Necessary optimality conditions for stationary nonlinear hydrodynamic disrupted problems in a bounded domain Текст научной статьи по специальности «Математика»

UDC 517.917

Necessary Optimality Conditions for Stationary Nonlinear Hydrodynamic Disrupted Problems in a Bounded Domain

Bale Bailly*, Gozo Yoro^, Richard Assui Kouassi*

* UFR de Mathématiques et Informatique, Université de Cocody 22 BP 582, Abidjan 22. Cote d'Ivoire

^ Laboratoire de Mathématiques et Informatique, UFR - SFA, Université d'Abobo — Adjamé 02 BP 801, Abidjan 02. Cote d'Ivoire

* Département de Mathématiques et Informatique, Institut National Polytechnique FHB BP 1083 Yamoussoukro Cote d'Ivoire

In the paper we establish the optimal necessary conditions for guaranteeing uniquely the resolution of boundary hydrodynamic problems in a bounded domain so that they could accurately describe the studied hydrodynamic phenomenon.

Key words and phrases: necessary conditions of optimality, Command, optimal command, uniqueness, disruption, linearization, nonlinear.

1. Introduction

The focus of our research on such problems lies in the fact that for nonlinear systems of the type of Navier-Stokes in a three-dimensional space, we can not find a class of spaces where we could uniquely solve the problem at the border. This class is found by the linearization of systems of Navier-Stokes. However linearized systems often do not describe accurately the movement of liquid (or fluid). An intermediate case of investigation was proposed in [1], i.e., to the linearized system, nonlinear terms are added, which may allow us to more accurately describe the movement of liquid (or fluid) and at the same time allow the resolution in a unique way of the nonlinear problem relative to the boundaries, obtained by disrupting our initial system.

Using the Hadamard theorem for infinitely small Lipschitz constant, satisfying the conditions of these disturbances, the obtained disrupted problem at the borders has a unique solution v = A(f, a) where a is the value of the velocity v at the border (with a = 0 for the studied problem), f is the second member of the perturbed obtained system, and A satisfies the Lipschitz conditions with respect to f, in the corresponding functional spaces.

For some smooth conditions on the Nemytsky-Hammerchtein operator and using the theorem of Hadamard about strong derivation of inverse functions, the operator A(f) is strongly differentiable in the sense of the corresponding f. This derivation is weaker than the Frechet derivation. But it is quite sufficient to establish the necessary conditions of optimality of problems relative to those equations.

2. Statement of the Problem of Optimal Control

The physical processes that find their applications in technique, are generally controlled. It means that they can be achieved in many ways at the mercy of man. Therefore, we must find the best control according to particular criteria, in other words, the optimal control of the process.

The flow of an incompressible viscous fluid in a not empty and bounded domain fi, is characterized by its velocity v = v(x) and pressure p = p(x).

Consider the associated system after disruption

v A v(x) + M(x,v(x)) + Jk(x,y)g(y,v(y))dy = VP(x) + f (x), (1)

n

div u(œ) = 0, (2)

v |an (x) = 0, x G tt c R3, (3)

and functional with the following form

Jk (f ) = / Pk (x, v(x)), f (œ))dœ, k = 0,1,...,si + S2, (4)

n

where Pk are Caratheodoric functions, that is they are measurable with regard to the triplet (x,v,f ) and continue with regard to the couple (v,f ) almost everywhere for all the elements x of tt; v is the kinematic coefficient of viscosity (or of tenacity) and it is considered to be constant. dtt = S is the border of the domain tt. In addition to that we have

IPk(x,v,f )| < Qk(x) + Ck (M2 + |/|2) , |V{vJ)Pk(x, v,f )| < Dk(x) + Ck (M2 + |/12)

with Pk, k = 0,1,..., si + s2, derivable with respect to the pair (v, f), Qk(x) G ¿i(tt), Dk(x) G L2(tt), Ck and Ck are constants. More, Pk, Pkv and Pkf verify the Lipschiz condition from the pair (v,f); s1 and s2 are non negative integers. According to [1] the following functions:

M : tt x R3 x R9 ^ R3

(x,C,V) ^ M (x,C,V),

g : tt x R3 x R9 ^ R3

(x,C,V) ^ 9(x,C,V), k : tt x tt ^ R9

(x,y) ^ k(x,y).

are measurable and satisfying the following conditions:

\\M(x,C,,v,)\\ < cq(HCII + N1) + di(x) (5)

lg(x,C,v)l < ci(\CII + N\) + d2(x) (6)

where di(x) G L2(tt), i = 1, 2.

