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Stabilization of uncertain systems via linear control. (English) Zbl 0554.93054
The paper gives two elementary results on the feedback stability of time- varying systems defined by \(\dot x=A(q)x+B(q)u\), where q is a time- varying parameter. (i) The stability of the system with a linear feedback \(u=Kx\) can be decided by a quadratic Lyapunov function if and only if the same is true of the system \[ \dot y=\left[ \begin{matrix} A\\ 0\end{matrix} \begin{matrix} B\\ 0\end{matrix} \right]y+\left[ \begin{matrix} 0\\ I\end{matrix} \right]u; \] (ii) If with \(u=p(x)\), where \(p(0)=0\) and p is continuously differentiable, the feedback system has a quadratic Lyapunov function, then with \(u=(\partial p/\partial x)_{x=0}^ x \)it also has a quadratic Lyapunov function.
Reviewer: S.Mossaheb

93D15 Stabilization of systems by feedback
93C99 Model systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C05 Linear systems in control theory
93D20 Asymptotic stability in control theory
34D20 Stability of solutions to ordinary differential equations
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