New G-Optimality Criteria for Multi-Dimensional Control Problem with Applications in Artificial Neural System

In this article, we aim to study a class of multi-dimensional control problems with first-order PDE constraints and obtain its optimality by deriving generalized necessary and sufficient optimality conditions. For this, we propose the concept of a generalized convex multi-dimensional integral functional as so-called G-convex functional. Using the proposed definition, we extend the concept of the G-KT point, state and prove the G-necessary optimality conditions for the problem mentioned earlier. After that, by imposing the hypothesis of G-convexity over-involved functionals, we derive the generalized sufficiency criteria of earlier established G-necessary optimality conditions, named G-sufficient optimality conditions. That ensures the optimality of a feasible solution to the problem under consideration. In addition, some applications are also constructed to demonstrate the utilization of primary results for the controlled behavior of an artificial neural system. Also, we provide an algorithm that illustrates the steps in addressing the control problem explored in this study.