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Homogeneous feedback controls for homogeneous systems. (English) Zbl 0877.93088
Summary: Let $$X(x)=\sum ^{n}_{j=2}a_{j}(x)\partial /\partial x_{j}$$ be a real analytic, n-dimensional vector field with $$X(0)=0$$ which is homogeneous of degree $$m\leq 0$$ with respect to a dilation $$\delta ^{r}_{\varepsilon}x= (\varepsilon ^{r_{1}}x_{1},\ldots,\varepsilon ^{r_{n}}x_{n})$$ and $$Y=\partial /\partial x_{1}$$. We show that if the control system $$\dot x=X(x)+uY$$ admits a continuous, asymptotically stabilizing, feedback control $$u(x)$$, then a one-dimensional dynamic extension of the system admits a homogeneous, asymptotically stabilizing (dynamic) feedback control $$\hat u(\tau,x)$$ such that the vector field $\hat W(\tau,x)=\alpha \tau^{1-m}\partial /\partial\tau+ \hat u(\tau,x)\partial /\partial x_{1}+\sum ^{n}_{j=2}a_{j}(x)\partial /\partial x_{j}$ describing the extended, controlled, system is homogeneous of degree m with respect to the extended dilation $\hat\delta^{r}_{\varepsilon}(\tau,x)= (\varepsilon\tau,\varepsilon^{r_{1}}x_{1},\ldots, \varepsilon^{r_{n}}x_{n}).$

MSC:
 93D15 Stabilization of systems by feedback
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References:
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