Homogeneous approximations of nonlinear control systems with output and weak algebraic equivalence

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In the paper, we consider nonlinear control systems that are linear with respect to controls with output; vector fields defining the system and the output are supposed to be real analytic. Following the algebraic approach, we consider series $S$ of iterated integrals corresponding to such systems. Iterated integrals form a free associative algebra, and all our constructions use its properties. First, we consider the set of all (formal) functions of such series $f(S)$ and define the set $N_S$ of terms of minimal order for all such functions. We introduce the definition of the maximal graded Lie generated left ideal ${\mathcal J}_S^{\rm max}$ which is orthogonal to the set $N_S$. We describe the relation between this maximal left ideal and the left ideal ${\mathcal J}_S$ generated by the core Lie subalgebra of the system which realizes the series. Namely, we show that ${\mathcal J}_S\subset {\mathcal J}_S^{\rm max}$. In particular, this implies that the graded Lie subalgebra that generates the left ideal ${\mathcal J}_S^{\rm max}$ has a finite codimension. Also, we give the algorithm which reduces the series $S$ to the triangular form and propose the definition of the homogeneous approximation for the series $S$. Namely, homogeneous approximation is a homogeneous series with components that are terms of minimal order in each component of this triangular form. We prove that the set $N_S$ coincides with the set of all shuffle polynomials of components of a homogeneous approximation. Unlike the case when the output is identical, the homogeneous approximation is not completely defined by the ideal ${\mathcal J}_S^{\rm max}$. In order to describe this property, we introduce two different concepts of equivalence of series: algebraic equivalence (when two series have the same homogeneous approximation) and weak algebraic equivalence (when two series have the same maximal left ideal and therefore have the same minimal realizing system). We prove that if two series are algebraically equivalent, then they are weakly algebraically equivalent. The examples show that in general the converse is not true.