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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 $\stackrel{˙}{x}=A\left(q\right)x+B\left(q\right)u$, 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

$\stackrel{˙}{y}=\left[\begin{array}{c}A\\ 0\end{array}\begin{array}{c}B\\ 0\end{array}\right]y+\left[\begin{array}{c}0\\ I\end{array}\right]u;$

(ii) If with $u=p\left(x\right)$, where $p\left(0\right)=0$ and p is continuously differentiable, the feedback system has a quadratic Lyapunov function, then with $u={\left(\partial p/\partial x\right)}_{x=0}^{x}$it also has a quadratic Lyapunov function.

Reviewer: S.Mossaheb

MSC:
 93D15 Stabilization of systems by feedback 93C99 Control systems, guided systems 93D05 Lyapunov and other classical stabilities of control systems 93C05 Linear control systems 93D20 Asymptotic stability of control systems 34D20 Stability of ODE