Continuous Classical Optimal Control of Triple Nonlinear Parabolic Partial Differential Equations

This paper concerns with the state and proof the existence and uniqueness theorem of triple state vector solution (TSVS) for the triple nonlinear parabolic partial differential equations (TNPPDEs) ,and triple state vector equations (TSVEs), under suitable assumptions. when the continuous classical triple control vector (CCTCV) is given by using the method of Galerkin (MGA). The existence theorem of a continuous classical optimal triple control vector (CCTOCV) for the continuous classical optimal control governing by the TNPPDEs under suitable conditions is proved.


Introduction
The subject of optimal control problem is divided in to two types, the relaxed and the classical optimal control problems, the first type is mostly studied in the last century, while the second one began to study in the beginning of this century. On other hand each of these two types are studied for systems governing by ordinary or partial differential equations. The optimal control problems play an important role in many fields in life problems, different examples for applications of such problems are studied in medicine [1], in aircraft [2], in electric power [3], in economic growth [4], and many other fields.
This role motivates many investigators in the recent years to interest about study the classical optimal control problems OPCTP that are governing by nonlinear ordinary ISSN: 0067-2904
All these investigations encourage us to seek about the OPCTP for triple nonlinear parabolic PDEs (TNLEPDEs). At first, our aim in this work is to state and to prove that the TNLEPDEs with a given CCTCV has a unique TSVS under a suitable conditions, by using the MGA with the compactness theorem (COMTH). The continuity of the Lipschitz operator between the TSVS, and the corresponding CCTCV are proved. Finally, we also prove theorem which ensures the existence CCTOCV for the TNLEPDEs.

Problem Description
Let , be a bounded open region with Lipschitz (LIP) boundary . Consider the following CCTOCP: The TSVEs is given by the following TNPPDEs: Let ⃗⃗ { ⃗ } . The notations ) , and ‖ ‖ refer to the inner product and the norm in ,respectively. The notations , and ‖ ‖ re the inner product and the norm in , the ( ⃗ ⃗) and ‖ ⃗‖ the inner product and the norm in , and ⃗ ⃗ = , ‖ ⃗‖ ‖ ‖ ‖ ‖ ‖ ‖ the inner product and the norm in ⃗⃗ and ⃗⃗ * is the dual of ⃗⃗ also the notations , will indicate to the convergence of a sequence is weakly and strongly respectively. The weak form (W.F) of the TSVEs (1-9) when ⃗ is given by , ‖ ‖ ‖ ‖ and . By substituting ((16)-(18)) in ((19) -(21)) and setting , and we get the following system, which has a unique solution ⃗ because of all the coefficient matrices are continuous.
The norm ⃗ ⃗⃗ is bounded: By using the same previous steps in (21), but with and ‖ ⃗ ‖ is positive, one can easily obtain that   42)) from ((34)-(36)) resp., to get that ((13)b-(15)b) are held. The strong convergence for ⃗ ⃗⃗ in : Substituting , in ((13)a-(15)a) , and then we add them together, on the other hand substitute , in ((19)a-(21)a) resp. and then we add them together, and integrat the three obtained equations from to T , one has , with . Using Lemma 1.2 in [11] for the terms in the L.H.S. of (43a&b), they become  Uniqueness of the solution: Let ⃗ ⃗⃗ be two TSVEs of ((13)a-(15)a), we subtract each equation from the other and then set ̅ , for , one obtains The BGIN is applied to give that ‖( ⃗ ̅ ⃗ ) ‖ , . Again, IBS of (50) w.r.t. from 0 to T, Assumptions (A-ii) of the R.H.S., one has