Compatibility of state constraints and dynamics for multiagent control systems

Area of Science:

• Control Theory
• Probability Theory
• Partial Differential Equations

Background:

• Multiagent systems with a large number of agents require statistical descriptions.
• State variables are represented by probability measures in Wasserstein space.
• The continuity equation governs the evolution of these probability measures.

Purpose of the Study:

• To establish a necessary and sufficient condition for state constraint compatibility.
• To analyze compatibility within the framework of controlled continuity equations in Wasserstein space.

Main Methods:

• Utilizing probability measures to describe agent density.
• Employing the Wasserstein space for state variable evolution.
• Characterizing the compatibility condition via a viscosity solution to a Hamilton-Jacobi-Bellman equation.

Main Results:

• A novel condition for state constraint compatibility in large multiagent systems is derived.
• The distance function to the constraint set is shown to be a viscosity supersolution.
• A new comparison theorem for evolutionary Hamilton-Jacobi equations in Wasserstein space is obtained.

Conclusions:

• The study provides a robust mathematical framework for state constraint compatibility in complex multiagent systems.
• The developed condition offers a precise criterion for system design and analysis.
• The findings advance the understanding of Hamilton-Jacobi equations in infinite-dimensional spaces.