Ensembles of affine-control systems with applications to Deep Learning

This thesis is devoted to the study of optimal control problems of ensembles of dynamical systems, where the dynamics has an affine dependence in the controls. By means of $\Gamma$-convergence arguments, we manage to approximate infinite ensembles with a sequence of growing-in-size finite ensembles. The advantage of this approach is that, under a suitable change of the states space, finite ensembles of control systems can be treated as a \textit{single} control system. Motivated by this fact, in the first part of the thesis we formulate a gradient flow equation on the space of admissible controls related to \textit{single} optimal control problems with end-point cost. Then, this is applied to the case of finite ensembles, where it is used to derive an implementable algorithm for the numerical resolution of ensemble optimal control problems. We also consider an iterative method based on the Pontryagin Maximum Principle. Finally, in the last part of the thesis, we formulate the task of the interpolation of a diffeomorphism with a Deep Neural Network as an ensemble optimal control problem. Therefore, we can take advantage the algorithms developed before to \textit{train} the network.