Consensus on complete Riemannian manifolds in finite time (Q1941105)

The present paper studies an extension of the consensus problem to complete Riemannian manifolds. Before stating the result, let us recall some terminology. For more details, the reader is suggested to consult the original source. Let \((M,g)\) be a complete Riemannian manifold and let \(d(\cdot,\cdot)\) be the distance induced by the metric \(g\). For a point \(e\in M\) the convexity radius \(r_e(M)\) of \(M\) at \(e\) is the largest number such that for any given \(0<r\leq r_e(M)\) the ball \(B_r=\{q\in M: d(e,q)<r\}\) is strongly convex, i.e., any two points \(p_1,p_2\in B_r\) are joined by a unique geodesic of length \(d(p_1,p_2)\) entirely contained in \(B_r\). A time-varying graph \(G_{\sigma(t)}\) with \(n\) vertices \(p_1(t),\dotsc,p_n(t)\in M\) is called a communication topology graph. The set of all such graphs is denoted by \({\mathbb Q}^n\). Given a decentralized dynamical system \[ \dot p_i=u_i(p_i,I_i(t))\in T_{p_i}M,\quad i\in\{1,\dotsc,n\}, \] of the vertices of \(G_{\sigma(t)}\), where \(I_i(t)=\{p_j: j\in N_i(G_{\sigma(t)})\}\) denotes the set of neighbors of the vertex \(p_i\), the set \(\{u_i\}\) is a consensus protocol on \(S\subset M\) if for any randomly chosen initial states \(p_i(0)\in S\) the trajectories \(\{p_i(t): t\in [0,\infty)\}\) stay in \(S\) and \(p_1(t)=\cdots=p_n(t)\) as \(t\to\infty\). If \(S=M\) the consensus is global and otherwise it is called local. The main local result of the paper under review is the following theorem: Given a complete Riemannian manifold \(M\), let \(r_e(M)\) be its convexity radius at \(e\in M\), and set the open ball \(B_r=\{q\in M: d(e,q)<r\}\) with \(0<r<r_e(M)\). Consider the following dynamical system, \[ \dot p_i=\sum_{j\in N_i(G_{\sigma(t)})}\frac{\exp_{p_i}^{-1}(p_j)}{\|\exp_{p_i}^{-1}(p_j)\|},\quad i\in\{1,\dotsc,n\}, \] where each \(p_i=p_i(t)\in B_r\), \(G_{\sigma(t)}\in{\mathbb Q}^n\) is a communication topology graph. If \(G_{\sigma(t)}\) is connected, all dynamical points \(p_i(t)\) achieve consensus globally on \(B_r\) in finite time. After some extra technical assumptions, the theorem above can be made global. The global consensus result is illustrated with some computational simulations on the two dimensional unit sphere.