Affiliations: [a] Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Politécnico de Bragança, Campus de Santa Apolónia, 5300-253 Bragança, Portugal | [b] Centro de Matemática (Faculdade de Ciências) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Correspondence:
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Corresponding author: Carlos Balsa, Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Politécnico de Bragança, Campus de Santa Apolónia, 5300-253 Bragança, Portugal. E-mail: balsa@ipb.pt.
Abstract: This work reports numerical explorations in advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z0 to the final destination zf, and (iii) the maximum absolute value umax that is imposed on the control variables. The latter consist in staircase controls, i.e., the control is written as a finite linear combination of characteristic functions on the real interval. The resulting optimization problems are solved numerically. The numerical results shows the existence nearly/quasi optimal control for the cases of N=1, N=2, N=3, and N=4 vortices.
Keywords: Point vortex, passive tracer, control problem, numerical optimization
