Revisiting the Complexity of Stability of Continuous and Hybrid Systems

TL;DR

This paper introduces a framework that characterizes the computational complexity of stability problems in nonlinear continuous and hybrid systems, providing decidability results for bounded cases and complexity bounds for unbounded cases.

Contribution

It offers a novel approach using first-order formulas and delta-decision problems to analyze the complexity of stability in nonlinear and hybrid systems.

Findings

Bounded stability problems are decidable with known upper complexity bounds.

Unbounded stability problems are generally undecidable, with specified degrees of unsolvability.

The framework precisely characterizes the complexity of various stability notions.

Abstract

We develop a framework to give 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 using first-order formulas over the real numbers, and reduce stability problems to the delta-decision problems of these formulas. The framework allows us to obtain a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the stability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we give upper bounds on their degrees of unsolvability.