Fuller problem

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

$Latex formula$
The solution is, for large enough T, bang-bang (i.e., with values alternately $Latex formula$), the switching times geometrically converging to a value $Latex formula$, and then the (trivial) singular arc $Latex formula$ and $Latex formula$. 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
$Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$.

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