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This is a category with all control problems that exhibit chattering behavior, i.e., the optimal solution only exists in the limit of infinite switching.

If an optimal control problem has sensitivity-seeking or path-constrained arcs, a bang-bang solution may have to switch infinitely often in a finite time interval to approximate the optimal solution. This behavior is referred to as 'chattering' in the optimal control community [Zelikin1994]Address: Basel Boston Berlin
Author: Zelikin, M.I.; Borisov, V.F.
Publisher: Birkh\"auser
Title: Theory of chattering control with applications to astronautics, robotics, economics and engineering
Year: 1994
Link to Google Scholar
. The first example of an optimal control problem exhibiting chattering behavior was given by Fuller.

In the engineering community chattering behavior is called Zeno's phenomenon. This refers to the great ancient philosopher Zeno of Elea. Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a pupil of Parmenides. He is mostly known for his 40 paradoxes, among which the most famous are

  • The Dichotomy.

Motion is impossible since that which is in locomotion must arrive at the half-way stage before it arrives at the goal.

  • The Arrow.

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.

  • The Achilles.

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Zeno of Elea was the first to draw attention to the apparent interpretational problems occurring whenever an infinite number of events have to take place in a finite time interval.


[Zelikin1994]Zelikin, M.I.; Borisov, V.F. (1994): Theory of chattering control with applications to astronautics, robotics, economics and engineering. (%edition%). Birkh\"auser, Basel Boston Berlin, %pages%Link to Google Scholar