# Necessary optimality conditions for stationary nonlinear hydrodynamic disrupted problems in a bounded domain.

код для вставкиСкачатьUDC 517.917 Necessary Optimality Conditions for Stationary Nonlinear Hydrodynamic Disrupted Problems in a Bounded Domain Balé 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 = (, ) where is the value of the velocity at the border (with = 0 for the studied problem), is the second member of the perturbed obtained system, and satisfies the Lipschitz conditions with respect to , in the corresponding functional spaces. For some smooth conditions on the Nemytsky–Hammerchtéin operator and using the theorem of Hadamard about strong derivation of inverse functions, the operator ( ) is strongly differentiable in the sense of the corresponding . This derivation is weaker than the Fréchet 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 Ω, is characterized by its velocity = () and pressure = (). 20 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 Consider the associated system after disruption M () + (, ()) + ∫︁ (, )(, ())d = ∇ () + (), (1) Ω div () = 0, (2) 3 |Ω () = 0, ∈Ω⊂R , (3) and functional with the following form ( ) = ∫︁ (, ()), ())d, = 0, 1, . . . , 1 + 2 , (4) Ω where are Caratheodoric functions, that is they are measurable with regard to the triplet (, , ) and continue with regard to the couple (, ) almost everywhere for all the elements of Ω; is the kinematic coefficient of viscosity (or of tenacity) and it is considered to be constant. Ω = is the border of the domain Ω. In addition to that we have (︀ )︀ | (, , )| 6 () + ||2 + | |2 , (︀ )︀ ̂︀ ||2 + | |2 |∇(, ) (, , )| 6 () + with , = 0, 1, . . . , 1 + 2 , derivable with respect to the pair (, ), () ∈ 1 (Ω), ̂︀ are constants. More, , and verify the Lipschiz () ∈ 2 (Ω), and condition from the pair (, ); 1 and 2 are non negative integers. According to [1] the following functions: : Ω × R3 × R9 (, , ) → R3 ↦ → (, , ), : Ω × R3 × R9 (, , ) → R3 ↦→ (, , ), : Ω×Ω (, ) → R9 ↦ → (, ). are measurable and satisfying the following conditions: ‖ (, , )‖ 6 0 (‖‖ + ‖‖) + 1 () |(, , )| 6 1 (‖‖ + ‖‖) + 2 () (5) (6) where () ∈ 2 (Ω), = 1, 2. Moreover and are continuously differentiable with respect to the correspondent (, ) almost at each fixed point ∈ Ω , and |′ | + |′ | 6 2 , |′ | + |′ | ∀ ∈ R3 , ∀ ∈ R9 3 6 3 , ∀ ∈ R , ∀ ∈ R 9 (7) (8) at almost every ∈ Ω, where is a constant for = 0, 1, 2, 3. The function defines a continuous integral operator 2 (Ω) → 2 (Ω), with the following form: ()() = ∫︁ Ω (, )(). (9) Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . 