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Stabilizability does not imply homogeneous stabilizability for controllable homogeneous systems. (English) Zbl 0879.93038

A control system \[ \dot x_i= f_i(x_1, \dots, x_n,u), \quad i=1, \dots,n \] is called homogeneous of degree \(\tau\) with respect to the dilation \(\delta^r_\varepsilon (x,u)= (\varepsilon^{r_1}\cdot x_1, \dots, \varepsilon^{r_n}\cdot x_n, \varepsilon^{r_{n+1}}\cdot u)\) if for some \(r_i>0\) and \(\tau \in (-\min \{r_j\},+ \infty)\) one has \(f_i(\varepsilon^{r_1}\cdot x_1, \dots, \varepsilon^{r_n}\cdot x_n, \varepsilon^{r_{n+1}}u) =\varepsilon^{\tau+r_i}\cdot f_i(x_1, \dots, x_n,u)\) for all \(i\). The system is called stabilizable by homogeneous feedback if there exists \(u(x)\in C^1(\mathbb{R}^n \backslash\{0\}, \mathbb{R}) \) such that \(u(\varepsilon^{r_1}\cdot x_1, \dots, \varepsilon^{r_n}\cdot x_n)= \varepsilon^{r_{n+1}} u(x_1, \dots, x_n)\) and the origin is a globally asymptotically stable equilibrium of the closed loop system \(dx/dt= f(x,u(x))\). The paper deals with stabilizability of the system \[ \begin{aligned} \dot x & =x+u \\ \dot y & =3y+ xu^2 \end{aligned} \] and of its augmentation \[ \begin{aligned} \dot x & = x+z \\ \dot y & =3y+ xz^2 \\ \dot z & =u \end{aligned} \] which are not stabilizable by homogeneous feedback.

MSC:

93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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