Criteria of stabilizability for switching-control systems with solvable linear approximations

We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant n-dimensional subsystems ẋ = Aix+ Bi(x)u (x ∈ R , t ∈ R+ and u ∈ R mi ), where i ∈ {1, . . . , N} and a switching signal σ(·) : R+ → {1, . . . , N} which orchestrates switching between these subsystems above, where Ai ∈ R, n ≥ 1, N ≥ 2, mi ≥ 1, and where Bi(·) : R → Ri satisfies the condition ‖Bi(x)‖ ≤ β‖x‖ ∀x ∈ R. We show that, if {A1, . . . , AN} generates a solvable Lie algebra over the field C of complex numbers and there exists an element A in the convex hull co{A1, . . . , AN} in R such that the affine system ẋ = Ax is exponentially stable, then there is a constant δ > 0 for which one can design “sufficiently many” piecewise-constant switching signals σ(t) so that the switching-control systems ẋ(t) = Aσ(t)x(t) + Bσ(t)(x(t))u(t), x(0) ∈ R n and t ∈ R+ are globally exponentially stable, for any measurable external inputs u(t) ∈ Rσ(t) with ‖u(t)‖ ≤ δ.