Pressureless Euler alignment system with control

TL;DR

This paper investigates a controlled pressureless Euler alignment system, analyzing control dynamics, critical thresholds for solution behavior, and proposing a finite volume scheme validated by numerical simulations.

Contribution

It introduces a control framework for the Euler alignment system, characterizes control dynamics as an approximation to optimal control, and develops a numerical scheme with validation.

Findings

Control dynamics approximate optimal control in the system.

Critical thresholds determine global regularity or blow-up.

Numerical simulations confirm theoretical predictions.

Abstract

We study a non-local hydrodynamic system with control. First we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that leading to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally we propose a finite volume scheme for numerical solutions of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.