Learning for control has achieved remarkable success in controlling complex dynamical systems such as autonomous vehicles and quadrupedal robots. The resulting controlled system often has a neural network (NN) in the loop which represents the system dynamics, control policy, or perception. While enjoying high empirical performance, NN dynamical systems suffer from the lack of formal guarantees which significantly limits the deployment of NNs or learning algorithms in safety-critical applications. This thesis aims to address this challenge by combining control theoretical analysis of dynamical systems and specialized optimization algorithm design for NNs. In the first part of the thesis, we focus on verifying the safety and stability of a NN dynamical system. A hierarchy of verification problems are considered: (i) isolated output range analysis of a NN, (ii) closed-loop reachability analysis, and (iii) closed-loop stability analysis of a NN dynamical system. Output range analysis of a NN concerns over-approximating the output of a NN given a bounded input set. This problem can be formulated as a global optimization problem which is NP-hard to solve, and even its convex relaxations are numerically challenging to solve when large NNs are considered. Verification of closed-loop properties such as reachability and stability of NN dynamical systems becomes even more challenging since the objectives are more complex and the system dynamics must be taken into account. For Problem (i), by utilizing the structure of NNs, we propose a salable operator splitting method for solving a linear program relaxation of the verification problem which achieves a preferable complexity-conservatism trade-off. We solve Problem (ii) and (iii) by combining NN output range analysis tools and specialized reachability (i.e., verifying unrolled NN dynamics in one-shot) or stability (i.e., synthesizing a Lyapunov function using cutting-plane methods) analysis frameworks.

In hope of combining robust model predictive control (MPC) and NN verification tools for safe control of NN dynamical systems, in the second part of the thesis, we propose a novel robust MPC method for uncertain linear dynamical systems with polytopic model uncertainty and bounded additive disturbances. This formulation allows abstracting nonlinear NN dynamics by polytopic uncertainties. Drawing tools from System Level Synthesis which transforms state feedback controller design into closed-loop system responses design, our proposed method can simultaneously search over robust linear time-varying state feedback controllers and bounds on the effects of model uncertainty. Extensive simulation shows that our proposed method achieves significantly reduced conservatism compared with existing robust MPC baselines.