21 In the same way, the following operators have been defined in [1]: : 21 (Ω) → 2 (Ω) ↦→ [ ()]() = (), (10) : 21 (Ω) → 2 (Ω) ↦→ [()]() = (), (11) [ ()]() = (, (), ∇()) (12) [()]() = (, (), ∇()) (13) and by the formulae: and the operators: ′ () : 21 (Ω) → 2 (Ω) ℎ() ↦→ [ ′ ()]() = ′ ()ℎ, and ′ : 21 (Ω) → 2 (Ω) ℎ() ↦→ [′ ()ℎ]() = ′ ()ℎ, depending on the parameter ∈ 21 (Ω) by the formulas: [ ′ ()]ℎ() = ′ (, (), ∇())ℎ() + 3 ∑︁ ′ =1 ℎ() (14) and [′ ()]ℎ() = ′ ℎ() + 3 ∑︁ =1 ′ ℎ() , (15) where ′ , ′ , ′ have for argument (, (), ∇()) (the notations ′ () and ′ () are in [2]). Let {︃ }︃ (︂ )︂ = ∈ 21 (Ω) : ∃! ∈ ′ ∘ 21 , ∃! ∈ , 1 () = (, ) , ‖‖ = ‖ ‖(︀ ∘1 )︀′ + ‖‖ . 2 Assuming that [1 ()]() = △ − ∇ , [1 ()]() ≡ (, (), ∇()) + ∫︁ (, )(, (), ∇()). Ω It has been prooved in [1] that the operator 1 : (, ‖ ‖ ) (︀ ∘ )︀′ → 21 × ↦ → 1 () = (, ) 22 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 ∘ is an isomorphism. Where 21 is the Hilbert space of vector functions, obtained by ∙ completing (Ω) according to the standard corresponding scalar product: (, ) = ∫︁ ( + ), Ω ∙ (Ω) is the set of infinitely differentiable vector functions and has been defined by ∈ ≡ { ∈ 2 () / ∃ ∈ 21 (Ω), = 0, | = } with ‖‖ = {‖‖21 (Ω) : ∈ 21 (Ω), = 0, | = }. In [1], it was shown that by choosing = (2 , 3 ) with the operators and , satisfying the conditions (5) − (8) and if there is a number 0 > 0 such that for any , we have 0 < < 0 then the problem ∫︁ ⎧ ⎪ () ≡ △() + (, (), ∇()) + (, )(, ()∇()) = ∇() + (), ⎪ ⎪ ⎨ Ω ⎪ div () = 0, ⎪ ⎪ ⎩ | () = () ∘ has a unique solution = (, ) for all ∈ 21 and ∈ and more: ∘ ∘ 1) : 21 × → 21 (Ω) is − continuous and − differentiable on 21 × ; ∘ 2) the operator is strongly differentiable on 21 × as a mapping on the space (21 (Ω), ), where is a weak topology in 21 (Ω). We also obtained in [1], that when the solution = (, ) is − continuous ∘ and − differentiable as a mapping from 21 × to , then is −continuous and − differentiable as a mapping from 2 (Ω) × in 21 (Ω). This is deduced from the (︀ ∘ )︀′ (︀ (︀ ∘ )︀′ )︀ continuity of from in 21 (Ω) and 2 (Ω) in 21 , this is (Ω) ⊂ 2 (Ω) ⊂ 21 . To obtain the result above stated, we had to show that the operators and are −continuous and −differentiable on 21 (Ω) and ′ = * . Similary, it was shown that, since is a continuous linear map and that the operators and satisfy the Lipschitz condition, then ∘ 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 from 0 , where 0 is a convex set in 2 (Ω), such that for the solution () of the system (1)–(3), depending of that command , constraints persist, which are given in the form of inequalities ( ) 6 0, = 1, 1 , (16) = 1 + 1, 1 + 2 , (17) given in the form of equalities ( ) = 0, and that in addition to this, the functional 0 ( ) takes the smallest possible value 0 ( ) = inf 0 ( ). 0 (18) Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . 