Optimal Control of Dynamic Systems Driven by Vector Measures: Theory and Applications

This book is about optimal control theory for finite-dimensional systems governed by deterministic and stochastic differential equations driven by vector measures. It covers a broad class of controls, including regular, relaxed, and vector-measure-determined controls, and considers both fully and partially observed control problems. The theory is developed for dynamic systems with ordinary and stochastic differential equations and includes results on the existence of optimal controls and necessary conditions for optimality. Computational algorithms are developed based on the optimality conditions, with numerical results presented to demonstrate the applicability of the theoretical results. The book is of interest to researchers in optimal control, applied functional analysis, and vector measures in control theory.

Format: Hardback
Length: 320 pages
Publication date: 14 September 2021
Publisher: Springer Nature Switzerland AG

Introduction:
Optimal control theory is a fundamental branch of mathematics that deals with the design of control systems for dynamic systems. It is a multidisciplinary field that combines mathematics, physics, and engineering to solve complex problems in various fields, such as robotics, aerospace, and finance. In recent years, there has been a growing interest in optimal control theory for dynamic systems driven by vector measures. Vector measures are a generalization of classical measures, which are used to define the state space of a system. They allow for the description of systems with complex geometries and non-linear dynamics, which are common in many real-world applications.
Motivation:
The motivation behind optimal control theory for dynamic systems driven by vector measures is to improve the performance of these systems. By designing optimal control systems, we can achieve desired behaviors, such as stability, performance, and efficiency, while minimizing the cost of control. Vector measures also provide a powerful tool for analyzing and understanding the behavior of complex systems, as they allow for the definition of complex controls, such as feedback controls and adaptive controls.
Scope of the Book:
This book is devoted to the development of optimal control theory for finite dimensional systems governed by deterministic and stochastic differential equations driven by vector measures. It deals with a broad class of controls, including regular controls (vector-valued measurable functions), relaxed controls (measure-valued functions), and controls determined by vector measures, where both fully and partially observed control problems are considered.
Chapter 1:
In this chapter, we introduce the basic concepts of optimal control theory for dynamic systems driven by vector measures. We discuss the definition of vector measures, the state space of a system, and the Hamiltonian and Lagrangian formulations of optimal control problems. We also introduce the concept of vector measures, which are a generalization of classical measures, and discuss their properties and applications.
Chapter 2:
In this chapter, we develop the theory of optimal control for dynamic systems governed by ordinary differential equations. We discuss the existence of optimal controls, the necessary conditions for optimality, and the computational algorithms for solving optimal control problems. We also present numerical examples to demonstrate the applicability of the theoretical results developed in the chapter.
Chapter 3:
In this chapter, we develop the theory of optimal control for dynamic systems governed by stochastic differential equations. We discuss the existence of optimal controls, the necessary conditions for optimality, and the computational algorithms for solving optimal control problems. We also present numerical examples to demonstrate the applicability of the theoretical results developed in the chapter.
Chapter 4:
In this chapter, we develop the theory of optimal control for dynamic systems driven by vector measures. We discuss the existence of optimal controls, the necessary conditions for optimality, and the computational algorithms for solving optimal control problems. We also present numerical examples to demonstrate the applicability of the theoretical results developed in the chapter.
Chapter 5:
In this chapter, we discuss the applications of optimal control theory for dynamic systems driven by vector measures. We discuss the control of systems with complex geometries, non-linear dynamics, and multiple inputs and outputs. We also present numerical examples to demonstrate the practical benefits of optimal control theory in real-world applications.
Conclusion:
In conclusion, this book presents optimal control theory for dynamic systems driven by vector measures. It covers a broad range of topics, including the theory of optimal control for ordinary and stochastic differential equations, the existence of optimal controls, and the computational algorithms for solving optimal control problems. The book is intended for researchers in optimal control or applied functional analysis interested in applications of vector measures to control theory, stochastic systems driven by vector measures, and related topics.

Weight: 670g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030821388
Edition number: 1st ed. 2021