Free-will arbitrary time optimization for multi-agent systems with time-varying function.

The article investigates the problem of free-will arbitrary-time optimization for multi-agent systems. This refers to an optimization algorithm that not only urges all agents to come to a consensus but also collaboratively minimizes the sum of their individual time-varying objective functions within a predesignated arbitrary time frame. Unlike fixed-time or finite-time optimization issues, the problem here allows for the specification of an arbitrary settling time. To address this issue, we developed distributed optimization strategies with arbitrary settling times for the single-integrator and double-integrator multi-agent systems. Given the strongly convex nature of the time-varying functions specific to each agent, our algorithms are crafted utilizing the zero-gradient-sum approach. Theoretical analysis shows that our proposed algorithms are effective in minimizing the collective objective function and ensuring consensus among all agents within a user-defined arbitrary time frame. Illustrative simulation examples validate these theoretical results.