23 Such control is called optimal. Definition 1. The function = ( ) is called generalized solution of system (1)– ∘ (3) in 21 (Ω) , if it satisfies the integral identity = 0, ∀ ≡ () ∈ 21 (Ω), where ]︂ ∫︁ ∫︁ [︂ ∫︁ = − d (, ()) + (, )(, ())d − () ()d = 0. (19) Ω Ω Ω Suppose that is sufficiently smooth. Then − )︂ ∫︁ ∑︁ ∫︁ ∑︁ ∫︁ 3 3 (︂ 3 ∑︁ 2 d = d − d = 2 Ω =1 Ω =1 = ∫︁ ∑︁ 3 Ω =1 3 ∑︁ Ω 2 2 d − ∫︁ =1 3 ∑︁ d = ⟨ Ω =1 (︂ )︂ , ⟩, M − ⟨ M , ⟩ = ⟨ M , ⟩ by condition (3). =1 Thus we obtain = ⟨ M + ( ′ () + ′ ()), ⟩ − ⟨, ⟩ = 0. (20) Theorem 1. Suppose that under the conditions of (4) (see [1]) is a solution of system (1)–(3), corresponding to the control () ∈ 0 , where (︂ )︂ () = () + () − () , (0 6 6 1), and () is a solution relative to the control () ∈ 0 . Then ⃦ ⃦ ‖ () − ()‖ 1 (Ω) 6 ⃦ () − ()⃦2 (Ω) . 2 (21) Proof. Remark that () = () − () satisfy the integral identity: ]︂ ∫︁ ∫︁ [︂ ∫︁ ∫︁ ∘ − − + ( − )d d + d = 0, ∀ ∈ 21 (Ω). (22) Ω Ω Ω Ω So, using condition (16) (see [1]) and the restriction (21) (see [1], for = 0), we obtain inequality (21). 3. Derivation of the Functional Consider the functional ( ) = ∫︁ (, ()), ())d, Ω Let’s prove that is differentiable in 2 (Ω). (23) 24 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 3.1. Formula for the Gradient of the Function Consider the problem (1)–(3) with a disrupted control ∈ 2 (Ω), which is linked to the solution () of the problem and the value of the functional ( ). Denote the variations by: = − , = − . We have ]︂ ∫︁ [︂ M =M ( ) = ( ) − ( ) = (, , ) − (, , ) d, Ω (, , ) − (, , ) = (, , ) − (, , ) + (, , ) − (, , ) = = ∫︁1 (, + , )d + (, , ) − (, , ) = 0 = (, , ) − (, , ) + (, , ) + ∫︁1 [︂ ]︂ (, , ̂︀ ) − (, , ) d = 0 = (, , ) − (, , ) + (, , ) + ∫︁1 [︂ ]︂ (, , ̂︀ ) − (, , ) d+ 0 [︂ ]︂ + (, , ) − (, , ) . Then M = ]︂ ∫︁ ∫︁ (, , ) − (, , ) d + (, , ) + (1 + 2 )d, ∫︁ [︂ Ω Ω where 1 = ∫︁1 [︂ ]︂ (, , ̂︀ ) − (, , ) d, Ω ̂︀ = + , 0 (24) [︂ ]︂ 2 = (, , ) − (, , ) . As the functions and satisfy respectively the integral identities (in this case we use relation (20)), so their difference = − also will satisfy the identity (20). Taking into account this fact, we have: M = ∫︁ [︂ ]︂ ∫︁ (, , ) − (, , ) d + (, , )d+ Ω Ω (︂ )︂ ∫︁ ∫︁ + ⟨ M + − + ( − )d , ⟩ − ⟨, ⟩ + (1 + 2 )d. (25) Ω Ω In the last expression, taking into account the conditions on and , also taking into account formulas (25) and (26) (see [1]), we rewrite the following: ∫︁ Ω (︂ )︂ − d = ∫︁ Ω (︂ )︂ (, ()) − (, ()) d = Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . = ∫︁ [︂ ∫︁1 ]︂ ∫︁ ∫︁ (, + ) d = (, ())()d + ()3 d = 0 Ω 25 Ω = ∫︁ Ω ()[ ′ ()]()()d + 3 = ()3 d, (26) Ω Ω where ∫︁ ∫︁1 [︂ ]︂ (, ()) ̂︀ − (, ()) ()d, ̂︀ = + , (27) 0 and ∫︁ Ω [︂ ∫︁ ]︂ [︂ ∫︁ (︂ )︂ ]︂ ∫︁ (, ) (, ()) − (, ()) d d = ( − )d d = () Ω Ω = ∫︁ () Ω = ∫︁ Ω (, [︂ ∫︁1 ]︂ ]︂ (, + ) d d = 0 Ω [︂ ∫︁ ]︂ [︂ ∫︁ ]︂ ∫︁ () (, ) (, ())()d d + () (, )4 d d. Ω ∫︁ [︂ ∫︁ Ω Ω Ω ]︂ ]︂ [︂ ∫︁ ∫︁ [︂ ∫︁ ()(, ) (, ())()d d = () (, ) (, ())()d d = Ω Ω Ω ∫︁ [︂ ∫︁ = Ω Ω ]︂ ]︂ ∫︁ [︂ ∫︁ ()(, )d (, ())()d = ()(, ) (, ())d ()d = Ω Ω Ω ∫︁ (′* () * )()()d, = Ω where ( * )() = ∫︁ Ω (, ) ()d. In what follows, as a matter of convenience, we will simply write but not . Thus, [︂ ∫︁ ]︂ ]︂ ∫︁ ∫︁ [︂ ∫︁ ∫︁ ′* * () (, )4 d ()d. (, )( − )d d = ( () )()d + Ω Ω Ω where 4 = Ω Ω ∫︁1 [︂ ]︂ (, ) ̂︀ − (, ) d. (28) 0 (, , ) − (, , ) = (, , ) + 5 , where 5 = ∫︁1 [︂ ]︂ (, , ̂︀) − (, , ) d, with ̂︀ = + . 