Controllability of Systems on Solvable Lie Groups: The Generic Case

Abstract

A Lie group G with Lie algebra \frak g is called SID-controllable if there exist A,B \in {\frak g} such that the (Single Input with Drift) control system \dot{g} = g(A + uB), u \in {\Bbb R}, is controllable. This is equivalent to saying that the semigroup S(A, \pm B) generated by \exp({\Bbb R}^{+} A) \cup \exp({\Bbb R} B) is all of G. This definition is due to Sachkov who also classified SID-controllable solvable Lie algebras, cf. [7]-[9], [11]. It turns out that SID-controllability is actually a property of the Lie algebra (rather than of a control system): if a solvable \frak g is SID-controllable, then a generic SID-system will be controllable. In this paper we generalize this result to systems with multiple inputs and drifts: G is I_{n}D_{m}-controllable if there exist “inputs” B_{1} \dots B_{n} and “drifts” A_{1] \dots A_{m} such that S(A_{1}, \dots, A_{m}; \pm B_{1}, \dots, \pm B_{n}) = G. A Lie algebra is called I_{n}D_{m}-controllable if the corresponding simply connected group has this property. We will show that every solvable Lie algebra has a generic controllability rank r \in {\Bbb N} and a generic controllability type (i,d) \in {\Bbb N} \times {\Bbb N}_{0} such that: (GCR) \frak g is not I_{r-1}-controllable, \frak g is I_{r}-controllable, and the latter is generic. In particular, \frak g is I_{n}D_{m}-controllable if n \geq r or n + m \gt r; (GCT) \frak g is I_{i}D_{d}-controllable, a generic I_{i}D_{d}-system is controllable, i + d = r, and i is minimal. Determination of these invariants is one of our goals. Our major tools will be reduction arguments which are of independent interest: we show that every solvable Lie algebra \frak g has a maximal ideal \frak i which is completely irrelevant for all controllability questions. Passing to the factoralgebra {\frak g} / {\frak i} and analyzing its structure is the key step in solving our problem.