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.
Highlights
In many fields, the optimal control problems play a significant role in life
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
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
Summary
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]. 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 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
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