ON SUBSEMIGROUPS OF SEMISIMPLE LIE-GROUPS

In this paper, we consider a real connected semisimple Lie group G and ask whether or not a subset S of G generates G as a semigroup. We dea l with the special case where S is infinitesimally generated, i.e. S = {expt X/X is an element of Sigma, t is an element of R(+)} for some s ubset Sigma of L, the Lie algebra of G. In the case where Sigma is a s ymmetric subset of L (i.e. Sigma = -Sigma), this is equivalent to the fact that S generates G as a group. It is also known, by an old result of Kuranishi, that S generates G as soon as Sigma is a symmetric subs et of L of the form {+/-X, +/-Y} for generic pairs (X, Y) in L x L. In the case where Sigma = {X, +/-Y}, almost nothing is known, except in the compact case where Kuranishi's result still holds. We deal with th e intermediate case where Sigma = {X, +/-Y}. This case is specially im portant in control theory, where such sets Sigma apear naturally throu gh control systems of the ''classical control-affine form x = X (x) uY(x)''. A theorem is proven, which is the final form of several resul ts in a series of papers of all of the authors. This theorem improves on all these results.