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Direct Methods in Control Problems

Peter Falb
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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Control theory aims to design and develop mathematical models for controlling systems that arise in a variety of processes, including engineering and machine control. Its goal is to provide methods and algorithms, preferably stable and optimal, for achieving a path from a starting point to a target state.

In this book the author notes that the field has focused in recent decades on what he calls indirect methods. His goal here is to reconsider direct methods and develop a unified approach that could build on past work and stimulate future research. His motivation is that indirect methods have sometimes proven to be inadequate. He also argues that the control theory literature has become fragmented, and finds value in identifying a more coherent unified theory.

The author gives a broad description of a direct method as one that determines the infimum of an appropriately selected functional by constructing a minimizing family, and showing that family has a limit that leads to a determination of the infimum of the functional. He argues that a key aspect of the direct approach is this construction of convergent minimizing sequences or families. He offers two general approaches that involve integration and approximation methods. In contrast, he characterizes indirect methods as those that reduce the control problem to one that can be described by a differential equation or systems of differential equations. Among the indirect methods he cites are dynamic programming, the method of viscosity solutions, stochastic control and filtering, and the classical variational approach.

In his approach, the author goes back to fundamentals of the minimization of functionals associated with dynamical control systems on Hilbert and Banach spaces, and necessary conditions for optimization of those functionals. He begins with notions from more standard functional analysis, and from there the level of mathematical sophistication increases steadily.

The author develops a new treatment of a general existence theory for optimal controls with the intent of proving continuity of the control, one that he calls “in principle, elementary”, but new in execution and one he believes is applicable more generally.

This is clearly a book aimed at specialists in control theory. It is more a monograph than a textbook with only the occasional exercise for the reader. The author makes it clear that he wants to describe the general principles and ideas of direct methods, but not any details of how to treat any specific control problems. As such, the examples he provides tend to be very general. 

Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from network modeling and speech recognition to structural reinforcement with ceramic fibers. He did his PhD work in dynamical systems and celestial mechanics.