Quadratic Embedding into Algebras and Global Stabilization for a Class of Nonlinear Control Systems

A class of nonlinear systems is considered in IR2 which system's function of has each component being a product of real powers of the state's entries. We call that an '?-algebraic' nonlinear systems. It is shown that every ?-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 origin in IR2 a globally asymptotically stable equilibrium point of the closed-loop system.