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 \dot{x}=A_ix+B_i(x)u\quad (x\in\mathbb{R}^n, t\in\mathbb{R}_+ \textrm{and} u\in\mathbb{R}^{m_i}),\qquad \textrm{where} i\in{1,...,N} and a switching signal \sigma(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N} which orchestrates switching between these subsystems above, where $A_i\in\mathbb{R}^{n\times n}, n\ge1, N\ge2, m_i\ge1$, and where B_i(\bcdot)\colon\mathbb{R}^n\rightarrow\mathbb{R}^{n\times m_i} satisfies the condition \|B_i(x)\|\le\bbbeta\|x\|\;\forall x\in\mathbb{R}^n. We show that, if ${A_1,...,A_N}$ generates a solvable Lie algebra over the field \mathbbm{C} of complex numbers and there exists an element \bbA in the convex hull $\mathrm{co}{A_1,...,A_N}$ in $\mathbb{R}^{n\times n}$ such that the affine system \dot{x}=\bbA x is exponentially stable, then there is a constant \bbdelta>0 for which one can design "sufficiently many" piecewise-constant switching signals $\sigma(t)$ so that the switching-control systems \dot{x}(t)=A_{\sigma(t)}x(t)+B_{\sigma(t)}(x(t))u(t),\quad x(0)\in\mathbb{R}^n\textrm{and} t\in\mathbb{R}_+ are globally exponentially stable, for any measurable external inputs $u(t)\in\mathbb{R}^{m_{\sigma(t)}}$ with |u(t)|\le\bbdelta.Comment: 32 pages; accepted by SIAM J Control & Opti