Theory of Chattering Control by M. Zelikin and V. Borisov
Book Details :
LanguageEnglish
Pages256
FormatPDF
Size9.1 MB


Theory of Chattering Control by M. Zelikin and V. Borisov



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The Author of Theory of Chattering Control PDF Book

M. Zelikin and  V. Borisov are the editors of Theory of Chattering Control PDF Book

Main Contents of Theory of Chattering Control PDF


  • Introduction
  • Fuller's Problem 
  • Second-Order Singular Extremals and Chattering 
  • The Ubiquity of Fuller's Phenomenon 
  • Higher-Order Singular Extremals
  • Applications 
  • Multidimensional Control and Chattering Modes

Preface to Theory of Chattering Control 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.

Theory of Chattering Control PDF Free download.

This phenomenon is called chattering. It has become increasingly clear that chattering is widespread. This Theory of Chattering Control by M. Zelikin and  V. Borisov 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. (Theory of Chattering Control by M. Zelikin and  V. Borisov)

Theory of Chattering Control PDF Free download.

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 found. (Theory of Chattering Control by M. Zelikin and  V. Borisov)

1 All these examples were two-dimensional with the one-parameter group of symmetries. In recent years, interest in Fuller's phenomenon has increased due to intense efforts to understand the substance of the concept of regular synthesis.

A regular synthesis is a definite system of axioms which pretends to be "typical" for the set of all optimal trajectories with a fixed terminal condition.

Theory of Chattering Control PDF Free download.

This system of axioms was proposed by V.G. Boltyanskii [1966) whose intention was to prove sufficient conditions for optimality. (Theory of Chattering Control by M. Zelikin and  V. Borisov)

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However, later on, typical examples of optimal synthesis were published which violate one or another of Boltyanskii's axioms.

2 Among these axioms the following one plays an important part in the proof of optimality: on any finite-time interval of a trajectory, there exists a finite number of switches.

The question of the ubiquity of Fuller's phenomenon is therefore very much to the point. (Theory of Chattering Control by M. Zelikin and  V. Borisov)


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