Optimal Control of a Passive Particle Advected by a Lamb–Oseen (Viscous) Vortex

This work concerns the optimal control of a passive particle in viscous flows. This is relevant since, while there are many studies on optimal control in inviscid flows, there is little to no work in this context for viscous flows, and viscosity cannot always be neglected. Furthermore, in many tasks...

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Bibliographic Details
Published in:Computation 2022-05, Vol.10 (6), p.87
Main Authors: Marques, Gil, Gama, Sílvio, Pereira, Fernando Lobo
Format: Article
Language:English
Subjects:
Control systems
Cost analysis
Dynamical systems
Energy costs
Inviscid flow
Kinematics
Lamb–Oseen vortex
Maximum principle
Optimal control
Optimization
Ordinary differential equations
Parameters
Pontryagin’s Maximum Principle
Time dependence
Velocity
Viscosity
Viscous flow
vortex methods
Vortex-motion
Vortices
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Summary:This work concerns the optimal control of a passive particle in viscous flows. This is relevant since, while there are many studies on optimal control in inviscid flows, there is little to no work in this context for viscous flows, and viscosity cannot always be neglected. Furthermore, in many tasks, there is a need to reduce the energy spent; thus, energy-optimal solutions to problems are important. The aim of this work is to investigate how to optimally move a passive particle advected by a Lamb–Oseen (viscous) vortex between two given points in space in a given time interval while minimising the energy spent on this process. We take a control acting only on the radial component of the motion, and, by using the Pontryagin’s Maximum Principle, we find an explicit time-dependent extremal. We also analyse how the energy cost changes with the viscosity of the flow.The problem is transformed into a parameter search problem with two parameters related to the radial and angular coordinates of the initial point. The energy cost of the process increases with viscosity as long as the passive particle maintains the number of full turns it makes around the vortex. However, the energy cost increases if the increase in viscosity forces the particle to make fewer full revolutions around the vortex.
ISSN:2079-3197
2079-3197
DOI:10.3390/computation10060087