Control of Cauchy Problem for a Laplacian Operator

In this paper we study the control of an ill-posed system relating to the Cauchy problem for an elliptical operator. The control of Cauchy systems for an elliptical operator has already been studied by many authors. But it still seems to be globally an open problem. Of all the studies that have been done on this problem, it is assumed that the set of admissible couple-state must be nonempty to make sense of the problem. This is the case of J. L. Lions in [6] who gave various examples of the admissible set to make a sense of the problem. O. Nakoulima in [9] uses the regularization-penalization method to approach the problem by a sequence of well-posed control problems, he obtains the convergence of the processus in a particular case of the admissible set. G. Mophou and O. Nakoulima in [10] do the same study and obtain the convergence of the processus when the interior of the admissible set is non empty. In this work, we give an approximate solution without an additional condition on the set of admissible couple-state.We propose a method which consists in associating with the singular control problem a "family" of controls of well posed problems. We propose as an alternative the stackelberg control which is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12]. Keywords: Systems Governed by PDEs, Stackelberg Control, Cauchy Problem, Cost Function 1. Statement of the Problem Let Ω be an open bounded subset of , with a boundary Γ of class C. Γ = e Γ ∪ i Γ with e Γ ∩ i Γ = ∅. The boundaries e Γ and i Γ are non-empty and of positive measure. Let us consider in Ω the state and the control (v1; v2) linked by the relations: ∆ 0 in Ω ,  on e Γ (1) With ∈ and (v1; v2)∈ e Γ × e Γ . Problem (1) is a Cauchy problem for an elliptic operator (here the Laplacian operator). In general, problem (1) does not admit a solution and there is the instability of the solution when it exists, see for instance [6], [11]. It is so an ill-posed problem. But, it is important to control the Cauchy data, considering the fact that, such problems come from many concrete situations. It is the case in gravimetry for instance. In the evolution case we have enzymatic reactions, see [4], and the bibliography of this work, the control of the transmission of electrical energy, the control of the form of plasmas. Let  and  be two no empty closed convex subsets of e Γ . We denote by !" , ; $ ∈ % x , ∆ 0 in Ω, ' e Γ , ( ( ' e Γ ) a subset of e Γ × e Γ × . We assume that * ∅ (2) It is obvious that is a closed convex subset of e Γ × e Γ × . A couple of control-state " , ; $ will be called “admissible couple” if it belongs in . To simplify the notation, we will write , ; instead of " , ; $. 