Global exponential stabilization on the n-dimensional sphere

In this paper, we show that the existence of centrally synergistic potential functions on the n-dimensional sphere, denoted by Sⁿ, is a sufficient condition for the global asymptotic stabilization of a point in Sⁿ. Additionally, if these functions decrease exponentially fast during flows and are bounded from above and from below by some polynomial function of the tracking error, then the reference point can be globally exponentially stabilized. We construct two kinds of centrally synergistic functions: the first kind consists of a finite family of potential functions on Sⁿ while the second kind consists of an uncountable number of potential functions on Sⁿ. While the former generates a simpler jump logic, the latter is optimal in the sense that it generates flows with minimal length.