Moreover M and g are continuously differentiable with respect to the correspondent ((, rj) almost at each fixed point x G tt , and

\M'C| + |M;| < C2, VC G R3, V^ G R9 (7)

Vj'cI + KI < C3, VC G R3, Vn G R9 (8)

at almost every x G tt, where q is a constant for i = 0,1, 2, 3.

The function K defines a continuous integral operator L2(tt) ^ L2(tt), with the following form:

(Ky)(x) = j k(x,y)<p(y)dy. (9)

In the same way, the following operators have been defined in [1]:

and

by the formulae:

and the operators:

and

N : W22 (fi) ^ ¿2(fi)

ti ^ [N (ti)](®) = N (ti),

G : Wl(fi) ^ L2 (fi)

ti ^ [G(ti)](®) = G(ti),

[W(v)](x) = M(x,v(x), Vv(x)) [G(u)](x) = g(x,v(x), Vv(x))

N'(ti) : Wl(fi) ^ L2 (fi)

h(x) ^ [N'(ti)](x) = N'(ti)h,

(10)

(11)

(12) (13)

G' : Wi(fi) ^ L2(fi)

h(x) ^ [G'(ti)h](x) = G'(ti)h,

depending on the parameter ti E W2 (fi) by the formulas:

3

[N'(ti)]h(x) = M'(x, ti(x), Vti(x))h(x) + E M'i ^ (14)

i=1

and

3

[G'(^)]h(s)= </c h(s) + E ^ 4 ^

(15)

¿=1

where M' , g', g' have for argument (x, ti(x), Vti(x)) (the notations N'(ti) and G'(ti) are in [2]). Let

U = | ti G (fi) : 31/ G , 31a G B, Mi(ti) = (f,ti)

IMk = 11/ll^y + 1Mb.

Assuming that [M2(ti)](x) = vAti - VP,

[Mi(§)](x) = M(x,ti(x), Vti(x)) + j k(x,y)g(y,ti(y), Vti(y))dy.

n

It has been prooved in [1] that the operator

Mi : (U, |||^) ^ (J2/ x B

ti ^ Mi(ti) = (f,ti)

,

is an isomorphism. Where J2X is the Hilbert space of vector functions, obtained by completing J(fi) according to the standard corresponding scalar product:

(u,r&) = J(w& + ux"dx)dx, n

J(fi) is the set of infinitely differentiable vector functions and B has been defined by a e B = {a e L2(S) / 3 a e w2(fi), diva = 0, a|s = a}

with ||a:||_B = Inf{IMIw.Kn) : a e W2(fi), diva = 0, a|s = a}.

In [1], it was shown that by choosing u = max(c2,c3) with the operators M and g, satisfying the conditions (5) — (8) and if there is a number w0 > 0 such that for any w, we have 0 < u <u0 then the problem

M(tf) = vA-d(x) + M(x, Vtf(x)) + Jk(x, y)g(y, ^(y)V^(y))dy = Vp(x) + f (x),

n

div tf(x) = 0, (x) = a(x)

o

has a unique solution § = A(f, a) for all f e J22 and a e B and more:

oo

1) A : J2 x B ^ ^22(fi) is s— continuous and s— differentiable on J2 x B;

o

2) the operator A is strongly differentiable on J2, x B as a mapping on the space (W22(fi), a), where a is a weak topology in W2, (fi).

We also obtained in [1], that when the solution § = A(f,a) is s— continuous

o

and s— differentiable as a mapping from J22 x B to U, then A is s—continuous and s— differentiable as a mapping from L2(fi) x B in W2, (fi). This is deduced from the

o o

continuity of A from U in ^2(fi) and L2(fi) in (J2)', this is (H(fi) C L2(fi) C (J2)').

To obtain the result above stated, we had to show that the operators N and G are s—continuous and s—differentiable on W2 (fi) and G' = K * G. Similary, it was shown that, since K is a continuous linear map and that the operators G and N satisfy the Lipschitz condition, then K o G also satisfies the Lipschitz condition.

Therefore, what conditions the command applied to the system (for disruption) should be submitted to, so that the associated solution to the command coud be unique?

