Here is a very classical example of a chattering phenomenon [1]:

The solution is, for large enough *T*, bang-bang (i.e., with values alternately ), the switching times geometrically converging to a value , and then the (trivial) *singular arc* and . These switches are not easy to reproduce numerically. We display in figure 1 the control, with a zoom on the entry point of the singular arc.

*Numerical simulations: problem fuller
Discretization: Gauss II with 1000 steps.
We take there* , ,

*and*.

Figure 1: Fuller problem: chattering control (with zoom); *x* and *v*.

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**References**

A.T. Fuller. Study of an optimum non-linear control system. *J. of Electronics and Control*, 15:63-71, 1963