Biased Solution of Differential Equations and their Usage in Control.

Given an ordinary differential equation (ODE) we show a way to build up a corresponding 'biased ODE', e. g. an ODE defined by the same vector field, to which a time-function, said 'bias', has been added, and for which the explicit solution can be directly written through an exponential-of-integral type formula. Such a result will be proved to be entailed by the exact quadratization (EQ) of the ODE, and for a class of exactly quadratizable ODEs, namely the {\it $\pi$-systems}. The significance of biased ODEs is that, as the bias goes to zero, for time going to infinity, they become the original, unbiased ODE, and thus their solutions become an originary solution. Such a property is relevant in control theory, and in particular for the problem of global stabilization of a nonlinear control system, since in this case the bias depends of the control, and the latter can in principle be used for, first, steering the bias to zero, and then tuned in the solution formula in order to attain the convergence to zero for the original system as well.