0 Taking into account formulas (26), (27), (29) and (25) we obtain (29) (30) 26 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 M = ∫︁ (, , ) d + Ω ∫︁ (, , )d+ Ω + ⟨ M + ′ (, ) + ′* () * , ⟩ − ⟨, ⟩+ )︂ ∫︁ ∫︁ ∫︁ (︂ ∫︁ + (1 + 2 + 5 )d + 3 d + 4 d. (31) Ω Ω Ω Ω Remark 1. The transformations in the formula (31) are true only for the functions , 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: M + ′ (()) + ′ (()) * = − . (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 M M = ∫︁ (, , ) d + Ω ∫︁ d + Ω + ∫︁ 3 d + Ω where = (1 + 2 + 5 )d+ Ω ∫︁ (︂ ∫︁ Ω ∫︁ ∫︁ )︂ 4 d d = Ω Ω (1 + 2 + 5 )d + Ω ]︂ (, , ) + d + , ∫︁ [︂ ∫︁ (︂ 3 + Ω ∫︁ )︂ 4 d d. Ω (︁ )︁ In assessing the balance of development, one can show that = ‖ ‖2 (Ω) . Due to the fact that satisfies the Lipschitz condition with respect to the group of arguments (, ) and using Theorem 1, we have ⃒ ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒∫︁ [︂ ∫︁1 (︂ )︂ ]︂ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ 1 d⃒ = ⃒ (, , ̂︀ ) − (, , ) d d⃒⃒ 6 ⃒ ⃒ ⃒ ⃒Ω ⃒ ⃒ ⃒Ω 0 ∫︁ ∫︁1 6 | (, , ̂︀ ) − (, , )| || dd 6 Ω 0 ∫︁ ∫︁1 6 = (︂ )︂ ‖̂︀ − ‖ 1 (Ω) + ‖ − ‖2 (Ω) || dd = 2 Ω 0 ∫︁ ∫︁1 1 ‖‖ 1 (Ω) || d = ‖‖ 1 (Ω) 2 2 2 Ω 0 1 ∫︁ || d 6 Ω 1 1 1 2 6 (Ω) 2 ‖‖ 1 (Ω) ‖‖ 1 (Ω) = (Ω) 2 ‖‖ 1 (Ω) 6 2 2 2 2 2 (︀ )︀ 1 1 2 6 (Ω) 2 ‖‖2 (Ω) = ‖‖2 (Ω) , 2 Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . 27 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒∫︁ [︂ ∫︁1 (︂ ⃒∫︁ )︂ ]︂ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ 5 d⃒ = ⃒ (, , ̂︀) − (, , ) d d⃒⃒ 6 ⃒ ⃒ ⃒ ⃒ ⃒Ω 0 ⃒Ω ⃒ ∫︁ ∫︁1 ⃒ ⃒ ⃒ ⃒ ⃒ (, , ̂︀) − (, , )⃒ | | dd 6 6 Ω 0 ∫︁ ∫︁1 (︂ ⃦ ⃦ ⃦ ⃦ ‖ − ‖ 1 (Ω) + ⃦̂︀ − ⃦ 6 2 2 (Ω) Ω 0 = ∫︁ ∫︁1 ⃦ ⃦ ⃦ ⃦ ⃦̂︀ − ⃦ 2 (Ω) Ω 0 1 2 = ‖ ‖2 (Ω) ∫︁ )︂ | | dd = ∫︁ ∫︁1 | | dd = ‖ ‖2 (Ω) dd = Ω 0 (︀ )︀ 1 1 2 | | d 6 (Ω) 2 ‖ ‖2 (Ω) = ‖ ‖2 (Ω) , 2 Ω and ⃒ ⃒ ⃒ ⃒ ⃒∫︁ ⃒ ⃒∫︁ (︂ ⃒ )︂ ⃒ ⃒ ⃒ ⃒ ⃒ 2 d⃒ = ⃒ (, , ) − (, , ) d⃒⃒ 6 ⃒ ⃒ ⃒ ⃒Ω ⃒ ⃒Ω ⃒ ∫︁ ⃒ ⃒ 6 ⃒ (, , ) − (, , )⃒ || d 6 Ω ∫︁ 6 )︂ (︂ ⃦ ⃦ ⃦ ⃦ ‖ − ‖ 1 (Ω) + − 2 (Ω) || d = 2 Ω = ∫︁ ∫︁ ∫︁ ⃦ ⃦ ⃦ ⃦ − 2 (Ω) || = ‖ ‖2 (Ω) || d = ‖ ‖2 (Ω) || d 6 Ω Ω Ω 1 1 1 1 2 6 (Ω) 2 ‖ ‖2 (Ω) ‖‖ 1 (Ω) 6 (Ω) 2 ‖ ‖2 (Ω) . 2 2 2 The other members are evaluated in the same way. For the variation of the functional M , we have finally ]︂ ∫︁ [︂ )︀ (︀ M = (, , ) + d + ‖ ‖2 (Ω) . Ω Let’s introduce the following function (, (), (), ()) = (, , , ) = (, , ) + . In this case, the formula for the variation will take the following form M = ∫︁ )︀(︀ )︀ (︀ )︀ (︀ , , , − d + ‖ ‖2 (Ω) . Ω So we’ve just proved the following theorem Theorem 2. Suppose that all the conditions of paragraph 1 [1] about functions and are satisfied, as well as the requirements of paragraph 1 about . 