272 Sadou Tao: Control of Cauchy Problem for a Laplacian Operator + e Γ ,./012 34-5.6 3-, .7 5895.1 -: ; e Γ < . (3) Then ≠ ∅ if z│ i Γ ∈ ( i Γ ), we build = solution of > ∆= = 0 in Ω = = 0 -, e Γ ; ? = on e Γ (4) The system (4) defines a unique ς ∈ @AB ( ). Therefore ς C e Γ ∈ @ ; e Γ < ⊂ ; e Γ < . Ehe triplet ;= C e Γ , , =< ∈ . See F6H p. 416. Let us consider now, a strictly convex cost functional J, defined for all admissible couple control-state ( , ; ) by I( , , ) = ‖ − ‖ 2 ) ( 2 Ω L +LM ‖ ‖2 ) ( 2 Ω L + LB ‖ ‖ 2 ) ( 2 Ω L (5) Where (N1, N2) ∈ RO∗ x RO∗ and ∈ (Ω ) the desired state. We are then interested in the problem: inf I( , ; ), ( , ; ) ∈ . (6) The assumption (2) and the structure of J, show easily that problem (6) has a unique solution (u1, u2; y) which we are going to characterize. The function J being differentiable, if (u1, u2; y) is the optimal control-state the first order EulerLagrange conditions gives. + :-Q R44 ( , ; ) ∈ , ∫Ω (2 − )( − 2)67 + ∫Γe T 8 ( − 8 )6 e Γ + ∫Γe T 8 ( − 8 )6 e Γ (7) Many authors have already studied Cauchy systems. In the parabolic and hyperbolic case we can cite M. Barry and O. Nakoulima in [1]; J. P. Kernevez [4]; M. Barry, O. Nakoulima and G. B. Ndiaye [2]. In the elliptic case we can cite J. L. Lions in [6]; O. Nakoulima in [9]; G. Mophou and O. Nakoulima in [10]. To get a system where state and control are independent, J. L. Lions proposed in [5] a method of approximation by penalization. He obtained the convergence of the process when  = L ( e Γ ) and  = L ( e Γ ). O. Nakoulima in [9] uses the regularization-penalization method. That means, he considered the control problem as a "singular" limit of a sequence of well-posed control problems. The convergence of the process is also obtained by the author in a particular case:  =  = (L ( e Γ )). O. Nakoulima and G. Mophou in [10] use a regularization method that consists of viewing a singular problem as a limit of a family of well-posed problems. Following this method and assuming that the interior of considered convex is -nonempty, they obtain a singular optimality system (S. O. S.) for the considered control problem. In this paper, we propose to give an approximate solution without an additional condition on the set of admissible couple-state. We associate with the control problem of the ill-posed problem (1) -(2), (5) (7) a family of hierarchical controls of well-posed problems. The rest of this paper is organized as follows. In section 2.1 and section 2.2, we establish respectively the optimality conditions of the follower and of the leader. In section 2.3, we study the equivalence of the problem studied in sections 2.1 and 2.2, and the problem (1), (5) (7). 2. Stackelberg Control of the Cauchy Problem for a Laplacian Operator In this part, we study two well-posed systems associated with the ill-posed problem (1). We propose as an alternative the Stackelberg control. The Stackelberg leadership model is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12]. This model involves two companies (controls) that compete on the market of the same product. The first (leader) to act must integrate the reaction of the other firm (followers) in the choices it makes in the amount of product that decides to put on the market. In this case, the control of Stackelberg consists of the hierarchical control of two objective functions, one by the follower and the other by the leader. More precisely, we consider the following systems: > ∆ = 0 in Ω = -, e Γ ; M = U on i Γ (8) > ∆ = 0 in Ω B = -, e Γ ; = U on i Γ (9) Where ( , ) ∈ (Ω ) × (Ω ), (U , U ) ∈ L( i Γ ) × L( i Γ ) with ( , ) defined