The Continuous Classical Boundary Optimal Control of Triple Nonlinear Elliptic Partial Differential Equations with State Constraints

Our aim in this work is to study the classical continuous boundary control vector problem for triple nonlinear partial differential equations of elliptic type involving a Neumann boundary control. At first, we prove that the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector have a unique "state" solution vector, by using the Minty-Browder Theorem. In addition, we prove the existence of a classical continuous boundary optimal control vector ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the triple adjoint equations related with the triple state equations. The Fréchet derivative is obtained. Finally we prove the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type through the Kuhn-Tucker-Lagrange's Multipliers theorem with equality and inequality constraints.


Introduction
In many fields, the optimal control problems play a significant role in life. Different examples of the applications of such problems are presented in medicine [1], aircraft industry [2], electric power production [3], economic growth [4], and many other fields. All these applications pushed many investigators to a higher level of interest in studying the optimal control problem for nonlinear ordinary differential equations [5], for different types of linear partial differential equations, including the hyperbolic, parabolic and elliptic [6][7][8], or for couple nonlinear partial differential equations of these three types [9][10][11]. While other authors [12,13] studied these three types but included a Neumann boundary control. More recently, optimal control problems were studied for triple partial differential equations of these three types [14][15][16]. Also, the optimal control problem involving Neumann boundary control for triple partial differential equations of parabolic type was also recently investigated [17]. All these investigations motivated us to seek the optimal control problem, involving Neumann boundary control ruled by the triple nonlinear partial differential equations of elliptic type. At first, our aim in this work is to prove that system of the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector, which has a unique "state" solution vector, by using the Minty-Browder Theorem. Then, we prove the existence of a classical continuous boundary optimal control vector, ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the system of the triple adjoint equations related with the triple state equations. At the end, we prove the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type through the Kuhn-Tucker-Lagrange's Multipliers with equality and inequality constraints.

Problem Description Let be a bounded and open connected subset in
with Lipshitz boundary . The optimal control problem is considered by the "state vector equation" which consists of the TNLEPDEs triple nonlinear elliptic partial differential equations with the Neumann boundary control.

Weak formulation of the triple state equations
To find the weak formulation of problem (1-6) , let ⃗ ⃗ ( ) ( ) ( ) * ( ) ( ( )) , with satisfy (4)-(6), respectively on }. By multiplying both sides of equations (1),(2) and (3) by , respectively, integrating both sides of each one of the obtained equations with respect to ӽ, and then using the generalize Green's theorem, we get By adding equations (11), (12) and (13), we get where . The following assumptions are useful to prove the existence theorem of a unique solution of the weak form (14).  (14) is strictly monotone, then for each fixed classical continuous boundary optimal control vector ⃗ ⃗ ⃗ , the weak form of (14) has a unique "state" solution vector ⃗ ⃗ ⃗ . Proof: It is clear that the existence of a unique solution of (14) is obtained after the usage of assumptions (I), then theorem (1) in reference [18] is applied.

Existence of the Classical Continuous Boundary Optimal Control Vector
In this section, the theorem of the existence of a classical continuous boundary optimal control vector under the suitable assumptions is proved. However, before proving it, it is necessary to deal with the following lemmas and assumptions. Lemma (1): If the assumption (I) is hold, the functions are Lipschitz continuous with respect to , res respectively, and if ( ) ( ) ( ) are bounded, then the mapping Proof: Assume that ⃗ ⃗ ⃗ ⃗ are two given controls, then there corresponding "state" solution vectors (of the weak form (14)) are ⃗ ⃗ . By subtracting the above three obtained weak forms from their corresponding ones in (14), putting ⃗⃗⃗⃗ ⃗ ⃗ and ⃗⃗⃗⃗ ⃗ ⃗ , with ⃗⃗⃗⃗ , then adding the obtained three equations, we get By using assumption A-(a, d) , taking the absolute value for both sides of (16), it becomes By using the Cauchy-Schwarz inequality and then the trace operator in the right side, on (17) , we obtain which gives

