Theory of Chattering Control

1. Front Matter

   Pages i-xv

   PDF

2. Introduction

   • Michail I. Zelikin, Vladimir Borisov

   Pages 1-18

3. Fuller’s Problem

   • Michail I. Zelikin, Vladimir Borisov

   Pages 19-37

4. Second Order Singular Extremals and Chattering

   • Michail I. Zelikin, Vladimir Borisov

   Pages 38-84

5. The Ubiquity of Fuller’s Phenomenon

   • Michail I. Zelikin, Vladimir Borisov

   Pages 85-104

6. Higher Order Singular Extremals

   • Michail I. Zelikin, Vladimir Borisov

   Pages 105-166

7. Applications

   • Michail I. Zelikin, Vladimir Borisov

   Pages 167-215

8. Multidimensional Control with Chattering

   • Michail I. Zelikin, Vladimir Borisov

   Pages 217-233

9. Epilogue

   • Michail I. Zelikin, Vladimir Borisov

   Pages 235-235

10. Back Matter

    Pages 236-244

    PDF

The common experience in solving control problems shows that optimal control as a function of time proves to be piecewise analytic, having a finite number of jumps (called switches) on any finite-time interval. Meanwhile there exists an old example proposed by A.T. Fuller [1961) in which optimal control has an infinite number of switches on a finite-time interval. This phenomenon is called chattering. It has become increasingly clear that chattering is widespread. This book is devoted to its exploration. Chattering obstructs the direct use of Pontryagin's maximum principle because of the lack of a nonzero-length interval with a continuous control function. That is why the common experience appears misleading. It is the hidden symmetry of Fuller's problem that allows the explicit solution. Namely, there exists a one-parameter group which respects the optimal trajectories of the problem. When published in 1961, Fuller's example incited curiosity, but it was considered only "interesting" and soon was forgotten. The second wave of attention to chattering was raised about 12 years later when several other examples with optimal chattering trajectories were 1 found. All these examples were two-dimensional with the one-parameter group of symmetries.

• Finite
• Lagrangian manifolds
• chattering control
• function
• optimal control
• theory of optimal control

"...a rich source of information."
— Mathematical Reviews

"This book is an excellent introduction to and survey of chattering optimal controls."
— SIAM Review

"The book is an interesting and modern study of some problems in control theory in the language of differential geometry."
— Zentralblatt Math

• Department of Mathematics, Moscow State University, Moscow, Russia

  Michail I. Zelikin

• Department of Mathematics, Moscow Technological Institute, Moscow, Russia

  Vladimir Borisov

Policies and ethics