Revisiting the Complexity of Stability of Continuous and Hybrid Systems

We develop a general framework for obtaining upper bounds on the “practical ” computational complexity of stability problems, for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems in first-order theories over the real numbers, and reduce stability problems to the δ-decision problems of their descriptions. The framework allows us to give a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the δ-stability problems are generally decidable, and give upper bounds on their complexity. The un-bounded versions are generally undecidable, for which we measure their degrees of unsolvability