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On Regularity Properties of Extremal Controls A. A. Agrachev

Summary: On Regularity Properties of Extremal Controls
A. A. Agrachev 
We prove some regularity properties of the optimal controls for the
smooth bracket generating systems with scalar control parameters, and
show that the Cantor sets cannot be the sets of switching points.
Thanks to papers by H.Sussmann we know some regularity properties
of optimal controls for general real-analytic systems, see [2],[3]. The same
author demonstrated in [2] that optimal controls for general C 1 -systems
do not possess any regularity properties. In this note, we show that the
situation is not so hopeless for the bracket generating systems and establish
a curious property of the sets of switching points, which is new for real-
analitic systems too.
Consider a control system
x = f(x) + ug(x); x 2 M; juj  1;
where M is a C 1 -manifold, f; g are C 1 -vector elds on M .
Let Lieff; gg be a Lie sub-algebra of the vector elds generated by f; g;
and L 0 (f; g) be an ideal in Lieff; gg generated by g. Suppose that
fv(x) : v 2 L 0 (f; g)g = T x M; 8x 2 M: (1)


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics