  This work addresses the problem of finding a consensus point in the state
space ($\mathbb{R}^2$) for a multi-agent system that is comprised of $N$
identical double integrator agents. It is assumed that each agent operates
under constrained control input (i.e., $|u_i(t)| \leq 1$ $\forall i = 1, \hdots
N$). Further, a fixed fuel budget is also assumed i.e., the total amount of
cumulative input that can be expended is limited by $\int_0^{t_f}|u(t)|dt \le
\beta$. First, the attainable set $\mathcal{A}(t,x_0,\beta)$ at time $t$, which
is the set of all states that an agent can attain starting from initial
conditions $x_0$ under the fuel budget constraints at time $t$ is computed for
every agent. This attainable set is a convex set for all $t\ge0$. Then the
minimum time to consensus is the minimum time $\bar{t}$ at which attainable
sets of all agents intersect, and the consensus point is the point of
intersection. A closed-form expression for the minimum time consensus point is
provided for the case of three agents. Then, using Helly's theorem, the
intersection will be non-empty at a time when all the $N \choose 3$ triplets of
agents have non-empty intersection. The computation of minimum time consensus
for all $N \choose 3$ triplets is performed independently and can be
distributed among all the $N$ agents. Finally, the overall minimum time to
consensus is given by the triplet that has the highest minimum time to
consensus. Further, the intersection of all the attainable sets of this triplet
gives the minimum time consensus point for all $N$ agents.