as in (7). Remark 2 Problems (8) and (9) are well-posed. If ; C e Γ ; C i Γ < = ( ; U ) ∈ ; e Γ < × ; i Γ < American Journal of Applied Mathematics 2020; 8(5): 271-277 273 and ( M C i Γ ; B C e Γ ) = (U , ) ∈ ; e Γ < × ; i Γ < for the trace theorem, see [5], [7] ( , ) ∈ @MB (Ω ) x @MB ;Ω <. (8) and (9) admit some unique solutions R,6 in @MB ;Ω <. We consider now the cost function define by I " , , V , V , (V ), (V )$ = W M (V ) − B (V )WXB(YZ) + ‖ − ‖XB(Y[) (10) Where ( , ) ∈ ( ) × ( ) is a solution of (8)-(9). And the minimization follow problem min I " , , V , V , (V ), (V )$; (U , U ) ∈ L2(Γ]) x L2(Γ]) (11) Remark 3 In a symmetrical way it is possible to introduce the functional cost I " , , V , V , (V ), (V )$ = W M (V ) − B (V )WXB(Y[) + ‖ − ‖XB(YZ) (12) The set being non-empty, we now show that the resolution of the problem (1) is equivalent to that of the problem (8) (11). Proposition 4 Assume that the problem (1) admits a solution. Then, solving problem (1) is equivalent to solve the problem (8) (11) and min I " , , U , U , (U ), (U )$ = 0, (U , U ) ∈ L2(Γ]) x L2(Γ]) (13) Proof Let ( , , ) be a solution of (1), such that V =  -, ]̂ R,6 V = -, ]̂ Then, we have I ( , , V , V , , ) = WV − WXB(YZ) + ‖ − ‖XB(Y[) = 0. Conversely, let ( , , V ∗, V ∗, ∗, ∗) be a solution of (8)-(11), we have 0 ≤ I ( , , V ∗, V ∗, ∗, ∗) ≤ I ( , , V , V , , ) = 0 (14) (14)b/04b.5 1hR1 ∗ = ∗ -, Γd R,6 ( ∗ ( = ( ∗ ( -, Γ] .e. 6.683. 1hR1, ∗ − ∗ b5 1h. 5-481b-, -: 1h. :-44-Vb,f 0Q-94./ > ∆( ∗ − ∗) = 0 in Ω ∗ − ∗ = 0 -, Γd; M∗ − B∗ = 0 on Γ] (15) :Q-/ (15), V. -91Rb, ∗ = ∗ = b, .∎ 2.1. Optimality System of the Follower Proposition 5 Eh. :8,31b-,R4 I b5 1Vb3. nâ1.R87 6b::.Q.,1bR94. R,6 51Qb3142 3-, .7. pQ--:.e. :bQ51 5h-V 1hR1 I b5 nâ1.R87 6b::.Q.,1bR94.. .1 (q , q ) ∈ ( ]̂) × ( ]̂), r ∈ R R,6 6.,-1. 92 (. , . )1h. 53R4RQ 0Q-6831 b, . .1 85 3-,5b6.Q 1h. R004b3R1b-,5 q → (q ) − (0) R,6 q → (q ) − (0) (16) 5-481b-, -: ∆t = 0 in Ω t = 0 -, Γd; uM = t on Γ] (17) ∆t = 0 in Ω uB = 0 -, Γd; t = q on Γ] (18) The applications defined in (16) are linear. The following calculations are performed I ( , , U + rq , U + rq , , ) = W M (U + rq ) − B (U + rq )WXB(YZ) + ‖ − (U + rq )‖XB(Y[) 274 Sadou Tao: Control of Cauchy Problem for a Laplacian Operator = WU − B V WXB YZ +vB Wq − B q + B 0)WXB(YZ) + ‖ − (U )‖XB(Y[) + vB ‖− (q ) + (0)‖XB(Y[) +λ;U − B (U ), q − B (q ) + B (0) <XB(YZ) +λ( − (U ), − (q ) + (0))XB(Y[) Hence lim v→y I (U + r q , q + r q , , , , ) − I (U , q , , , , ) r = ;U − B (U ), q − B (q ) + B (0) <XB(YZ)+ ( − (U ), − (q ) + (0))XB(Y[) This last result shows that I is Gâteaux differentiable and 6I ( , , U , U , , ) ∙ (q , q )= ;U − B (U ), q − B (q ) + B (0) <XB(YZ)+ ( − (U ), − (q ) + (0))XB(Y[) (19) For ({ , { ) ∈ ( ]̂) × ( ]̂) and λ ∈ R V. hR . 