Two-point problem for systems satisfying the controllability condition with Lie brackets of the second order (Q521077)

Consider a class of systems described by \[ \dot x =\sum_{i=1}^mu_if_i(x) \] with state \(x=(x_1,\dots,x_n)^T\in\mathbb{R}^n\), input \(u=(u_1,\dots,u_m)^T\in\mathbb{R}^m\) and smooth mapping \(f_i(x)\) from \(\mathbb{R}^n\) into \(\mathbb{R}^n\), \(m < n\). Suppose the system is controllable with the Lie brackets up to the second order. The goal is to find an admissible control \(u(t)\in\mathbb{R}^m, t\in [0,\varepsilon]\) for such a system moving the states from the initial state \(x^0\) to the final state \(x^1\) for a period of time \(\varepsilon\). The solution is based on the usage of trigonometric polynomials with coefficients computed by the expansion of solutions into the Volterra series. The original problem is reduced to a system of algebraic equations of the third order (Theorem 1). The novelty concerns local solvability of this system of algebraic equations leading to at least one real solution (Theorem 2). Its proof is based on the principle of nonzero rotation which justifies the usage of trigonometric controls. Several numerical examples are supplied.