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

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    We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant nn-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 Ai∈RnΓ—n,nβ‰₯1,Nβ‰₯2,miβ‰₯1A_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 A1,...,AN{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 coA1,...,AN\mathrm{co}{A_1,...,A_N} in RnΓ—n\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 Οƒ(t)\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)∈RmΟƒ(t)u(t)\in\mathbb{R}^{m_{\sigma(t)}} with |u(t)|\le\bbdelta.Comment: 32 pages; accepted by SIAM J Control & Opti
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