Series method in nonlinear time optimality

We develop the formal series technique for analysis of the time-optimal control problem. We introduce a concept of equivalence of symmetric control systems in the sense of time optimality. Our approach is based on the consideration of a free algebra of iterated integrals and structures induced in this algebra by control systems. It is close to a homogeneous approximation problem and, in essence, yields that under certain conditions a homogeneous approximation of a system is equivalent to this system in the sense of time optimality.