The work presented in this thesis lies at the intersection of Machine Learning (ML) and control theory.
The first part is dedicated to the design of Deep Neural Network (DNN) architectures with specific numerical properties, inspired by dynamical systems models. First, leveraging the structure of Hamiltonian systems and employing various time-discretization methods, we establish a framework for a class of DNNs, named H-DNNs, that are well-behaved in terms of training and forward propagation. We investigate the impact of different discretization schemes on numerical properties of the resulting DNNs. Moreover, we address the issues of vanishing and exploding gradients during weight optimization, formally proving that a broad set of H-DNNs ensures non-vanishing gradients irrespective of the deep of the neural network. Next, we exploit the structure of contracting and passive systems for designing a new class of DNNs with these properties, which are crucial for robust learning and control of dynamical systems. Importantly, this parametrization is unconstrained, enabling learning for a large number of parameters. We validate our approaches on relevant benchmarks in ML and control, including image classification tasks and system identification for stable nonlinear systems.
The second part of this thesis focuses on designing DNN-based control policies for nonlinear systems while guaranteeing closed-loop stability and minimizing arbitrary cost functions. We consider a general pre-stabilized nonlinear system and develop parametrizations of all and only stabilizing control policies in both the state-feedback and output-feedback settings. Moreover, we discuss the connection between this approach, nonlinear Youla parametrizations, and the internal model control framework. By exploiting the relationship between DNN models and dynamical systems, we show how to map the optimal control problem into a learning problem within a rich class of freely parametrized stable operators, allowing for gradient descent and backpropagation for controller design. Finally, leveraging the compositional properties of port-Hamiltonian systems, we introduce a class of distributed DNN control policies, based on a more general class of H-DNNs, capable of stabilizing a network of port-Hamiltonian systems regardless of the chosen DNN parameters. The effectiveness of the proposed approaches is demonstrated by considering various tasks in swarm robotics.