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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

##### MSC:
 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