The problem is to choose a command f from U0, where U0 is a convex set in L2(fi), such that for the solution v(x) of the system (1)-(3), depending of that command f, constraints persist, which are given in the form of inequalities

Jk(f) < 0, k = 1,si, (16)

given in the form of equalities

Jk (f ) = 0, k = Si + 1,si + s2, (17)

and that in addition to this, the functional J0(f) takes the smallest possible value

Jo (7) = inf Jo(f). (18)

Uo

Such control is called optimal.

Definition 1. The function v = A(f) is called generalized solution of system (1)-

o

(3) in W22 (fi) , if it satisfies the integral identity 1 = 0, Vp = p(x) G J2 (fi), where

I = v Jvx(pxdx J M(x, v{x)) + Jk(x, y)g(y, v(y))dy - f(x)

Q Q

p(x)dx = 0. (19)

Suppose that p is sufficiently smooth. Then

f ( dvi f ^ vid2<Pi j f ^ d<fi

flS7jdx = VJ l^-d^rdx^dx =

Q 3=lK 3 Q 3=1 3 ¿Q 3=1 3

f 3 d 2 f 3 d ( d \ = ",/E^Sd® Jvi E a^:ds = ^( Ap*- ^M,

n j'=2 J a n ^=2 ' V J

3

^ A Pi, vi) = (^ A P,v) by condition (3).

3 = 1

Thus we obtain

1= (V A p + (N'(v) + KG'(v))p, v) - (p, f) = 0. (20)

Theorem 1. Suppose that under the conditions of (4) (see [1]) v is a solution of system (1)-(3), corresponding to the control f(x) G U0, where

r(x) = J(x) + £ (f(x) - J(x)^ , (0 < £ < 1),

and v£(x) is a solution relative to the control f£(x) G U0. Then

llvE(x) - v(x)llwin ^C ||r(x) - J(x)\\h2{n) . (21)

Proof. Remark that Sv(x) = vs(x) - v(x) satisfy the integral identity:

v / pxbvx - p

Me -M + k(g£ - g)dy

dx + pSfdx = 0, Vp G Jï(iï). (22)

So, using condition (16) (see [1]) and the restriction (21) (see [1], for a = 0), we obtain inequality (21). □

3. Derivation of the Functional

Consider the functional

J (f) = JP (x> v(x)), f(x))dx, (23)

n

Let's prove that J is differentiable in L2(fi).

3.1. Formula for the Gradient of the Function

Consider the problem (1)-(3) with a disrupted control fs <E L2(fi), which is linked to the solution v£(x) of the problem and the value of the functional J(fe).

Denote the variations by: öv = v£ — v, Sf = f£ — f. We have

A J = A J(f ) = J(f£) — J(f) =

P (x,ve,f £) — P (x,v,f )

dx,

P (x, Ve, f£) — P (x,v, f ) = P (x, 1/Je) — P (x,v, fe) + P (x,v, fe) — P (x,v, f )

1

= y Pv (x,v + ïïôvj + P (x,v,fe ) — P (x,vj) =

0

1

= P(x, v, fe) — P(x, V, f ) + Pv(x, V, f )Sv +

Pv (x,v,f e) — Pv (x,v,f )

=

P(x, v, fe) — P(x,v, f ) + Pv(x,v, f )Sv +

Pv (x,v,f £ ) — Pv (x,v,fe)

+

Pv (x,v,f£) — Pv (x,v,f )

ÖV.

Then

A J =

P (x,vj e) — P (x,vj )

d^ + J Pv(x,v,f )5vdx + J (r1 + r2)dx, n n

where

n

Pv (x,v,f £) — Pv (x,v,f e)

¿wd^, v = v + "9ÖV,

(24)

r2 =

Pv (x,v,fe) — Pv (x,vj )

As the functions v and v satisfy respectively the integral identities (in this case we use relation (20)), so their difference 5v = v£ — v also will satisfy the identity (20). Taking into account this fact, we have:

A J =

n

P (x,vj £) — P (x,vj )

d^ + J Pv (x,v,f )övdx+ n

+ {v A y + Jm £ — M + j k(g£ — g)dyj , Sv) — {<p, Sf) + j(ri + r2)dx. (25)

n n

In the last expression, taking into account the conditions on M and g, also taking into account formulas (25) and (26) (see [1]), we rewrite the following:

