Project Abstract

Optimal control theory is a powerful mathematical framework that has been widely applied to a diverse range of dynamical systems, from engineering and economics to biology and social sciences. This project aims to explore the fundamental principles of optimal control theory and its practical applications in addressing complex challenges within various domains. At the core of this project is the recognition that many real-world systems, whether they are physical, biological, or socioeconomic, can be modeled as dynamical systems governed by a set of governing equations. These systems often face the challenge of optimizing their performance or behavior under certain constraints, such as resource limitations, environmental factors, or desired outcomes. Optimal control theory provides a systematic approach to identify the optimal control strategies that can steer these dynamical systems towards their desired states while minimizing or maximizing specific objectives. The project begins by delving into the theoretical foundations of optimal control theory, examining the key concepts and mathematical tools used to analyze and solve optimal control problems. This includes understanding the necessary conditions for optimality, such as the principle of optimality and the Pontryagin's maximum principle, as well as exploring the various numerical techniques employed to find the optimal control policies, such as dynamic programming, the Hamiltonian approach, and direct optimization methods. With a solid grasp of the theoretical underpinnings, the project then explores the diverse applications of optimal control theory in dynamical systems. One area of focus is the application of optimal control in engineering systems, where it can be used to optimize the design and operation of various devices and processes, such as robotic systems, aerospace vehicles, and energy systems. The project will investigate how optimal control can be used to minimize energy consumption, improve efficiency, and enhance the performance of these engineered systems. Another key application domain is in the field of economics and finance, where optimal control theory can be employed to model and optimize the dynamics of financial markets, investment portfolios, and macroeconomic policies. The project will examine how optimal control can be used to guide decision-making in these complex systems, addressing challenges such as portfolio optimization, resource allocation, and policy implementation. The project will also explore the application of optimal control theory in biological and social systems, where it can be used to understand and manipulate the dynamics of complex phenomena, such as ecological systems, epidemiological models, and social networks. By leveraging the power of optimal control, the project will demonstrate how these systems can be optimized to achieve desired outcomes, whether it's maximizing the preservation of endangered species, minimizing the spread of infectious diseases, or enhancing the effectiveness of social interventions. Throughout the project, the emphasis will be on developing a deeper understanding of the theoretical foundations of optimal control theory and its wide-ranging applications in dynamical systems. By exploring case studies, implementing numerical simulations, and analyzing real-world data, the project aims to provide valuable insights into the potential of optimal control theory to address complex challenges and drive innovation across various domains.

Project Overview