  In control theory, the term chattering is used to refer to strong
oscillations of controls, such as an infinite number of switchings over a
compact interval of times. In this paper we focus on three typical occurences
of chattering: the Fuller phenomenon, referring to situations where an optimal
control switches an infinite number of times over a compact set; the Robbins
phenomenon, concerning optimal control problems with state constraints, meaning
that the optimal trajectory touches the boundary of the constraint set an
infinite number of times over a compact time interval; the Zeno phenomenon,
referring as well to an infinite number of switchings over a compact set, for
hybrid optimal control problems. From the practical point of view, when trying
to compute an optimal trajectory, for instance by means of a shooting method,
chattering may be a serious obstacle to convergence.
  In this paper we propose a general regularization procedure, by adding an
appropriate penalization of the total variation. This produces a quasi-optimal
control, and we prove that the family of quasi-optimal solutions converges to
the optimal solution of the initial problem as the penalization tends to zero.
Under additional assumptions, we also quantify the quasi-optimality property by
determining a speed of convergence of the costs.