J <p\M£ — Mj d^ ^y y\M (x,v£(x)) — M (œ,û(œ)Mdœ nn

1

1

p

Mv (x,v + tid v)5vdti

dx = J pMv(x, v(x))5v(x)dx + J p(x)r3dx = n n

= J p(x)[N'(v)](x)5v(x)dx + f p(x) r3dx, (26)

where

and

3=

Mv(x,v(x)) - Mv(x, v(x))

5v(x)dti, v = v + ti5v,

(27)

p

J %e - 9)dV = J p(x) J k(x, y)(g(y, v£(y)) - g(y, v(y))^jdy

dx =

Jp(x) Jk(x,y

gv( y, v + tidv)5vdti

d

dx =

f p(X)

k(x, y)gv(y, v(y))5v(y)dy

dx + J p(x) Jk(x, y)rAdy dx

n n

Jip(x) Jk(x,y)gv(y, v(y))Sv(y)dy

dx =

Jip(x)k(x, y)dx

9v(y, v(y))5v(y)dy =

J<p(x)k(x, y)gv(y, v(y))Sv(y)dy J¥>(y)k(y,x)gv (x, v(x))dy

dx =

Sv(x)dx =

= J( G'* (v) K*<pT )(x)Sv(x)dx,

where (K*pT)(x) = k(y,x)pT(x)dx. In what follows, as a matter of convenience,

n

we will simply write p but not pT. Thus,

Jp(x) Jk(x, y)(ge -g)dy dx = J(G'*(v)K*p)(x)5vdx + J Jk(x, y)r4dy

nn where

rA =

9v( y, v) - gv ( y, v)

5 vdti.

P(x, v, n - P(x, V, f) = P(x, V, f)5f + T5,

where

5

Pf (x,v, f) -Pf (x,v, f)

5fdti, with f = f + ti5f.

p(x)dx.

(28)

(29)

Taking into account formulas (26), (27), (29) and (25) we obtain

1

1

1

1

1

A J =

(x,v,f )ôfdx + f Pv (x,v,f )5vdx+

+ {v A y + N'(x,v)<p + G'*(v)KSv) - {<p, Sf )+

+

J(r\ + r2 + r5)dx + J r3pdx + f( Jkridyjpdx. (31) n n n n

Remark 1. The transformations in the formula (31) are true only for the functions <p, sufficiently "smooth" and are issued only by the evidence of obtaining the conjugate form of the problem. For the following transformations, consider the following conjugate problem:

v A y + N'(v(x))<p + G'(v(y))K*<p = -Pv.

(32)

From the existence (see Theorem 2 (see [3, p. 54] and Theorem of Hadamard) of the solution of the conjugate problem (32), we finally have the expression for A J

A J = JPf (x, v, f )Sfdx + Jpôf dx + J(r1 + r2 + r5)dx+

n

+ /r3^dx + J ^ Jkr4dy^ipdx = J

nn

Pf (x,v,f) + <p

Sf dx + R,

where

R = J(r\ + r2 + r5)d^ + / (r3 + J kridyjpdx. n n n

In assessing the balance of development, one can show that R = o \\L{2.

Due to the fact that Pv satisfies the Lipschitz condition with respect to the group of arguments (v,f) and using Theorem 1, we have

/ r1dx =

J n

dx

<

I J(pv (x,v,n - Pv (X,v,f)] Svdiï

n o

1

flPv(X,g,f e) - Pv(X,1J, fd^d^ ^

no

1

/L( ^ - ^W) + 11f£ - f ^ <n)) H d#dx =

no2 1

J J L§ \\M\w2i<n) H àx = -L \\H\wi <n) / H dx <

1

'2 \'ÙJ ' ' — " " "2 1

n o n

< 1 L(mesQ)2 \\H\w2i<n) \\H\w2i<n) = -L(mesQ)2 \\5vfwif<n) <

1

< -L(mesU)2c \H\l2<n) = o( \\Sv\\L2<n) )

/ r55dx =\

\ n

1

/[/(

Pf (x,v, f) -Pf (x,v, f) U/dti

dx

<

<

Pf (x,v, f) -Pf (x,v, f) \5f\ dtid^ <

n 0

< / IH ll^-^(n) +

f -f

\5 f\ dtid^ =

n0 1

L

f -f

n0

l2 (n)

1 1

l2 (n) 1

\S.f\ dtid^ = / ¡M ll5fllL2(n) dtid^ =

n0

2L ll5fllL2(n) / № dx < 2L(mesfi)2c HSf^^ = °( Wh2(n) ),

and

/ r2dx =\

J n

/(

p.<*,*,n -P.<*,*.7)W

<

<J \PV (x,v, fe) -Pv (x,v, f)\ Hd^ < n

< /Vll« -vllwi(Q) +1|r -7HL2{n)) Hdx =

n2 2

= fL ||r - 7|L2(n) N dx = L\ lM^n) H d^ = L lMlL2(n) /H d^ <

n

n

< \h(rnesfi)1 ll<J/lL2(n) HHU2(n) < \L(me1c lMl2

¿2(n) .

