Regularization of chattering phenomena via bounded variation control

TL;DR

This paper introduces a regularization method for control problems exhibiting chattering phenomena, using total variation penalization to approximate optimal controls and ensure convergence to true solutions.

Contribution

It proposes a novel regularization approach that penalizes total variation to handle chattering in optimal control, with proven convergence and quantification of quasi-optimality.

Findings

The regularization produces quasi-optimal controls that approximate true solutions.

Convergence of quasi-optimal solutions to the true optimal control is established.

Under certain conditions, the convergence rate of the costs is quantified.

Abstract

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.