Formal analysis of the continuous dynamics of cyber-physical systems using theorem proving

Abstract Transform methods, such as the Laplace and the Fourier transforms, are widely used for analyzing the continuous dynamics of the physical components of Cyber–physical Systems (CPS). Traditionally, the transform methods based analysis of CPS is conducted using paper-and-pencil proof methods, computer-based simulations or computer algebra systems. However, all these methods cannot capture the continuous aspects of physical systems in their true form and thus unable to provide a complete analysis, which poses a serious threat to the safety of CPS. To overcome these limitations, we propose to use higher-order-logic theorem proving to reason about the dynamical behavior of CPS, based on the Laplace and the Fourier transforms, which ensures the absolute accuracy of this analysis. For this purpose, this paper presents a higher-order-logic formalization of the Laplace and the Fourier transforms, including the verification of their classical properties and uniqueness. This formalization plays a vital role in formally verifying the solutions of differential equations in both the time and the frequency domain and thus facilitates formal dynamical analysis of CPS. For illustration, we formally analyze an industrial robot and an equalizer using the HOL Light  theorem prover.