Inverse Optimal Safety Filters

Control barrier function quadratic programs (QP) safety filters are pointwise minimizers of the control effort at a given state, i.e., myopically optimal at each time. But are they optimal over the entire infinite time horizon? What does it mean for a controlled system to be "optimally safe" as opposed to, conventionally "optimally stable?" When disturbances, deterministic and stochastic, have unknown upper bounds, how should safety be defined to allow a graceful degradation under disturbances? Can safety filters be designed to guarantee such weaker safety properties as well as the optimality of safety over the infinite time horizon? We pose and answer these questions for general systems affine in control and disturbances and illustrate the answers using several examples. In the process, using the existing QP safety filters, as well as more general safety-ensuring feedbacks, we generate entire families of safety filters that are optimal over the infinite horizon although they are conservative (favoring safety over "liveness") relative to the standard QP.