Review of Path Planning, Search, and Area Coverage Methods for Multi-Agent Systems with Emphasis on Mathematical and Ergodic Approaches

This paper provides a structured overview of path planning, search, and area coverage problems, with a focus on multi-agent autonomous systems and mathematically grounded approaches. The evolution of path planning is reviewed, from classical graph-based, sampling-based, and artificial potential field methods to recent centralized and decentralized multi-agent pathfinding (MAPF) strategies. Search methods are examined from early Koopman models to adaptive real-time algorithms, while area coverage approaches are discussed from distributed Voronoi-based strategies to modern ergodic methods such as SMC and HEDAC. The interconnections between path planning, search, and coverage are highlighted, and a comparative analysis of the reviewed methods is presented, emphasizing their advantages, limitations, and suitability for operation in uncertain environments with limited resources. KeyWords: Path Planning, Search, Area Coverage, Multi-Agent Systems, Mathematical Modeling.