, изложенной М.Ф. Сухининым в своей
монографии . Сформулированы и доказаны теоремы для случая, когда фазовые ограничения и смешанные ограничения суть линейные функции траектории и управления в
бесконечномерном локально выпуклом отделимом пространстве с нормой.
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