Automation is becoming pervasive in everyday life, and many automated systems, such as unmanned aerial systems, autonomous cars, and many types of robots, are complex and safety-critical. Formal verification tools are essential for providing performance and safety guarantees for these systems. In particular, reachability analysis has previously been successfully applied to small scale control systems with general nonlinear dynamics under the influence of disturbances. Its exponentially scaling computational complexity, however, makes analyzing more complex, large scale systems intractable. Alleviating computation burden is in general a primary challenge in formal verification.

This thesis presents ways to tackle this "curse of dimensionality" from multiple fronts, bringing tractable verification of complex, practical systems such as unmanned aerial systems, autonomous cars and robots, and biological systems closer to reality. The theoretical contributions pertain to Hamilton-Jacobi (HJ) reachability analysis, with applications to unmanned aerial system. In addition, this thesis also explores two frontiers of HJ reachability by combining the formal guarantees of reachability with the computational advantages of optimization and machine learning, and with fast motion planning algorithms commonly used in robotics. The potential and benefits of the theoretical advances are demonstrated in numerous practical applications.




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