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Practical Methods for Optimal Control Using Nonlinear Programming

John T. Betts
Publication Date: 
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[Reviewed by
Brian Borchers
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In optimal control problems, we have a system governed by ordinary or partial differential equations with an input that is under our control.  The goal is to optimize the input function to produce a desired output or to minimize some functional of the output.  A classical approach involves the Euler-Lagrange conditions from the calculus of variations.  The optimality condition for a stationary point of a functional is expressed as a differential equation.  In practice, these equations must often be discretized and solved by numerical methods.  This is referred to as the "optimize then discretize" approach.
This book focuses on the alternative "discretize then optimize" approach in which the ordinary or partial differential equation describing the system and the control inputs are first discretized.  Then, nonlinear programming methods are used to optimize the output of the system subject to constraints that ensure that differential equations are satisfied.  The optimization problems that arise can be extremely large because of the large number of state variables for the discretized system of differential equations, but the Jacobian and Hessian matrices that arise are typically sparse. Thus specialized methods for large and sparse problems are required.
The book begins with two chapters that introduce optimization methods including sequential quadratic programming and then discuss how these methods can be applied to sparse large-scale problems.  In the following chapters, the author introduces the discretize then optimize approach and shows how it can be applied to optimal control problems, least-squares parameter estimation problems, and problems involving delay differential equations.  The last half of the book consists of a large collection of worked examples and test problems.  These problems were solved using the author's proprietary software package, SOS.
This book will be of interest to graduate students and researchers who are using the discretize then optimize approach to solve differential equation constrained optimization problems. 

Brian Borchers is a professor of mathematics at New Mexico Tech and the editor of MAA Reviews.