Quadratic embedding into algebras and global stabilization for a class of nonlinear control systems

A class of nonlinear systems is considered in IR² whose system´s function is, component-wise, given by the product of real powers of the state´s entries. We call it the class of ´Π-algebraic´ systems. It is shown that every Π-algebraic nonlinear system undergoes a quadratic embedding into a suitable (non associative) algebra. This means that a product can be defined in the state-space, which makes the latter a non associative algebra, whose associated quadratic differential equation has a subset of entries of its solution equal to the solution of the original nonlinear system. We also study a related control problem, where for a meaningful subclass of the considered systems it is shown that a state-feedback regulator can be build up, having exponential performance, which makes the y-axe in IR² a globally asymptotically stable equilibrium set of the closed-loop system.