Mathematics

Let G be a real connected Lie group and L its Lie algebra (the set of all right-invariant vector fields on G). For right-invariant control systems on G (1) dx/dt = A(x) + u B(x), x \in G, u \in R, where A, B \in L, we are interested in the (global) cotrollability property: when any pair of points in G can be connected by a trajectory of system (1)? This is equivalent to the following property: the subsemigroup of G generated by the set exp(A + R B) coincides with the Lie group G. It turns out that for solvable simply connected Lie groups G the property of controllability of systems (1) depends primarily not on vector fields A, B, but on the Lie group G, i.e., on the Lie algebra L. We call a Lie algebra L controllable if it contains elements A, B such that the corresponding system (1) is controllable. The talk will be devoted to the following questions: Description of controllable solvable Lie algebras (a recent work of D.Mittenhuber), A complete list of low-dimensional controllable Lie algebras (up to dim 6).