Robust Decidability of Sampled-Data Control of Nonlinear Systems with Temporal Logic Specifications

Abstract: This paper explores the theoretical limits of using discrete abstractions for nonlinear control synthesis. More specifically, we consider the problem of deciding continuous-time control with temporal logic specifications. We prove that sampled-data control of nonlinear systems with temporal logic specifications is robustly decidable in the sense that, given a continuous-time nonlinear control system and a temporal logic formula, one can algorithmically decide whether there exists a robust sampled-data control strategy to realize this specification when the right-hand side of the system is slightly perturbed by a small disturbance. If the answer is positive, one can then construct a (potentially less) robust sampled-data control strategy that realizes the same specification. The result is proved by constructing a robustly complete abstraction of the original continuous-time control system using sufficiently small discretization parameters. We illustrate the result with a nonlinear control example.