Triangular systems: A global extension of the Coron-Praly theorem on the existence of feedback-integrator stabilisers
This paper deals with single-input, nonlinear smooth systems in the so-called triangular form \[ \begin{cases} \dot x_1= f_1(x_1,x_2)\\ \dot x_2= f_2(x_1,x_2,x_3)\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ \dot x_n= f_n(x_1,x_2,\dots, x_n,v)\end{cases}\tag{1} \] The problem is to find a function \(u(x_1,\dots, x_{n+1})\) such that the origin of \(\mathbb{R}^{n+1}\) becomes a globally asymptotically stable equilibrium for (1), with \(v= x_{n+1}\) and the additional equation \(\dot x_{n+1}= u\). The author establishes sufficient conditions for the solution to this problem. The feedback law \(u\) turns out to be continuous, and of class \(C^1\) on \(\mathbb{R}^{n+1}\backslash\{0\}\). The conditions are satisfied, for instance, in the case where the \(f_i\)'s are polynomials.
scientific article; zbMATH DE number 1057556