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We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admits solutions. Moreover, resorting to a -convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.
Fukao et al. (Fri,) studied this question.