Here is a very classical example of a chattering phenomenon [1]:
\(
Min \int_0^T x^2(t) dt; \quad \ddot x(t) = u(t) \in [-1,1].
\)
The solution is, for large enough T, bang-bang (i.e., with values alternately \(\pm 1\)), the switching times geometrically converging to a value \(\tau>0\), and then the (trivial) singular arc \(x=0\) and \(u=0\). 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 \(T=3.5\), \(x(0)=0, \dot x(0)=1\), \(x(T)=\dot x(T)=0\) and \(u(t) \in [-10^{-2},10^{-2}]\).
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