Geometric Optimal Control

1. Front Matter

   Pages i-xix

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2. The Calculus of Variations: A Historical Perspective

   • Heinz Schättler, Urszula Ledzewicz

   Pages 1-81

3. The Pontryagin Maximum Principle: From Necessary Conditions to the Construction of an Optimal Solution

   • Heinz Schättler, Urszula Ledzewicz

   Pages 83-194

4. Reachable Sets of Linear Time-Invariant Systems: From Convex Sets to the Bang-Bang Theorem

   • Heinz Schättler, Urszula Ledzewicz

   Pages 195-230

5. The High-Order Maximum Principle: From Approximations of Reachable Sets to High-Order Necessary Conditions for Optimality

   • Heinz Schättler, Urszula Ledzewicz

   Pages 231-321

6. The Method of Characteristics: A Geometric Approach to Sufficient Conditions for a Local Minimum

   • Heinz Schättler, Urszula Ledzewicz

   Pages 323-423

7. Synthesis of Optimal Controlled Trajectories: From Local to Global Solutions

   • Heinz Schättler, Urszula Ledzewicz

   Pages 425-484

8. Control-Affine Systems in Low Dimensions: From Small-Time Reachable Sets to Time-Optimal Syntheses

   • Heinz Schättler, Urszula Ledzewicz

   Pages 485-568

9. Back Matter

   Pages 569-640

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This book gives a comprehensive treatment of the fundamental necessary and sufficient conditions for optimality for finite-dimensional, deterministic, optimal control problems. The emphasis is on the geometric aspects of the theory and on illustrating how these methods can be used to solve optimal control problems. It provides tools and techniques that go well beyond standard procedures and can be used to obtain a full understanding of the global structure of solutions for the underlying problem. The text includes a large number and variety of fully worked out examples that range from the classical problem of minimum surfaces of revolution to cancer treatment for novel therapy approaches. All these examples, in one way or the other, illustrate the power of geometric techniques and methods. The versatile text contains material on different levels ranging from the introductory and elementary to the advanced. Parts of the text can be viewed as a comprehensive textbook for both advanced undergraduate and all level graduate courses on optimal control in both mathematics and engineering departments. The text moves smoothly from the more introductory topics to those parts that are in a monograph style were advanced topics are presented. While the presentation is mathematically rigorous, it is carried out in a tutorial style that makes the text accessible to a wide audience of researchers and students from various fields, including the mathematical sciences and engineering. Heinz Schättler is an Associate Professor at Washington University in St. Louis in the Department of Electrical and Systems Engineering, Urszula Ledzewicz is a Distinguished Research Professor at Southern Illinois University Edwardsville in the Department of Mathematics and Statistics.

• Lie bracket computations
• Pontryagin Maximum Principle
• calculus of variations
• geometric optimal control
• reachable sets
• ordinary differential equations

From the reviews:

“The monograph under review concerns finite-dimensional deterministic optimal control problems. … The main body of the book is divided into seven chapters. … The variety of fully solved examples that illustrate the theory makes this text a strong educational asset. The book is recommended as a comprehensive textbook for both advanced undergraduate and all levels of graduate courses on optimal control in mathematics and engineering.” (Ovidiu Cârjă, zbMATH, Vol. 1276, 2014)

“The book presents a comprehensive treatment of both necessary and sufficient conditions for optimal control using geometric approach … . The book is of interest to senior and graduate students in engineering and mathematics, and scientists and engineers working in academic and industrial organizations. … The book is a valuable addition to some of the recent books on this ever-green field of Optimal Control … .” (D. Subbaram Naidu, Amazon.com, September, 2013)

“Grown out of well-tested lecture notes, large parts of this volume are suitable as a comprehensive textbook at an advanced undergraduate or at the graduate level, either in mathematics or in engineering … . this most readable text provides a rich and versatile resource which is suitable as a textbook in various settings, is a valuable reference for theory, and which provides a very large collection of model examples that are analyzed completely using state-of-the-art methods.” (Matthias Kawski, Mathematical Reviews, February, 2013)

“Schättler (electrical and systems engineering, Washington Univ.) and Ledzewicz (mathematics and statistics, Southern Illinois Univ.) use a geometric approach to present the theory of optimal control. … authors have developed a general approach that can be applied to a wide variety of control problems. … This book may be of interest to graduate students and researchers working in this area. Summing Up: Recommended.Graduate students and researchers/faculty.” (B. Borchers, Choice, Vol. 50 (5), January, 2013)

• , Electrical and Systems Engineering, Washington University, Saint Louis, USA

  Heinz Schättler

• , Mathematics and Statistics, Southern Illinois University Edwardsvill, Edwardsville, USA

  Urszula Ledzewicz

Urszula Ledzewicz is a Distinguished Research Professor in the Department of Mathematics and Statistics at Southern Illinois University. Heinz Schaettler is an Associate Professor at Washington University.

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