6I ( , , U + r{ , U + r{ , , ) ∙ (q , q ) =;U + r{ − B (U + r{ ), q − B (q ) + B (0) <XB(YZ) + ( − (U + r{ ), − (q ) + (0))XB(Y[) = ;U − B (U ), q − B (q ) + B (0) <XB(YZ) + ;r{ − B ({ ) + B (0), q − B (q ) + B (0) <XB(YZ) + ( − (U ), − (q ) + (0))XB(Y[) +λ (− ({ ) + (0), − (U ) + (0))XB(Y[) Which give lim v→y(|B( M, B,}MOv~M ,}BOv~B, M, B)∙(M,B) v -|B"}M,}B, M, B, M, B$∙(M,B) v ) = ( ( ({ ) − ( ( (0) − ( ( ({ ) + ( ( (0), ( ( (q ) − ( ( (0) − ( ( (q ) + ( ( (0)XB(YZ) + (− ({ ) + (0), − (U ) + (0))XB(Y[) and so 6I ( , , U , U , , ) ∙ ({ , { ) = ;{ − B ({ ) + B (0), q − B (q ) + B (0) <XB(YZ) + (− ({ ) + (0), − (U ) + (0))XB(Y[). This last result shows that I is twice Gâteaux differentiable. We have 6I ( , , U , U , , ) ∙ (q , q ) ∙ (q , q ) = q − ( ( (q ) + ( ( (0)XB(YZ) +‖− (q ) + (0)‖XB(YZ) American Journal of Applied Mathematics 2020; 8(5): 271-277 275 5 6I , , U , U , , ) ∙ (q , q ) ∙ (q , q ) ≥ 0, then I is convex. In addition, we have 6I ( , , U , U , , ) ∙ (q , q ) ∙ (q , q ) = 0 Implies that q − B (q ) + B (0) = 0 -, Γ] R,6 (q ) − (0) = 0 -, Γd . From (18) and the identity (q ) − (0) = 0 -, Γd , the uniqueness of the Cauchy problem gives (q ) − (0) = 0 b,  R,6 1h85 q = 0. F From the identity q − B (q ) + B (0) = 0 -, Γ] V. hR . q1 = 0. Finally I is strictly convex. ∎ The strict convexity of I leads to the uniqueness of the solution of prob-lem (11). We introduce now some adjoint systems of (8) and (9) define by ∆0 = 0 in Ω 0 = − -, Γd; M = −U + B on Γ] (20) ∆0 = 0 in Ω B = − -, Γd; 0 = −U + B on Γ] (21) We obtain the optimality system below Proposition 6 (U ∗ , U ∗ ) is an optimal solution of (13) if and only if it exists ( ∗, ∗ ) satisfying (8)(9) and (0 ∗, 0 ∗) satisfying (20)-(21) such as the triplet (U ∗ , U ∗ ), ( ∗, ∗ ), (0 ∗, 0 ∗ ) is the solution of the optimality systems: ∆0 ∗ = 0 in Ω ∗ = -, Γd; M∗ = −U ∗ on Γ] (22) ∆0 = 0 in Ω B = -, Γd; ∗ = U ∗on Γ] (23) ∆0 ∗ = 0 in Ω 0 ∗ = − ∗ -, Γd; M∗ = −U ∗ + B∗ on Γ] (24) ∆0 ∗ = 0 in Ω B∗ = − ∗ -, Γd; 0 ∗ = −U ∗ + B∗ on Γ] (25) and (0 ∗ ( = 0 -, Γ]; (0 ∗ ( = 0 -, Γ]. pQ--:. .1 (U , U , ), (q , q ) ∈ ( ]̂) × ( ]̂), λ ∈ R. Q-/ (19) V. hR . 6I " , ,U , U , , $ ∙ (q , q ) = ;U − B (U ), q − B (q ) + B (0) <XB(YZ) + ( − (U ), − (q ) + (0))XB(Y[). According to the Euler optimality conditions, (U ∗, U ∗) is the optimal solution of (13) if and only if, (q , q ) ∈ ( ]̂) × ( ]̂), 6I " , ,U ∗ , U ∗ , ∗, ∗ $ ∙ (q , q ) ≥ 0. that is, ∀(q , q ) ∈ ( ]̂) × ( ]̂), U ∗ − ( ∗ ( , q − ( ( (q ) + ( ( (0)XB(YZ) +( − ∗, − (q ) + (0))XB(Y[) ≥ 0. Hence U ∗ − ( ∗ ( , q XB(YZ) + U ∗ − ( ∗ ( − ( ( (q ) + ( ( (0)XB(YZ) +( − ∗, − (q ) + (0))XB(Y[) ≥ 0, 5 R Q.5841, 1R4b,f into account (24) and (25) we have ;− M∗ , q <XB(YZ) + ;0 ∗, B (q ) − B (0)<XB(YZ) (26) -−; B∗ , (q ) − (0)<XB(Y[) ≥ 0. Multiplying the first equation of (25) by t the solution of (18) And integrating by parts we have − (02∗ ( Γ. t 65 +  02∗ (t ( Γb 65 =  (02∗ ( Γb t 65. (26) become ∀(q , q ) ∈ L2(Γb)x L2(Γb), −(0 ∗ ( , q XB(YZ) + (0 ∗ ( , q XB(YZ) = 0. 276 Sadou Tao: Control of Cauchy Problem for a Laplacian Operator b,R442 V. hR . (01∗ ( 0 -, Γ] R,6 (02∗ ( = 0 -, Γ]. ∎ 2.2. Optimality System of the Leader Consider the cost function I " , ,U ∗, U ∗ , ∗, ∗ $ = M ‖ ∗ − ‖XB() + B ‖ ∗ − ‖XB() + LM ‖ ‖XB(Y[) + LB ‖ ‖XB(Y[) (27) Vh.Q. (T , T ) R,6 6 are defined in (5),  R,6  RQ. 5-/. Q.R45 583h 1hR1  +  = 1, R,6 1h. 0Q-94./ inf I " , ,U ∗, U ∗ , ∗, ∗ $, " , ,$ ∈ ( d̂) × ( d̂) (28) pQ-0-5b1b-, 7 Eh.Q. .7b51. (8 ∗, 8 ∗) ∈ ( d̂) × ( d̂) 8,b8. 