28 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 Then the functional ( ) is differentiable with respect to , and its derivatives at )︀ (︀ the point are expressed by the formulae ( ) = , , , . 4. Necessary Conditions of Optimality Let = () ∈ 0 , with () an optimal control. Consider an arbitrary command (), with = () ∈ 0 . Let’s find the variation of the optimal control in the direction of ( − ) as follows: (︀ )︀ () = () + () − () (33) (︀ )︀ = − = − , In the variations, is always the same and ∈ 0 . This is satisfied for example, when ∈ [0, 1], because 0 is a convex set. 4.1. First Variation of the Functionals Consider a family of functions , = 0, 1 + 2 , where (, , , ) = (, , ) + . (34) The functions are solutions of the conjugate problems. So M = ∫︁ )︀(︀ )︀ (︀ )︀ (︀ , , , − d + ‖ ‖2 (Ω) , = 0, 1 + 2 , (35) Ω the first variation of the functional ( ) at point is determined as follows: = ( ) = lim →0 M ( ) . (36) As − = ( − ) and the norm ‖ − ‖2 (Ω) for any fixed , are fixed finite quantities, then ⎧ ⎫ ⎨∫︁ ⎬ )︀ (︀ 1 = lim , , , ( − )d + () = →0 ⎩ ⎭ Ω = ∫︁ )︀ (︀ , , , ( − )d (37) Ω where ∘ = ∫︁ ( − )d, (38) Ω ∘ where is the function with the arguments related to optimal control . 4.2. Establishment of the Necessary Conditions of Optimality Let be a set of parameters: {︀ }︀ = ( − ), > 0, ∈ 0 (39) Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . 29 {︀ }︀ Or simply , = ( − ) , giving the variation of the optimal control = + ( − ). )︀ (︀ Then is the family of variations of the functionals = 0 , 1 , . . . , 1 +2 , where ∘ ∫︁ = ( − )d, = 0, 1 + 2 . (40) Ω All kinds of , whose form looks like the family of variation of the functional { } = 1 ⊂ 1 +2 +1 . Let’s show that 1 is a cone in 1 +2 +1 with its apex at the zero point. It is clear that = 0 ∈ 1 +2 +1 corresponds to the family = {0}, with = 0. We have implicitly 0 ∈ 1 . }︀ {︀ Consider the family (︀ = ( − ) . For that )︀ family, there is a vector of variation of the functionals = 0 , 1 , . . . , 1 +2 ∈ 1 . )︀ (︀ Consider = 0 , 1 , . . . , 1 +2 , where = ∫︁ Ω ∘ ( − )d, > 0. Consider the family = {( − )} too. Such a family is admissible, like the corresponding vector of variation of the functionals ∈ 1 . Moreover, it is clear that = , as = , we conclude that, ∈ 1 and 1 is a cone. We now show that the cone 1 is convex. For this it is sufficient to show that ∀1 ,{︀2 ∈ 1 }︀their sum 1 + 2 ∈ 1 . Consider {︀1 generated }︀ by the family 1 = 1 ( − ) , and 1 generated by the family 2 = 2 ( − ) . Consider the set {︀ }︀ 1 + 2 = ( − ) , or = 1 + 2 , = 1 + (1 − ) 2 , 0 6 = 1 6 1. 