Max-Plus Methods for Nonlinear Control and Estimation

1. Front Matter

   Pages i-xiv

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2. Introduction

   Pages 1-10

3. Max-Plus Analysis

   Pages 11-30

4. Dynamic Programming and Viscosity Solutions

   Pages 31-56

5. Max-Plus Eigenvector Method for the Infinite Time-Horizon Problem

   Pages 57-96

6. Max-Plus Eigenvector Method Error Analysis

   Pages 97-127

7. A Semigroup Construction Method

   Pages 129-141

8. Curse-of-Dimensionality-Free Method

   Pages 143-181

9. Finite Time-Horizon Application: Nonlinear Filtering

   Pages 183-195

10. Mixed L_∞/L₂ Criteria

    Pages 197-215

11. Back Matter

    Pages 217-245

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The control and estimation of continuous-time/continuous-space nonlinear systems continues to be a challenging problem, and this is one of the c- tral foci of this book. A common approach is to use dynamic programming; this typically leads to solution of the control or estimation problem via the solution of a corresponding Hamilton–Jacobi (HJ) partial di?erential eq- tion (PDE). This approach has the advantage of producing the “optimal” control. (The term “optimal” has a somewhat more complex meaning in the class of H problems. However, we will freely use the term for such controllers ? throughout, and this meaning will be made more precise when it is not ob- ous. )Thus,insolvingthecontrol/estimationproblem,wewillbesolvingsome nonlinear HJ PDEs. One might note that a second focus of the book is the solution of a class of HJ PDEs whose viscosity solutions have interpretations as value functions of associated control problems. Note that we will brie?y discuss the notion of viscosity solution of a nonlinear HJ PDE, and indicate that this solution has the property that it is the correct weak solution of the PDE. By correct weak solution in this context, we mean that it is the solution that is the value function of the associated control (or estimation) problem. The viscosity solution is also the correct weak solution in many PDE classes not considered here, and references to further literature on this subject will be given.

• Analysis
• GUM
• algorithm
• algorithms
• construction
• linearity
• numerical methods
• operator
• programming
• partial differential equations

"As is well known, control and estimation of continuous time and continuous space nonlinear systems are difficult to handle. One frequently uses dynamic programming methods to treat such problems. Central to this approach are solutions of the associated Hamilton-Jacobi-Bellman (HJB) equations. The book under review is motivated by solutions of such problems.... The entire book is divided into nine chapters and an appendix. Chapter 1 provides an introduction, and states why the problem to be studied and methods are needed. Concepts of max-plus methods are also provided.... All in all, this is a nicely written book that introduces important topics."   —Mathematical Reviews

"This outstanding book discusses new theoretical approaches to the study of the control and estimation of continuous-time/continuous-space nonlinear systems to be a challenging problem...This book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of the max-plus algebra and some basics of functional analysis over the max-plus algebra...The volume is primarily addressed to applied mathematicians...However, the book will also be useful for scientists from the application areas, in particular, applied scientists from engineering and physics." (Jan Lovisek, Zentralblatt MATH, Vol. 1103 (5), 2007)

• Department of Mechanical and Aerospace Engineering and Department of Mathematics, University of California at San Diego, La Jolla, USA

  William M. McEneaney

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