Optimal control theory of fourth order differential inclusions

Abstract

Optimal control theory is a field of study in mathematics and engineering that focuses on finding the most effective method of controlling a system in order to accomplish a desired result. The process involves mathematically modeling the system, devising a control strategy that optimizes a specific objective function, and implementing the control method. Optimal control theory is applied in various disciplines such as engineering, economics, biology. Optimization challenges are prevalent in several fields of research and engineering. We consistently strive for optimal solutions, maximizing resource utilization and minimizing effort to get desired results. In the field of classical mechanics, one example of a problem we may encounter is finding the path that minimizes the amount of time it takes to go. Nevertheless, traditional differential equations sometimes encounter difficulties in accurately representing the intricacy of real-world systems. Fourth-order differential inclusions are introduced as a strong tool to address optimization issues that include higher-order dynamics. The fundamental concept behind the use of fourth-order (DFIs) for optimization is to identify the state trajectory (expressed by a function) that minimizes a certain cost functional. We use the notion of locally adjoint mappings (LAMs) to deduce optimality criteria. These mappings establish a connection between changes in the system's condition and fluctuations in the cost functional. Mathematicians may determine the necessary and sufficient conditions for a trajectory to be optimum by examining these links. The use of fourth-order (DFIs) in optimization is not limited to mechanics. They are used in fields such as optimal control theory, where engineers develop control systems for dynamic processes. In this context, the term "inclusion" refers to the description of how a system behaves when subjected to different control inputs. The objective of the optimization issue is to find the control strategy that minimizes a certain cost, such as fuel consumption.