Stabilization of solutions of the controlled non-local continuity equation
Aleksei Volkov
TL;DR
The paper addresses stabilizing the evolution of the spatial distribution of an infinite system of interacting particles governed by the controlled non-local continuity equation $\partial_t m_t + \operatorname{div}(f(x,m_t,u) m_t) = 0$ on the Wasserstein space. It extends Clarke's control-Lyapunov framework to measure-valued dynamics by developing an inf-convolution-based proximal scheme and a proximal subgradient calculus in $\mathscr{P}_2(\mathbb{R}^d)$, culminating in a control-Lyapunov pair (CLP) $ (\phi,\psi) $ that yields both local and global stabilization results. Local stabilization is achieved via a feedback $k_\kappa^\varepsilon$ implementing an extremal shift along an optimal transport plan, ensuring a monotone decrease of the proximal Lyapunov function $\phi_\kappa$ and entry into a neighborhood of the stabilization target in finite time. Global stabilization is built by patching a hierarchy of local controllers across scales into a global, $s$-stabilizing feedback $\hat{k}$, which drives trajectories to the target with vanishing bound $M(R)$ as the initial data's radius $R$ decreases. The results provide a rigorous, constructive pathway to stabilization for non-local PDEs on probability measures using variational and proximal tools in Wasserstein spaces.
Abstract
Non-local continuity equation describes an infinite system of identical particles, which interact with each other through the common field. Solution of this equation is a probability measure that stands for spatial distribution of particles. The paper is concerned with stabilization of this solution in the case of controlled dynamic. By generalizing methods used control-Lyapunov function to the case of Wasserstein spaces, we construct a feedback strategy that provides local stabilization, i.e. leads the trajectory to a small neighbourhood of stabilization target. Based on this strategy, we construct a feedback that makes global stabilization, i.e. leads the trajectory infinitely close to stabilization target.