Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow

Abstract

In this work we are interested in controlling the displacement of particles in interaction with N point vortices, in a two-dimensional fluid and neglecting the viscous diffusion. We want to drive a passive particle from an initial point to a final point, both given a priori, in a given finite time, the control being due to the possibility of impulsion in any direction of the plane. For the energy cost, the candidates as minimizers are given by the normal extremals of the Pontryagin Maximum Principle (PMP). The transcription of the PMP gives us a set of nonlinear equations to solve, the so-called shooting equations. We introduce these shooting equations and present numerical computations in the cases of N = 1, 2, 3 and 4 point vortices. In the integrable case N = 1, we give complete quadratures of the normal extremals.

A Numerical Algorithm for Optimal Control Problems with a Viscous Point Vortex

“…The rewriting of Pontryagin's maximum principle yields a set of nonlinear equations to be solved, called the shooting equations. In [8], the shooting equations are introduced and numerical calculations are presented in the cases of N = 1, 2, 3 and 4 perfect point vortices.…”