  We consider the multi-task coordination problem for multi-agent systems under
the following objectives: 1. collision avoidance; 2. connectivity maintenance;
3. convergence to desired destinations. The paper focuses on the safety
guaranteed region of multi-task coordination (SG-RMTC), i.e., the set of
initial states from which all trajectories converge to the desired
configuration, while at the same time achieve the multi-task coordination and
avoid unsafe sets. In contrast to estimating the domain of attraction via
Lyapunov functions, the main underlying idea is to employ the sublevel sets of
Lyapunov-like barrier functions to approximate the SG-RMTC. Rather than using
fixed Lyapunov-like barrier functions, a systematic way is proposed to search
an optimal Lyapunov-like barrier function such that the under-estimate of
SG-RMTC is maximized. Numerical examples illustrate the effectiveness of the
proposed method.
