Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality

In the paper we consider nonlinear systems depending linearly on control with one-dimensional output. As is well known, under the analyticity requirement, the output can be expressed as a series of iterated integrals of controls with scalar coefficients. Since iterated integrals generate a free associative algebra, algebraic and combinatorial tools can be applied. Developing ideas of the algebraic approach to homogeneous approximation of nonlinear control systems, we propose the definition of a homogeneous approximation for series of iterated integrals with one-dimensional coefficients and study its algebraic properties. In addition, we describe relations between the homogeneous approximation of one-dimensional series of iterated integrals and an approximation in the sense of time optimality. Namely, we give conditions under which the optimal time and optimal controls for the minimal realization of the initial series are approximated by the optimal time and optimal controls for the minimal realization of its homogeneous approximation in a neighborhood of the origin