This dissertation develops high-performance safe control algorithms for autonomous systems under deterministic and stochastic uncertainties. The research is divided into two main parts: deterministic and stochastic control systems.We focus on constructing safety certificates for unknown linear and nonlinear optimal control systems in the deterministic domain. We introduce an online method to develop control barrier certificates (CBCs) that expand the domain of attraction (DoA) without compromising performance. By formulating a feasible optimization problem using a relaxed algebraic Riccati equation (ARE) for linear systems and a relaxed Hamilton-Jacobi-Bellman (HJB) equation for nonlinear systems, alongside safety constraints, we identify the maximum barrier-certified region—called safe optimal DoA—where stability and safety coexist. To address the need for complete system dynamics knowledge, we propose an online data-driven approach employing a safe off-policy reinforcement learning algorithm, which learns a safe optimal policy while using a different exploratory policy for data collection.Building upon these results, we incorporate disturbances using the $H_{\infty}$ control framework to attenuate unknown disturbances while ensuring safety and optimality. We unify the robustness of CBCs with $H_{\infty}$ control methods to construct a robust and safe optimal DoA. A feasible optimization problem is developed using the relaxed game algebraic Riccati equation (GARE), solved iteratively via a sum-of-squares (SOS)-based safe policy iteration algorithm. To demonstrate practical applicability, we develop a LiDAR-based model predictive control (MPC) framework that incorporates control barrier functions (CBFs). We reduce computational complexity by synthesizing CBFs from clustered LiDAR data and integrating them into the MPC framework while ensuring safety and recursive feasibility. We validate this approach through simulations and experiments on a unicycle-type robot.In the stochastic domain, we synthesize risk-aware safe optimal controllers for partially unknown linear systems under additive Gaussian noise. By utilizing Conditional Value-at-Risk (CVaR) in the one-step cost function, we account for extremely low-probability events without excessive conservatism. Safety is guaranteed with high probability by imposing chance constraints. An online data-driven quadratic programming optimization simultaneously and safely learns the unknown dynamics and controls the system, tightening safety constraints as model confidence increases. We extend this framework to a fully risk-aware MPC for chance-constrained discrete-time linear systems with process noise, incorporating CVaR in both constraints and cost function. This approach ensures constraint satisfaction and performance optimization across the spectrum of risk assessments in stochastic environments. Recursive feasibility and risk-aware exponential stability are established through theoretical analysis.Finally, we present a data-driven risk-aware MPC framework where the mean and covariance of the noise are unknown and estimated online. We provide a computationally efficient solution to the multi-stage CVaR optimization problem using dual representations and data-driven ambiguity sets, casting it as a tractable semidefinite programming (SDP) problem. Recursive feasibility and risk-aware exponential stability are demonstrated, with numerical examples illustrating the efficacy of the proposed methods.Overall, this dissertation addresses challenges in unknown dynamics, disturbances, risk assessment, and computational tractability, providing robust and efficient solutions for safe optimal control in both deterministic and stochastic settings.
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