Introduction Bressan SINGULARITIES OF STABILIZING FEEDBACKS

Rend. Sem. Mat. Univ. Pol. Torino Vol. 56, 4 1998

A. Bressan SINGULARITIES OF STABILIZING FEEDBACKS

1. Introduction

This paper is concerned with the stabilization problem for a control system of the form ˙x = f x, u, u ∈ K , 1 assuming that the set of control values K ⊂ m is compact and that the map f : n × m 7→ n is smooth. It is well known [6] that, even if every initial state ¯x ∈ n can be steered to the origin by an open-loop control u = u ¯x t , there may not exist a continuous feedback control u = Ux which locally stabilizes the system 1. One is thus forced to look for a stabilizing feedback within a class of discontinuous functions. However, this leads to a theoretical difficulty, because, when the function U is discontinuous, the differential equation ˙x = f x, Ux 2 may not have any Carath´eodory solution. To cope with this problem, two approaches are possi- ble. I On one hand, one may choose to work with completely arbitrary feedback controls U . In this case, to make sense of the evolution equation 2, one must introduce a suitable definition of “generalized solution” for discontinuous O.D.E. For such solutions, a general existence theorem should be available. II On the other hand, one may try to solve the stabilization problem within a particular class of feedback controls U whose discontinuities are sufficiently tame. In this case, it will suffice to consider solutions of 2 in the usual Carath´eodory sense. The first approach is more in the spirit of [7], while the second was taken in [1]. In the present note we will briefly survey various definitions of generalized solutions found in the liter- ature [2, 11, 12, 13, 14], discussing their possible application to problems of feedback stabiliza- tion. In the last sections, we will consider particular classes of discontinuous vector fields which always admit Carath´eodory solutions [3, 5, 16], and outline some research directions related to the second approach. In the following,  and ∂ denote the closure and the boundary of a set , while B ε is the open ball centered at the origin with radius ε. To fix the ideas, two model problems will be considered. Asymptotic Stabilization AS. Construct a feedback u = Ux, defined on n \ {0}, such that every trajectory of 2 either tends to the origin as t → ∞ or else reaches the origin in finite time. Suboptimal Controllability SOC. Consider the minimum time function T ¯x . = min {t : there exists a trajectory of 1 with x0 = ¯x, xt = 0} . 3 87 88 A. Bressan Call Rτ . = {x : T x ≤ τ} the set of points that can be steered to the origin within time τ . For a given ε 0, we want to construct a feedback u = Ux, defined on a neighborhood V of Rτ , with the following property. For every ¯x ∈ V , every trajectory of 2 starting at ¯x reaches a point inside B ε within time T ¯x + ε. Notice that we are not concerned here with time optimal feedbacks, but only with subopti- mal ones. Indeed, already for systems on 2 , an accurate description of all generic singularities of a time optimal feedback involves the classification of a large number of singular points [4, 15]. In higher dimensions, an ever growing number of different singularities can arise, and time op- timal feedbacks may exhibit pathological behaviors. A complete classification thus appears to be an enormous task, if at all possible. By working with suboptimal feedbacks, we expect that such bad behaviors can be avoided. One can thus hope to construct suboptimal feedback controls having a much smaller set of singularities.

2. Nonexistence of continous stabilizing feedbacks