583h 1hR1 ∀" , ,$ ∈ ( d̂) × ( d̂), (.) I (8 ∗, 8 ∗ , U ∗ , U ∗ , ∗, ∗ ) ≤ I " , ,U ∗, U ∗ , ∗, ∗ $ (29) pQ--:. ( d̂) × ( d̂) is closed convex. I is coercive and strictly convex, then the problem (29) holds true. ∎ Consider again one adjoint problem of (8) and (9) define by ∆0 = ∗ − in Ω 0 = 0 -, Γd; M = 0 on Γ] (30) ∆0 = ∗ − in Ω B = 0 -, Γd; 0 = 0 on Γ] (31) We obtain the optimality system below Proposition 8 (8 ∗, 8 ∗) is an optimal solution of (28) if and only if it exists ( ∗; ∗) satisfying (8) (9) and (0 ∗; 0 ∗) satisfying (30)-(31) such as the triplet {(8 ∗; 8 ∗), ( ∗; ∗), (0 ∗; 0 ∗)} is the solution of the optimality systems: ∆ ∗ = 0 in Ω ∗ = 8 ∗ -, Γd; M∗ = U ∗ on Γ] (32) ∆ ∗ = 0 in Ω B∗ = 8 ∗ -, Γd; ∗ = U ∗ on Γ] (33) ∆0 ∗ = ∗ − in Ω 0 ∗ = 0 -, Γd; M∗ = 0 on Γ] (34) ∆0 ∗ = ∗ − in Ω B = 0 -, Γd; 0 ∗ = 0 on Γ] (35) And - (01∗ ( + T181∗ = 0 -, Γ.;  02∗ + T282∗ =0 on Γd (36) Proof. Let (φ , φ ) ∈ 2(^.)× 2(^.) , λ ∈ R and denote by (.,.) the scalar Product in L 2 . Let us consider the applications φ → z (φ ) − z (0) and φ → z (φ ) − z (0) (37) solutions of ∆t = 0 in Ω t = φ -, Γd; uM = 0 on Γ] (38) ∆t = 0 in Ω uB = φ -, Γd; t = 0 on Γ] (39) The applications define in (37) are linear. We have ∀"q , q ,$ ∈ ( d̂) × ( d̂), 6I " , , $ ∙ "q , q ,$ =  ( ∗( ) − , ∗(q ) − ∗(0))XB() + ( ∗( ) − , ∗(q ) − ∗(0))XB() +T " , q , $XB([)+ T " , q , $XB([) According to the Euler optimality conditions, (8 ∗; 8 ∗) is an optimal solution of (28) if and only if, ∀"q , q ,$ ∈ ( d̂) × ( d̂), 6I (8 ∗; 8 ∗) ∙ "q , q ,$ ≥ 0. That gives ∀"q , q ,$ ∈ ( d̂) × ( d̂), (40)  ( ∗(8 ∗) − , ∗(q ) − ∗(0))XB() + ( ∗(8 ∗) − , ∗(q ) − ∗(0))XB() +T "8 ∗, q , $XB([)+ T "8 ∗ , q , $XB([) ≥ 0. Multiplying the firsts equations of (30) and (31) respectively by t and t (solutions of (38) and (39) and integrating by parts we obtain " ∗ (8 ∗) − , ∗ (q ) − ∗ (0)$XB()=;− M∗ , q <XB([) (41) And " ∗ (8 ∗) − , ∗ (q ) − ∗ (0)$XB()=(0 ∗, q )XB([) (42) From (40)-(42) we obtain ∀"q , q ,$ ∈ ( d̂) × ( d̂), ;− M∗ +T 8 ∗, q <XB([)+( 0 ∗ +T 8 ∗ , q )XB([) = 0. American Journal of Applied Mathematics 2020; 8(5): 271-277 277 and finally, we have − (0 ∗ ( + T 8 ∗ 0 -, d̂ R,6  0 ∗ + T 8 ∗ = 0 -, d̂ . 2.3. Equivalence to the Problem (1); (5)-(7) In this section, we will show that the optimality system (32)-(36) makes it possible to calculate the solution of problem (1), (5) (7). Let (8 , 8 , y) be the unique solution of problem (1), (5) (7) and ( , ) ∈ ( d̂) × ( d̂). J2 being the cost function defined in (10) and (8 ∗ , 8 ∗ ); (U , ∗ U ∗ ); ( ∗ , ∗ ) defines respectively in proposition 6 and proposition 8, we have I (8 ∗, 8 ∗ , U ∗ , U ∗ , ∗, ∗ ) ≤ I "8 , 8 ,U ∗ , U ∗ , ∗, ∗ $ ≤ I "8 , 8 ,U ∗ , U ∗ , 2, 2 $ = 0 From proposition 4 we conclude that 8 ∗ = 8 , 8 ∗ = 8 ,; and ∗ = ∗ = 2


Statement of the Problem
Let Ω be an open bounded subset of , with a boundary Γ of class C 2 . Problem (1) is a Cauchy problem for an elliptic operator (here the Laplacian operator). In general, problem (1) does not admit a solution and there is the instability of the solution when it exists, see for instance [6], [11]. It is so an ill-posed problem. But, it is important to control the Cauchy data, considering the fact that, such problems come from many concrete situations. It is the case in gravimetry for instance.
