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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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