Quaternary Boundary Optimal Control Problem for Quaternary Nonlinear Elliptic System with Constraints

Background: Boundary optimal control problems governed by nonlinear elliptic systems are complex, involving equality and inequality constraints. Objective: This paper examines a quaternary boundary optimal control vector problem (QBOCVP) regulated by a quaternary nonlinear elliptic system (QNES) and subject to equality and inequality constraints (EINC). Methods: A weak formulation of the QBOCVP is developed, along with a mathematical representation of the quaternary adjoint equations (QAEs) associated with the QNES. Results: An existence theorem for a QBOCV addressing the constrained problem is established and rigorously proven under appropriate assumptions. The QAEs corresponding to the QNES are mathematically formulated. The Fréchet derivative (FD) of the cost function (CF) and the EINC is also derived. Furthermore, the necessary condition theorem (NCTH) and the sufficient condition theorem (SCTH) for optimality are presented and proved. Conclusions: This work provides a rigorous analysis of the QBOCVP with EINC controlled by QNES. It establishes the existence theorem and optimality conditions, providing a theoretical framework for addressing such constrained problems.