Advancements in engineering and technologies are confronted with unprecedented challenges to meet often strict safety requirements in its various forms. Barrier methods have been successfully implemented in safety-critical control tasks to enforce safety. Nonetheless, most of the existing work in the literature trades off between performance and safety by relaxing performance objectives or compromising safety or are mostly limited to certain classes of dynamical systems and constraints. The objective of the proposed research is to confront the trade-off between safety restrictions and performance through designing the appropriate mathematical tools used to develop provably safe and robust optimal control and planning for general safety-critical dynamical systems and path constraints.
In this thesis, aiming to develop algorithms that efficiently achieve safety and optimality simultaneously, I first build on the foundational work of Control Barrier Functions (CBFs) within an optimal control framework. Realizing the limitations of the current form of CBFs, I then pursue the design of embedded barrier states (BaS) as a means of integrating safety into performance objectives. The proposed technique is subsequently used with various robust control, optimal control and motion planning frameworks where it is shown to be effective, efficient and flexible substantially overcoming the limitations of existing work in the literature. The proposed idea is integrated with various techniques such as the nonlinear quadratic regulators NLQR), the State-Dependent Riccati Equation (SDRE) to solve the safety-critical infinite horizon optimal control problem, the differential dynamic programming (DDP), model predictive control (MPC) and min-max game theoretic optimal control to develop novel algorithms that produce safety-aware and robust control and decisions. Additionally, the proposed model-based frameworks are extended to the data-drive case in which the dynamics of the control system is learned using Gaussian processes to provide probabilistic safety guarantees. Finally, utilizing recent advances in distributed optimization, the optimal control techniques are applied to solve large multi-agent systems.