Optimality of the sufficient gain condition

Determine whether the gain condition \(\alpha/\gamma>nM/2\) is optimal under the stated assumptions for exact finite-time consensus of the intrinsic signum–gradient protocol on undirected tree graphs.

Background

The paper proves exact finite-time consensus for every Filippov solution when the fixed-gain ratio satisfies $\alpha/\gamma>nM/2$, where $n$ is the number of agents and $M$ uniformly bounds the local Riemannian gradients. The resulting settling-time estimate uses additional conservative inequalities, including a weighted Cauchy–Schwarz bound and the uniform gradient bound. Consequently, the authors explicitly leave unresolved whether this sufficient threshold can be reduced or is necessary under the stated assumptions.