Here is a very classical example of a chattering phenomenon :
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.
A.T. Fuller. Study of an optimum non-linear control system. J. of Electronics and Control, 15:63-71, 1963