The other members are evaluated in the same way. For the variation of the functional A J, we have finally

A J =

Pf (x,v, f) + p

5fdx + o( H<5/lL2(n)).

Let's introduce the following function

H v(x), f(x),p(x)) d= H (x,v,f, p) = P (x,v, f) + pf.

In this case, the formula for the variation will take the following form

_ _

-w{x,v,f,p){ r -f )ds + 0 Hl5fllL2(n)).

So we've just proved the following theorem

1

1

Theorem 2. Suppose that all the conditions of paragraph 1 [1] about functions M and g are satisfied, as well as the requirements of paragraph 1 about P.

28 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19-32 Then the functional J(f ) is differentiable with respect to f, and its derivatives at

____Q^ _

the point f are expressed by the formulae Jf (f ) = -^j (x,v, f, p) .

4. Necessary Conditions of Optimality

Let f = f (x) E U0, with f (x) an optimal control. Consider an arbitrary command f (x), with f = f (x) E Uo. _ _

Let's find the variation f£ of the optimal control f in the direction of (f - f ) as follows: _ _

f£(x) = f (x) + e{f (x) -f(x))

Sf = T - 1 = e(f - J),

In the variations, s is always the same and fs E U0. This is satisfied for example, when e E [0,1], because U0 is a convex set.

4.1. First Variation of the Functionals

Consider a family of functions Hk, k = 0, si + s2, where

Hk (x,v,f,(pk ) = Pk(x,v,f) + f. (34)

The functions (fik are solutions of the conjugate problems. So

/q ^ _____

{x, V, f, yk) (fe — /)d^ + 0( \\5f ||L2(n^, k = 0,si + (35)

n

the first variation 5Jk of the functional Jk(f) at point f is determined as follows:

5Jk = 5Jk (f) = lim ^^. (36)

As f — f = e(f — f) and the norm ||/ — f ||L2(n) for any fixed f, are fixed finite quantities, then

5Jk = lino 1 \j^j- {x,vj,vk)e(f — J)dx + o(e) \ =

jjff ix,v,f,^k )(f — f)d^ (37)

where

àJk = / ^ (f — f )dx, (38)

n

o _

where dHk is the function Hk with the arguments related to optimal control f.

4.2. Establishment of the Necessary Conditions of Optimality

Let 7 be a set of parameters:

7 = {Kf — 7), A > 0, f G Uo} (39)

Or simply , 7 = {X(f — /)}, giving the variation of the optimal control

r = f + e A(f — 7).

Then 7 is the family of variations of the functionals Z7 = [ÔJ7 , ÔJ7, ..., ÔJ] + ), where

o

ÔJk = \j ^(f — J)dx, k = 0, si + S2. (40)

n

All kinds of 7, whose form looks like the family of variation of the functional {Z7} = Ki c ES1+S2+i. Let's show that Ki is a cone in ES1+S2+i with its apex at the zero point.

It is clear that Z7 = 0 G ^S1+S2+i corresponds to the family 7 = {0}, with A = 0. We have implicitly 0 G Ki.

Consider the family 7 = {A(/ — /)}. For that family, there is a vector of variation

of the functionals Z7 = [ÔJ7 , ôf-[, ..., ÔJ]1+S2) G Ki.

Consider AZ7 = [aôJg , aôJ7, ..., aôJ]1+S2), where

o

aôJl = aA f-qJ-(f — f)dx, a> 0. n *

Consider the family a7 = {aA(f — f)} too.

Such a family is admissible, like the corresponding vector of variation of the functionals Za7 G Ki. Moreover, it is clear that Za7 = aZ7, as ÔJa7 = aSJ^, we conclude that, aZ7 G Ki and Ki is a cone.