Assumption (II):
Assume that ԏ_P1 ,ԏ_P2 ,ԏ_P3 on Ψ×R and ԏ_P4 ,ԏ_P5 ,ԏ_P6 on Ψ×D are of the Carathéodory type, then the following are satisfied for each P=0,1,2: where ( ) ( ) and Lemma (2): If assumption (II) is held, then the functional Ԏ_P ( d ) is continuous on (L_2 (∂Ψ))^3 for each P=0,1,2.Proof: For any , we set To prove the continuity for any one of the above two integrals, the used technique will be similar. Thus, it is enough to prove one of them, which is in this case the second integral. Hence, let ⃗ ( ) , with , then from assumption (II), we have Then, the ( ⃗ ) is continuous on ( ( )) (by using Proposition (1) in reference [19]). Hence,

Theorem (2):
If the assumptions (I) and (II) are hold, ⃗ ⃗ , are not dependent on respectively , and are bounded functions , so that, , and , for . are not dependent on respectively .
Also, from the assumptions on ( ) and ( ) ( ) and Lemma (2) , the integrals ( ) and ( ) are continuous with respect to and , respectively, but ( ) , ( ) is convex with respect to , then ( ) is weakly lower semicontinuous with respect to , i.e.
By the same manner, and for each , we get the following two convergences: From the above inequalities, one gets that ( ⃗ ) ( ) is weakly lower semicontinuous with respect to ( ⃗ ⃗ ). Thus ( ⃗ ) ( ⃗ ) , and ⃗ is a continuous classical boundary optimal control vector .

Boundary Optimal Control Vector
The following assumptions are useful in this section to derive the Fréchet derivative of the Hamiltonian.

Assumption (III) a)
are of the Carathéodory type on and satisfy are of the Carathéodory type on and satisfy ( ) One can easily prove that the weak form (30), with fixed continuous classical boundary optimal control vector ⃗ ⃗ ⃗ has a unique "state" solution vector ⃗ ⃗ by applying the same manner employed in the proof of theorem (3). Now, by setting once the solution in the weak forms of the state equations (11) and once again the solution , then subtracting the obtained weak form from the other one , we obtain ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ) (31) The same above substituting and subtracting are repeated but from a side with the solutions and in the weak form of equation (12) and from thither side with the solutions and in the weak form of the state equation (13), respectively, to obtain ( ) ( From the assumptions on and by using Proposition (2) in reference [19], the Fréchet derivative of exists. Hence, from Lemma (1) and the Minkowski inequality, (34) becomes Now, from the assumptions on , Proposition (2) in reference [19], and then using the result of Lemma (1), we have where ( ⃗⃗⃗⃗ ) , as ⃗⃗⃗⃗ By substituting (36) in the above equality, we get where ̃ ( ⃗⃗⃗⃗ ) ̃ ( ⃗⃗⃗⃗ ) ̃ ( ⃗⃗⃗⃗ ) ̃ ( ⃗⃗⃗⃗ ) ̃ ( ⃗⃗⃗⃗ ) , as ⃗⃗⃗⃗ . But from the definition of the Fréchet derivative of , one gets ⃗⃗⃗⃗⃗ ( ⃗ ) ⃗⃗⃗⃗ ∫ ⃗ ⃗ ⃗⃗⃗⃗ , where ⃗ ⃗ is defined above.
(38) Note: In the proof of the above theorem, we have found the Fréchet derivative for the functional , so the same technique is used to find the Fréchet derivative for and .

Conclusions
The existence and uniqueness theorem for the "state" solution vector of the triple nonlinear partial differential equations of elliptic type is proved successfully, when the classical continuous boundary control vector is given. The proof of the existence of the classical continuous boundary control vector, ruling by the considered triple nonlinear partial differential equations of elliptic type, is demonstrated with the equality and inequality constraints. The studying of the existence solution of the triple adjoint equations related with the triple nonlinear partial differential equations of elliptic type is demonstrated with the equality and inequality constraints. Finally, the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type, through the Kuhn-Tucker-Lagrange's Multiplires with equality and inequality constraints, is demonstrated.