  In this paper, multi-agent systems minimizing a sum of objective functions,
where each component is only known to a particular node, is considered for
continuous-time dynamics with time-varying interconnection topologies. Assuming
that each node can observe a convex solution set of its optimization component,
and the intersection of all such sets is nonempty, the considered optimization
problem is converted to an intersection computation problem. By a simple
distributed control rule, the considered multi-agent system with
continuous-time dynamics achieves not only a consensus, but also an optimal
agreement within the optimal solution set of the overall optimization
objective. Directed and bidirectional communications are studied, respectively,
and connectivity conditions are given to ensure a global optimal consensus. In
this way, the corresponding intersection computation problem is solved by the
proposed decentralized continuous-time algorithm. We establish several
important properties of the distance functions with respect to the global
optimal solution set and a class of invariant sets with the help of convex and
non-smooth analysis.