We now show that the cone Ki is convex. For this it is sufficient to show that V Z71, Z72 G Ki their sum Z71 + Z72 G Ki. Consider Z11 generated by the family 7i = {Ai(/ — /)}, and Z71 generated by the family 72 = {A2(/ — /)}. Consider the set _

7i + 72 = {A(/ — /)} , or A = Ai + A2,

f = $xfi + (1 — #x)f2, 0 = ^^ < 1.

As U is convex, the set 7i + 72 is admissible. So is the correspondent vector of variation of the functionals Z71+72 G Ki.

In expression (40) for dJ^1 +72, k = 0, si + s2, consider the expression

o / \ o A f^(f — 7)dx = (Ai + A2) i^(&xfi — (1 — #x)f2 — 7)dx =

df " J' — ' "2)J df

n n

' dHk , -i , f dHk , r2

= AiJ ~WU — f)dx + A2J iff (f — f)dx.

nn

Thus, 5J71+12 = 5 J11 + 5J+72, k = 0, si + S2, then Zl1 + Z72 = Z71+72 G Ki and the cone Ki is convex.

Definition 2. The contsraints at the point f, part of restrictions Jk(f) ^ 0, for which Jk(f) = 0 are called active. Those for which Jk(f) < 0 are called inactive at that point.

To begin, suppose that all the restrictions (16) are active. Consider the set

K- = {c e ES1+S2+i : c = (co,ci, ..., cai, 0, ..., 0), a < 0, i = 07sT} a negative angle in ES1+S2+i. It is clear that K- is a cone in ES1+S2+i.

Lemma 1. The cone K\, built for optimal control and the cone K- are divided in ESl+S2+i by the hyperplane r, defined by the nontrivial functional I*:

S1+S2

I* = (lo,h,...,ls 1+S2) e {ES1+S2+i)* = ES1+S2+i, £ llk | > 0, for lk > 0, k = 0^,

k=0

and the rests lk, k = si + 1, si + s2 may have any sign. The condition of separation of K\ and K- takes the following form,

(l*,Zy) ES1+S2+1 > (l*,c) ES1+S2+1, VZ1 e Ku Vc e K-. (41)

This follows from the known theorem (see [4], p.224 or [5], 3.1).

Theorem 3. Let X be a normed space, U a convex set in X, u* e U a local minimum point in the problem

J0(u) ^ inf, Ji(u) < 0, i = 1, si, Ji(u) = 0, i = s\ + 1,s 1 + s2, u e U,

where Ji, i = 0, s 1 + s2, s-differentiable at the point u* and Ji, i = si + 1, s 1 + s2 continuous in the neighborhood of the point u*.

Then there are numbers 10,l\, l2, ..., lS1+S2 such that

I* = (l0 ,h, k,..., ¿S1+S2 ) = 0, I0 > 0, 0,..., IS1 > 0,

(Cu(u*, l*),u — u*) > 0, Vu e U, liJi(u*) = 0, i = 1,s 1 + S2

here C u(u*, I*) = l0J0(u*)+hJ[ (u*)+..+ S1+S,J'S 1+s2(u*) the gradient of the function C(u, 1*) with variable u eU at the point u = u*.

Then, using inequality (41) in which c^ 0, we obtain

(I*, Z7)E^1+s2+1 > 0, VZ7 e Ku (42)

That is, for any family 7 like in (39).

Inequality (3) is well demonstrated, assuming that all restrictions (16) are actives. Now consider the general case.

Let I = {k : 1 < k < s\,Jk(/) = 0} be the set of all constraints at the point f among all the restrictions like (16). The other constraints in the formula (16) are inactive at the point f, that is Jk(/) < 0, 1 < k < s 1, but k <e I. So thanks to the continuity of the functional Jk with respect to the control f for small disrupted controls, non-active constraints are not affected. Therefore, we can not take account of them. In this case we will examine variations of functionals only for the active constraints and their vectorial variations

= {8J0i, {SJ2}kei, SJZ +S2,..., ^S71+S2} .

Let's build the cone Ki = {Z7} c EdtmI+S1+i. The corresponding cone is Kf = {c G EdmI+S1+i : c = (co, {Ck}kei, 0, ..., 0), Ck < 0} , and by taking the above steps till (42), we obtain

S1+S2

lo6J7 + E lkSJ7 + E lk$Jk > 0, VZ7 G K. (43)

kei k=S1+i

Let lk = 0 for all k : 1 < k ^ s 1, k G L Then (43) takes the form of (42).

