Multi/Single-Stage Sliding Manifold Approaches for Prescribed-Time Distributed Optimization

The prescribed-time convergence mechanism has garnered significant attention within the fields of optimization and control, primarily attributed to its ability for precise manipulation of target completion times. This article formulates sliding manifolds with prescribed-time stability, based on which two modified zero-gradient-sum (ZGS) algorithms are established. One of the optimization algorithms is developed based on a multistage structural framework and another is based on a single-stage one. Contrary to the necessary customization of multistage structures in the prescribed-time distributed optimization, this study bridges the gap on the single-stage structured prescribed-time ZGS algorithm. Different from the existing multistage algorithms, the proposed optimization algorithms are proven to converge to the optimal solution within a prescribed-time under more mild conditions. Besides, for the first time, a singularity-free prescribed-time distributed optimization algorithm based on time-varying scaling function is proposed. Furthermore, the robustness of the singularity-free algorithm in rejecting external disturbances is also analyzed. The simulation involving a rendezvous formation problem is elaborated to demonstrate the singularity-free prescribed-time convergence and enhanced robustness of the proposed approaches.