Carlos Balsa is grateful to the Foundation for Science and Technology
(FCT, Portugal) for financial support through national funds FCT/MCTES (PIDDAC) to CeDRI
(UIDB/05757/2020 and UIDP/05757/2020) and SusTEC (LA/P/0007/2021).
M. Victoria Otero-Espinar is partially supported by the Ministerio de Ciencia e Innovaci´on, Agencia
Estatal de Investigación (Spain), grant PID2020-115155GB-I00.
Sílvio Gama is partially supported by (i) CMUP, member of LASI, which is financed by national
funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference
UIDB/00144/2020, and (ii) project SNAP NORTE-01-0145- FEDER-000085, co-financed by the
European Regional Development Fund (ERDF) through the North Portugal Regional Operational
Programme (NORTE2020) under Portugal 2020 Partnership Agreement.
In this article, a simplified mathematical model is presented to depict
the process of collecting ocean debris. The responsible autonomous vehicles for
transporting the trash are represented as passive particles, while the ocean current
is simulated by the movement of point vortices on a sphere. To ensure the
autonomy of the vehicles, a system of piecewise constant controls is employed,
using a limited number of predetermined switching points that determine their
trajectories. Each control incurs an energy cost that is aimed to be minimized.
This minimization is achieved by solving a nonlinear optimization problem on
the spherical surface. The initial findings indicate the existence of multiple possible
trajectories for autonomous vehicles.