Thus, for all inactive contraints Jk(f) < 0 corresponding to lk = 0, and for the active contraints Jk (f) = 0,

IkJk (f) = 0,k = 1,s 1 + S2 (44)

so the condition (42) is verified too. From this we deduce the necessary conditions of optimality of the control.

^—>S1 +S2

Let's introduce the following function: tf = tf(x) = Efc=0 lk''k(x), where 'k(x) is the solution of the conjugate problem (32). Multiplying (32) by lk and making a summation for all k = 0,s 1 + s2, then the function tf(x) will be a solution of the problem:

S1+S2

z/ A tf(x) — N '(x, v(x))tf(x) + G (y, v(y))K *tf = — E lkpkv ■ (45)

k=o

Let's introduce the functions

S1+S2

H(x, v, f, tf) = E lkHk (x, v(x), f(x), tf(x)). (46)

k=o

Using the formula (32) for Hk and taking into account the introduced function ^(x), we can write formula (44) in expanded form:

S1+S2

n(x, v, f, tf) = U(x, v(x), f(x), tf(x)) = E 1 kpk(x, V(x), f(x)) + tf(x)f(x). (47)

k=o

Let's consider the family 7 = {f — f} for all f G L2(H). To this family we associate the variation vector of the functional

Z7 = {ÔJ1, ÔJ7 ,...,SJl+S2 ) G Ki,

S1+S2

and inequality (43) persists: E IkdJ^ ^ 0.

fc=0

Replacing 5J2 by their respective expressions from (40) using formulas (47), we obtain

(x, v(x), J(x), *(x))(f - J)dx > 0, VfEUo. (48)

n

So we have just proved the following theorem:

Theorem 4 (The principle of linearized minimum). Suppose that all of the

conditions of theorems 1 and 2 are satisfied. Then, for the optimal control f(x) £ U0

it is necessary that there exists a nontrivial vector

Sl+S2

I* = {l0 ,h,..., lsl+s^ , E i lk i > 0,

k=0

where lk > 0 for k = 0, si and the conditions (48) as well as the conditions IkJk (/) = 0 k = 1, s i + s2 (conditions (45) ),

are satisfied; where function v(x) is the solution of the problem (1)-(3), ^(x) is the

solution of the conjugate problem, (46) associated to f(x) and the function % is defined

in (47).

References

1. Yoro G., Bailly B., Assui K. R. Unicité et dérivabilité de la solution des systumes hydrodynamiques stationnaires non lineaires et perturbes // Revist. — 2010. — Vol. 15, B. — Pp. 91-107.

2. Operateurs integraux dans les espaces des fonctions additionnables / M. A. Crasno-celski, P. P. Zabrehiko, E. N. Poustilnik, P. E. Sobolevski. — M.: Naouka, 1966. — 496 p.

3. Ladyjenskana O. A. Les questions mathematiques de la dynamique de la tenacite des liquides non compressibles. — M.: Naouka, 1970.

4. Vassilev F. P. Methodes numeriques pour la resolution des problumes des extrema. — M.: Naouka, 1980. — 520 p.

5. Souhinine M. F. La regle du multiplicateur de Lagrange dans les espaces localement convexes // Revue scientifique de mathematiques de Siberie. — 1982. — Vol. 23, No 4. — Pp. 153-165.

УДК 517.917

Необходимые условия оптимальности для стационарной нелинейной возмущённой задачи гидродинамики в ограниченной области

о , ,

Бале Байлли*, Гозо Иорот, Ришар Ассюй Куасси*

* Факультет математики и информатики Университет Кокоди Абиджан Кот Д'Ивуар, 22 BP 582 Абиджан 22

^ Кафедра математики и информатики

Университет Абобо — Аджамэ Абиджан Кот Д'Ивуар, 02 Вр 801 Абиджан 02 UFR — SFA

* Кафедра математики и информатики Политехнический институт им. Феликса Уфует Буагни Ямуссукро Кот д'Ивуар, а/я 1083, Ямуссукро

Цель работы состоит в том, чтобы установить оптимальные необходимые условия, которые могут позволить нам решить задачу относительно границ данной области. В предлагаемой статье исследуется частный случай, а именно, в линеаризованную систему добавлены нелинейные члены, позволяющие более точно описать движение жидкости, и вместе с тем допускающие однозначную разрешимость полученной нелинейной возмущённой краевой задачи.

Ключевые слова: необходимые условия оптимальности, команда, оптимальная команда, уникальность, возмущение, линеаризация, нелинейность.