Abstract
This study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on {mathbb {R}}^d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.
Highlights
In classical control theory, a single agent controls a dynamics x(t) ∈ F(x(t)), x(0) = x0, (1.1)where F : Rd ⇒ Rd is a set valued map, associating with each x ∈ Rd the subsetF(x) of Rd of the admissible velocities from x
The evolution of the controlled multi-agent system can be represented by the following two-scale dynamics
In the framework of mean field approximation of multi agent system, starting from a control problem for a large number of the agents, the problem is recasted in the framework of probability measures
Summary
In the framework of mean field approximation of multi agent system, starting from a control problem for a large number of the (discrete) agents, the problem is recasted in the framework of probability measures (see the recent [15] or the preprint [12] for -convergence results for optimal control problems with nonlocal dynamics). This procedure reduces the dimensionality and the number of equations, possibly leading to a simpler and treatable problem from the point of view of numerics.
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