1 + 2 As 0 is convex, the set 1 + 2 is admissible. So is the correspondent vector of variation of the functionals 1 +2 ∈ 1 . In expression (40) for 1 +2 , = 0, 1 + 2 , consider the expression ∘ ∫︁ ( − )d = (︂ 1 + 2 Ω )︂ ∫︁ ∘ ( 1 − (1 − ) 2 − )d = Ω ∘ = 1 ∫︁ Ω Thus, 1 +2 = 1 + +2 , the cone 1 is convex. 1 ( − )d + 2 ∘ ∫︁ 2 ( − )d. Ω = 0, 1 + 2 , then 1 + 2 = 1 +2 ∈ 1 and Definition 2. The contsraints at the point , part of restrictions ( ) 6 0, for which ( ) = 0 are called active. Those for which ( ) < 0 are called inactive at that point. 30 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 To begin, suppose that all the restrictions (16) are active. Consider the set {︀ }︀ 1− = ∈ 1 +2 +1 : = (0 , 1 , . . . , 1 , 0, . . . , 0), < 0, = 0, 1 a negative angle in 1 +2 +1 . It is clear that 1− is a cone in 1 +2 +1 . Lemma 1. The cone 1 , built for optimal control and the cone 1− are divided in 1 +2 +1 by the hyperplane Γ, defined by the nontrivial functional * : 1∑︁ +2 (︀ )︀* * = (0 , 1 , . . . , 1 +2 ) ∈ 1 +2 +1 = 1 +2 +1 , | | > 0, for > 0, = 0, 1 , =0 and the rests , = 1 + 1, 1 + 2 may have any sign. The condition of separation of 1 and 1− takes the following form ∀ ∈ 1 , ∀ ∈ 1− . ⟨* , ⟩ 1 +2 +1 > ⟨* , ⟩ 1 +2 +1 , (41) This follows from the known theorem (see [4], p.224 or [5], 3.1). Theorem 3. Let be a normed space, 1 a convex set in , * ∈ 1 a local minimum point in the problem 0 () → inf, () 6 0, () = 0, = 1, 1 , = 1 + 1, 1 + 2 , ∈ 1 , where , = 0, 1 + 2 , -differentiable at the point * and , = 1 + 1, 1 + 2 continuous in the neighborhood of the point * . Then there are numbers 0 , 1 , 2 , . . . , 1 +2 such that * = (0 , 1 , 2 , . . . , 1 +2 ) ̸= 0, * * * ⟨ℒ( , ), − ⟩ > 0, ∀ ∈ 1 , 1 > 0, . . . , 1 > 0, 0 > 0, * ( ) = 0, = 1, 1 + 2 here ℒ(* , * ) = 0 0′ (* )+1 1′ (* )+. . .+1 +2 ′ 1 +2 (* ) the gradient of the function ℒ(, * ) with variable ∈ 1 at the point = * . Then, using inequality (41) in which → 0, we obtain ⟨* , ⟩ 1 +2 +1 > 0, ∀ ∈ 1 , (42) That is, for any family like in (39). Inequality (3) is well demonstrated, assuming that all restrictions (16) are actives. Now consider the general case. Let = { : 1 6 6 1 , ( ) = 0} be the set of all constraints at the point among all the restrictions like (16). The other constraints in the formula (16) are inactive at the point , that is ( ) < 0, 1 6 6 1 , but ̸∈ . So thanks to the continuity of the functional with respect to the control 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 {︀ }︀ = 0 , { }∈ , 1 +2 , . . . , 1 +2 . Bailly Balé, Yoro Gozo, Assui Kouassi Richard Necessary Optimality Condi . . . 