Formally Analyzing Continuous Aspects of Cyber-Physical Systems Modeled by Homogeneous Linear Differential Equations

Traditionally, the continuous aspects of cyber-physical systems (CPS), usually modeled by differential equations, are analyzed using paper-and-pencil proof methods, computer based numerical methods or computer algebra systems. All these methods are error-prone and thus the analysis cannot be termed as accurate, which poses a serious threat to the accuracy of the cyber-physical systems. To guarantee the correctness of analysis, we propose to use higher-order-logic theorem proving to reason about the correctness of solutions of differential equations. This paper presents a formalization framework to express homogeneous linear differential equation of arbitrary order and formally verify their solutions within the sound core of the higher-order-logic theorem prover HOL4. In order to illustrate the usefulness of the proposed formalization, we utilize it to formally verify a couple of CPS used in the domain of bio-medicine, namely, a heart pacemaker and a fluid-filled catheter.