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 behaviour 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.

诸如Laplace和Fourier变换之类的变换方法被广泛用于分析网络物理系统（CPS）物理组件的连续动力学。传统上，基于变换方法的CPS分析是使用纸笔校验方法，基于计算机的模拟或计算机代数系统进行的。但是，所有这些方法都无法以其真实形式捕获物理系统的连续方面，因此无法提供完整的分析，这对CPS的安全性构成了严重威胁。为了克服这些限制，我们建议使用高阶逻辑定理，基于Laplace和Fourier变换证明CPS的动力学行为，以确保分析的绝对准确性。以此目的，本文介绍了Laplace和Fourier变换的高阶逻辑形式化，包括对它们经典性质和唯一性的验证。这种形式化在形式验证时域和频域中的微分方程解中起着至关重要的作用，因此有助于对CPS进行形式动力学分析。为了说明起见，我们使用HOL Light 定理证明者。