Maximum Principle and Dynamic Programming Viscosity Solution Approach : From Open-Loop to Closed-Loop / by Bing Sun, Bao-Zhu Guo, Zhen-Zhen Tao.

Format:

Book

Author/Creator:

Sun, Bing.

Contributor:

Guo, Bao-Zhu.

Tao, Zhen-Zhen.

Series:

Systems & Control: Foundations & Applications, 2324-9757

Language:

English

Subjects (All):

System theory.

Control theory.

Mathematical optimization.

Calculus of variations.

Mathematics--Data processing.

Mathematics.

Systems Theory, Control.

Calculus of Variations and Optimization.

Optimization.

Computational Mathematics and Numerical Analysis.

Local Subjects:

Systems Theory, Control.

Calculus of Variations and Optimization.

Optimization.

Computational Mathematics and Numerical Analysis.

Physical Description:

1 online resource (697 pages)

Edition:

1st ed. 2025.

Place of Publication:

Singapore : Springer Nature Singapore : Imprint: Birkhäuser, 2025.

Summary:

This book is concerned with optimal control problems of dynamical systems described by partial differential equations (PDEs). The content covers the theory and numerical algorithms, starting with open-loop control and ending with closed-loop control. It includes Pontryagin’s maximum principle and the Bellman dynamic programming principle based on the notion of viscosity solution. The Bellman dynamic programming method can produce the optimal control in feedback form, making it more appealing for online implementations and robustness. The determination of the optimal feedback control law is of fundamental importance in optimal control and can be argued as the Holy Grail of control theory. The book is organized into five chapters. Chapter 1 presents necessary mathematical knowledge. Chapters 2 and 3 (Part 1) focus on the open-loop control while Chapter 4 and 5 (Part 2) focus on the closed-loop control. In this monograph, we incorporate the notion of viscosity solution of PDE with dynamic programming approach. The dynamic programming viscosity solution (DPVS) approach is then used to investigate optimal control problems. In each problem, the optimal feedback law is synthesized and numerically demonstrated. The last chapter presents multiple algorithms for the DPVS approach, including an upwind finite-difference scheme with the convergence proof. It is worth noting that the dynamic systems considered are primarily of technical or biologic origin, which is a highlight of the book. This book is systematic and self-contained. It can serve the expert as a ready reference for control theory of infinite-dimensional systems. These chapters taken together would also make a one-semester course for graduate with first courses in PDE-constrained optimal control.

Contents:

- 1. Mathematical Preliminary

Part I: Open-Loop Control

2. Dubovitskii-Milyutin Approach

3. Spectral Method

Part II: Closed-Loop Control

4. Dynamic Programming Viscosity Solution Approach

5. Multiple Algorithms for DPVS Approach.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

981-9657-39-3

OCLC:

1526863893