A Converse Control Lyapunov Theorem for Joint Safety and Stability
This provides a theoretical foundation for joint safety and stability in control systems, addressing a key challenge in robotics and autonomous systems, though it is incremental as it builds on existing converse Lyapunov literature.
The paper tackles the problem of certifying both asymptotic stability and safety in control systems by proving that a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to a single smooth Lyapunov function, with a PDE characterization ensuring exact certification of the safe set.
We show that the existence of a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to the existence of a single smooth Lyapunov function that certifies both asymptotic stability and safety. This characterization complements existing literature on converse Lyapunov functions by establishing a partial differential equation (PDE) characterization with prescribed boundary conditions on the safe set, ensuring that the safe set is exactly certified by this Lyapunov function. The result also implies that if a safety and stability specification cannot be certified by a single Lyapunov function, then any pair of control Lyapunov and control barrier functions necessarily leads to a conflict and cannot be satisfied simultaneously in a robust sense.