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

Abstract

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.

Solvability for Optimal Classical Continuous Control Problem Controlling by Quaternary Hyperbolic Boundary Value Problem

“…𝑦 4 (𝑥, 0) = 𝑦 4 0 (𝑥),and 𝑦 4𝑡 (𝑥, 0) = 𝑦 4 1 (𝑥), in Ω (12) where (𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 ) ∈ 𝐿 2 (Q)=(𝐿 2 (Q)) 4 is a function given vector for each (𝑥 1 , 𝑥 2 ) ∈ Ω , 𝑢 ⃗ = (𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 ) ∈ 𝐿 2 (Q) is a given classical continuous control quaternary vector and the corresponding quaternary state vector solution is 𝑦 = (𝑦 1 , 𝑦 2 , 𝑦 3 , 𝑦 4 ) ∈ 𝐻 2 (Ω) = (𝐻 2 (Ω)) 4 .…”

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems

“…The classical continuous optimal control problem ) CCOCP) dominated by nonlinear parabolic or elliptic or hyperbolic PDEs are studied in [9][10][11] respectively (resp.). Later, the study of the CCOCPs dominated by the three types of nonlinear PDEs is generalized in [12][13][14] to deal with CCOCPs dominating by coupling NLPDEs of these types resp., and then these studies are generalized also to deal with CCOCPs dominated by triple and NLPDEs of the three types [15][16][17] .…”

The Classical Continuous Optimal Control for Quaternary parabolic boundary value problem

“…These problems are usually ruled either by ordinary differential equations or by partial differential equations (PDEs), in particular, optimal control problems are studied for systems that are ruling by PDEs of elliptic or parabolic or hyperbolic type are achived and considered ISSN: 0067-2904 in [5 -10]. On the other hand, the authors in [11][12][13] disscsed and achieved the optimal control problems for systems ruling by triple PDEs of the previous three types. In this work, we look at the optimal control problems that is formed by QLPBVPs.…”