In the evolution case we have enzymatic reactions, see [4], and the bibliography of this work, the control of the transmission of electrical energy, the control of the form of plasmas. Let and be two no empty closed convex subsets of e Γ . We denote by We assume that It is obvious that is a closed convex subset of A couple of control-state " , ; $ will be called "admissible couple" if it belongs in . To simplify the notation, we will write , ; instead of " , ; $. (3) The system (4) defines a unique ς ∈ @ A B ( ). Therefore ς C e Γ ∈ @ ; e Γ < ⊂ ; e Γ < . Ehe triplet ;= C e Γ , , =< ∈ . See F6H p. 416. Let us consider now, a strictly convex cost functional J, defined for all admissible couple control-state ( , ; ) by We are then interested in the problem: inf I( , ; ), ( , ; ) ∈ .
The assumption (2) and the structure of J, show easily that problem (6) has a unique solution (u 1 , u 2 ; y) which we are going to characterize. The function J being differentiable, if (u 1 , u 2 ; y) is the optimal control-state the first order Euler-Lagrange conditions gives.
Many authors have already studied Cauchy systems.
In the elliptic case we can cite J. L. Lions in [6]; O. Nakoulima in [9]; G.
Mophou and O. Nakoulima in [10]. To get a system where state and control are independent, J. L. Lions proposed in [5] a method of approximation by penalization. He obtained the convergence of the process when = L 2 ( e Γ ) and O. Nakoulima in [9] uses the regularization-penalization method. That means, he considered the control problem as a "singular" limit of a sequence of well-posed control problems. The convergence of the process is also obtained by the author in a particular case: = = (L 2 ( e Γ )) + .
O. Nakoulima and G. Mophou in [10] use a regularization method that consists of viewing a singular problem as a limit of a family of well-posed problems. Following this method and assuming that the interior of considered convex is -nonempty, they obtain a singular optimality system (S. O. S.) for the considered control problem.
In this paper, we propose to give an approximate solution without an additional condition on the set of admissible couple-state.
We associate with the control problem of the ill-posed problem (1) -(2), (5) -(7) a family of hierarchical controls of well-posed problems.
The rest of this paper is organized as follows. In section 2.1 and section 2.2, we establish respectively the optimality conditions of the follower and of the leader. In section 2.3, we study the equivalence of the problem studied in sections 2.1 and 2.2, and the problem (1), (5) -(7).

Stackelberg Control of the Cauchy Problem for a Laplacian Operator
In this part, we study two well-posed systems associated with the ill-posed problem (1). We propose as an alternative the Stackelberg control. The Stackelberg leadership model is a multiple-objective optimization approach proposed by H. Von Stackelberg in [12]. This model involves two companies (controls) that compete on the market of the same product. The first (leader) to act must integrate the reaction of the other firm (followers) in the choices it makes in the amount of product that decides to put on the market. In this case, the control of Stackelberg consists of the hierarchical control of two objective functions, one by the follower and the other by the leader.
We consider now the cost function define by Where ( , ) ∈ ( ) × ( ) is a solution of (8)- (9). And the minimization follow problem Remark 3 In a symmetrical way it is possible to introduce the functional cost The set being non-empty, we now show that the resolution of the problem (1) is equivalent to that of the problem (8) - (11). Proposition 4 Assume that the problem (1) admits a solution. Then, solving problem (1) is equivalent to solve the problem (8) - (11) and Proof Let ( , , ) be a solution of (1), such that V = -, ^] R,6 V = -, ^] Then, we have Conversely, let ( , , V * , V * , * , * ) be a solution of (8) The applications defined in (16)   This last result shows that I is twice Gâteaux differentiable.
We have
From (18) and the identity (q ) − (0) = 0 -, Γ d , the uniqueness of the Cauchy problem gives Finally I is strictly convex. ∎ The strict convexity of I leads to the uniqueness of the solution of prob-lem (11).

Conclusion
The present article presents an alternative possibility to study the optimal control problem (6) and to obtain an optimality condition where state and control are independent, different from the regularization-penalization method introduced by O. Nakoulima in [9] or the regularization method due to O. Nakoulima and G. Mophou in [10].