Optimal control of gradient flows via the Weighted Energy-Dissipation method

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 $\Gamma$-convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.