We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant n-dimensional subsystems <(x)over dot> = A(i)x + B-i(x) u (x is an element of R-n, t is an element of R+, and u is an element of R-mi), where i is an element of {1, ... , N} and a switching signal sigma(.): R+ -> {1, ... , N} which orchestrates switching between these subsystems above, where A(i) is an element of R-nxn, n >= 1, N >= 2, m(i) >= 1, and where B-i(.): R-n -> R-nxmi satisfies the condition parallel to B-i(x)parallel to = <= beta parallel to x parallel to for all x is an element of R-n. We show that, if {A(1), ..., A(N)} generates a solvable Lie algebra over the field C of complex numbers and there exists an element A in the convex hull co{A(1), ..., A(N)} in R-nxn such that the affine system <(x)over dot> = Ax is exponentially stable, then there is a constant delta > 0 for which one can design "sufficiently many" piecewise-constant switching signals sigma(t) so that the switching-control systems <(x)over dot>(t) = A(sigma(t))x(t) + B-sigma(t)(x(t))u(t), x(0) is an element of R-n and t is an element of R+, are globally exponentially stable for any measurable external inputs u(t) is an element of R-m sigma(t) with parallel to u(t)parallel to <= delta.