On the geometric construction of a stabilizing time-invariant state feedback controller for the nonholonomic integrator (Q2063837)

The classical Brockett integrator is revisited. This is the first example of a nonlinear control system that is fully controllable but for which no stabilizing continuous time-invariant state feedback exists [\textit{R. W. Brockett}, Prog. Math. 27, 181--191 (1983; Zbl 0528.93051)]. A natural and elementary geometric construction is used to construct a stabilizing time-invariant state feedback law. This feedback is smooth almost everywhere except for discontinuities along the \(z\)-axis and non-differentiability on the \(xy\)-plane. The corresponding closed-loop trajectories exhibit uniform exponential convergence to the origin. To illustrate the proposed approach, the closed-loop state trajectory is calculated for the initial state \( (0, 0, 1)\).