Nonsmooth robust stabilization of a class of two-dimensional nonlinear systems (Q1306787)

The paper considers the \(\gamma\)-parameterized family of two dimensional systems \[ \dot x_1=f_\gamma(x_1,x_2),\quad \dot x_2=u \] where \(\gamma\) is a real parameter, \(f_\gamma:\mathbb{R}^2\to\mathbb{R}\) is real analytic on a neighborhood of the origin and \(u\) is a scalar control; \(\gamma\in [-\zeta_0,\zeta_1]\) with \(\zeta_0>0\), \(\zeta_1>0\). It is required to find a state feedback law \(v:\mathbb{R}^2\to\mathbb{R}\) which is at least \(C^0\) on a neighborhood of 0, \(v(0) = 0\), that stabilizes the entire family, being independent of \(\gamma\). Several cases are studied according to the sign of the numbers \[ a(\gamma)=\frac{\partial f_\gamma}{\partial x_1} (0,0),\quad b(\gamma)=\frac{\partial f_\gamma}{\partial x_2} (0,0). \]