31 +1 +1 Let’s build the cone 1 = { } ⊂ . The corresponding cone is {︀ }︀ 1− = ∈ +1 +1 : = (0 , { }∈ , 0, . . . , 0), < 0 , and by taking the above steps till (42), we obtain 0 0 + ∑︁ + 1∑︁ +2 > 0, ∀ ∈ 1 . (43) =1 +1 ∈ Let = 0 for all : 1 6 6 1 , ̸∈ . Then (43) takes the form of (42). Thus, for all inactive contraints ( ) < 0 corresponding to = 0, and for the active contraints ( ) = 0, ( ) = 0, = 1, 1 + 2 (44) so the condition (42) is verified too. From this we deduce the necessary conditions of optimality of the control. ∑︁1 +2 (), where () Let’s introduce the following function: Ψ = Ψ() = =0 is the solution of the conjugate problem (32). Multiplying (32) by and making a summation for all = 0, 1 + 2 , then the function Ψ() will be a solution of the problem: ′ ′ * M Ψ() − (, ())Ψ() + (, ()) Ψ = − 1∑︁ +2 . (45) =0 Let’s introduce the functions ℋ(, , , Ψ) = 1∑︁ +2 (, (), (), Ψ()). (46) =0 Using the formula (32) for and taking into account the introduced function (), we can write formula (44) in expanded form: ℋ(, , , Ψ) ≡ ℋ(, (), (), Ψ()) = 1∑︁ +2 (, (), ()) + Ψ() (). (47) =0 Let’s consider the family = { − } for all ∈ 2 (Ω). To this family we associate the variation vector of the functional (︀ )︀ = 0 , 1 , . . . , 1 +2 ∈ 1 , and inequality (43) persists: 1∑︁ +2 > 0. =0 Replacing by their respective expressions from (40) using formulas (47), we obtain ∫︁ ℋ (48) (, (), (), Ψ())( − )d > 0, ∀ ∈ 0 . Ω 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 () ∈ 0 32 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 2, 2012. Pp. 19–32 it is necessary that there exists a nontrivial vector (︀ )︀ * = 0 , 1 , . . . , 1 +2 , 1∑︁ +2 | | > 0, =0 where > 0 for = 0, 1 and the conditions (48) as well as the conditions ( ) = 0 = 1, 1 + 2 (conditions (45)), are satisfied; where function () is the solution of the problem (1)–(3), Ψ() is the solution of the conjugate problem (46) associated to () and the function ℋ is defined in (47). References 1. Yoro G., Bailly B., Assui K. R. Unicité et dérivabilité de la solution des systumes hydrodynamiques stationnaires non linéaires et perturbés // Revist. — 2010. — Vol. 15, B. — Pp. 91–107. 2. Opérateurs intégraux dans les espaces des fonctions additionnables / M. A. Crasnocelski, P. P. Zabrehiko, E. N. Poustilnik, P. E. Sobolevski. — M.: Naouka, 1966. — 496 p. 3. Ladyjenskana O. A. Les questions mathématiques de la dynamique de la ténacité des liquides non compressibles. — M.: Naouka, 1970. 4. Vassilev F. P. Méthodes numériques pour la résolution des problumes des extrema. — M.: Naouka, 1980. — 520 p. 5. Souhinine M. F. La régle du multiplicateur de Lagrange dans les espaces localement convexes // Revue scientifique de mathématiques de Sibérie. — 1982. — Vol. 23, No 4. — Pp. 153–165. УДК 517.917 Необходимые условия оптимальности для стационарной нелинейной возмущённой задачи гидродинамики в ограниченной области * Бале Байлли , Гозо Йоро† , 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.

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