This paper proposes a novel interpretation of the hybrid Deep Learning (DL) models in the form of an auxiliary semiclassical Optimal Control Problem (OCP) with a switched structure. The proposed equivalent reformulation of the problem of training a hybrid deep neural network allows to apply a widely developed solution methodology for OCPs to the trainable design of the network under consideration. In particular, we consider the reduced gradient method for a concrete numerical solution of the obtained OCP. We also establish the non-effectiveness of the generic necessary optimality conditions for a possible numerical treatment of the auxiliary OCP in the hybrid DL framework.

This paper is devoted to a novel approximation technique for optimal feedback control. We study a constrained Optimal Control Problem (OCP) in the context of a closed-loop dynamic system described by Ordinary Differential Equations (ODEs). The approximate optimal feedback control design we develop involves a specific$\beta$-relaxation technique (see [1]). We present here an initial research related to explicit approximations of the optimal control processes governed by the closed-loop dynamic systems. The obtained mathematical result has a potential to be extended to some concrete numerical solution algorithms for optimal feedback control problems.