Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem

-e focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). -e Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.


Introduction
e subject of optimal control problem (OCP) plays a basic role in many real life problems in different branches of sciences; for example, in, medicine [1], engineering and social sciences [2], biology [3], ecology [4], electric power [5], aerospace [6], and many other branches.
is role encouraged many researchers to go deeply into studying the OCPS governed by differential equations (deqs). Such OCP problems are studied at the beginning for the systems which are controlled by nonlinear ordinary deqs (nodeqs) [7] or by linear deqs (lpdeqs) [8]. Later great interests have been made to study this subject but for systems which are controlled by pdeqs of elliptic type (ET) [9], or of hyperbolic type (HT) [10], or of parabolic type (PT) [11], or by couple of npdeqs of ET [12], or of PT [13], or of HT [14].
Recently, the attention in this subject is magnified to deal with more general types as the studying the CCCOPCVE controlling by TBVP of ET [15], and of PT [16]. All these studies motivated us to look deep inside the CCCOPCVE controlled by TNHBVP.
In this work and at first, we give a mathematical description for the CCCOPCVE, and then the TNHBVP is written in its weak form (wkf ), and the existence and uniqueness theorem of the Stvs for the TNHBVP using the Galm with the Aubin compactness theorem is proved under appropriate hypotheses when the CCCVE is given. Under reasonable hypotheses, the objective function and the EQVC and INEQVC are proved continuous. e proof of the existence theorem of a CCCOPCVE governed by the TNHBVP is achieved. Under a certain hypotheses, the study of the existence theorem for a unique Advs of the TAEqs associated with the considered Tsteqs is done. e Fréchet Derivative (Frde.) of the HAM is found. Finally, the Necoop and the Sucoop theorems for the CCCOPCVE are proved.
Assumption A. k i is of the Carathéodory type (Caraty.) on Π × (R × ∁ i ) and satisfies the following conditions for (x, t) ∈ Π and ∀i � 1, 2, 3: Journal of Applied Mathematics

e Solution of the State Equations.
In this part, the existence theorem of a unique solution for triple nonlinear hyperbolic partial differential equations (TNLHPDEQs) under Assumption A is proved when the control vector is given, and the following proposition will be needed.

Journal of Applied Mathematics
where en, corresponding to the sequence Υ → n , there exists a sequence of the following approximation problems, i.e., for each υ → n � (υ 1n , υ 2n , υ 3n ) ⊂ Υ → n , and n � 1, 2, . . .: which has a sequence of unique solution ψ → n . Substituting υ in � ψ int for i � 1, 2, 3 in (25)-(27), respectively, adding the three obtained equations together, and employing Lemma 1.2 in [18] for the first term of the LHS, to get (33) which is given by Or (33) can be rewritten as (34) which is where Appling the Belman-Gronwall (BGin) inequality, the abovementioned inequality gives ∀t ∈ [0, T]: (37) and en, the Aubin compactness theorem [18] can be applied here to get that ψ → n ⟶ ψ → in (L 2 (Π)) 3 . Now, multiplying both sides of (27) and (29), and (31) by integrating on, finally integrating by parts twice the 1 st term of each one of the obtained three equations, led to Journal of Applied Mathematics 5 Now, for each i � 1, 2, 3, we have the following convergences: On the other hand, since and w in is measurable in E, so from Assumption (A-I) and Proposition 1, the integral (45) □ Now, from these convergences and (24) and (25), for (i � 1), we can passage to the limits in (39) and (40) to get Using these values in (47) (for i � 1), using integration by parts twice for the first terms in the LHS of the obtained equation, yields to which give that ψ 1 is a solution of (10) (a.e. on I).
Also, similar way can be used but for i � 2, 3 with the pairs (12) and (47) and (14) and (48), respectively, to get the same result.
For i � 1, integrating both sides of (10) on [0, T] after multiplying it by χ 1 (t), using integrating by parts for the first term in the LHS of the obtained equation, then subtracting this obtained equation from (47), we get . Also, for i � 2, 3 and by using (12) and (14), we can use a similar way to get the same result.
From the last two cases, easily we can get the ICs (11) and (13), and (15).
Uniqueness of the solution: let ψ → � (ψ 1 , ψ 2 , ψ 3 ) and ψ → � (ψ 1 , ψ 2 , ψ 3 ) be two solutions of the wkf (10)-(15), i.e., ψ i and ψ i (for each i � 1, 2, 3) are satisfied the wkf (10)-(15), subtracting each equality from the other and letting Adding these three equations, using Lemma 1.2 in ref. [18] on the first term in LHS of the obtained equation which will be positive, integrating both sides from 0 to t, using the initial conditions, the Lipshctiz property on the RHS, and lastly applying the B-G inequality, to get , the solution is unique. (50)

Lemma 1. In addition to Assumption A, if the functions k i (for each i � 1, 2, 3) is Lip. with respect to y i and ω i , and if the control vector is bounded, then the operator ω
are their corresponding states' solutions which satisfy the wkf of (10)- (15), Substituting υ i � Δψ it for each i � 1, 2, 3 in (51), (53), and (55), respectively, adding the obtained three equations together, and using the same way that we used to get (32), a similar equation can be obtained but with Δψ � �→ in state of ψ → n , and then integration of both sides on [0, t], using the Lip. property on k i , i � 1, 2, 3, with respect to each dependent variable, yields Journal of Applied Mathematics 7 Using the definitions of the norms and the relations between them, we get where Applying the BGin, with L 2 � L 2 e L 1 , we get □ From the abovementioned three inequalities, the Lip. continuity of the operator ω → ↦ ψ → easily obtained.

e Existence of a Classical Optimal Control.
is section concerned with proving the existence theorem with a CCOPCV satisfying the EQVC and INEQVC is studied.
Hence, the following assumption and lemmas will be needed.
Proof. From the Assumption on ∁ i ⊂ R ∀i � 1, 2, 3 and Egorov's theorem, one obtains that W Since A , ∀ρ, and we get that (71) us, ω → is a CCOPCV.
Assumption C. Assume for r � 0, . . . , q and i � 1, 2, 3, the functions k i , k iψ i , k iω i , m r i ψ i , and m r i ω i are defined and are of where Note: for simplicity, in the following theorem, we will drop the index k from the functions m li and M l . Also, we assume the Assums. (A), (B), and (C) are considered.

Theorem 3. Consider the TAEqs ξ
and the Ham is given by en, the Frde. of G is defined by where κ r m ri and ξ i � q r�0 κ r ξ li , for each i � 1, 2, 3.

Necessary and Sufficient Conditions for Optimality.
is section deals with the theorems for the Necoop necessary under certain hypotheses which are proved as follows: Journal of Applied Mathematics