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Industrial control systems (ICS) are increasingly targeted by sophisticated attacks on sensors and actuators, necessitating advanced frameworks that enable proactive mitigation. This paper introduces HyTwin, a formal framework that models both adversarial actions and corresponding mitigation strategies through digital twin-based interventions. HyTwin leverages differential dynamic logic (dL) to represent the temporal evolution of attacks and quantify the mitigation horizon, a critical parameter enabling precise reasoning about when and how to deploy fail-safe mechanisms during ongoing attacks. Our approach integrates temporal semantics with attack models to dynamically engage fail-safe controls. This work provides a rigorous framework for designing proactive countermeasures that preserve system safety, ensuring robustness in adversarial scenarios. The proposed framework establishes a foundation for advancing ICS security through verifiable temporal reasoning and contributes to bridging gaps between theoretical modeling and real-world industrial applications. 

Definition packages in theorem provers provide users with means of defining and organizing concepts of interest. This system description presents a new definition package for the hybrid systems theorem prover KeYmaera X based on differential dynamic logic (dL). The package adds KeYmaera X support for user-defined smooth functions whose graphs can be implicitly characterized by dL formulas. Notably, this makes it possible to implicitly characterize functions, such as the exponential and trigonometric functions, as solutions of differential equations and then prove properties of those functions using dL's differential equation reasoning principles. Trustworthiness of the package is achieved by minimally extending KeYmaera X's soundness-critical kernel with a single axiom scheme that expands function occurrences with their implicit characterization. Users are provided with a high-level interface for defining functions and non-soundness-critical tactics that automate low-level reasoning over implicit characterizations in hybrid system proofs. 

Switched systems are known to exhibit subtle (in)stability behaviors requiring system designers to carefully analyze the stability of closed-loop systems that arise from their proposed switching control laws. This paper presents a formal approach for verifying switched system stability that blends classical ideas from the controls and verification literature using differential dynamic logic (dL), a logic for deductive verification of hybrid systems. From controls, we use standard stability notions for various classes of switching mechanisms and their corresponding Lyapunov function-based analysis techniques. From verification, we use dL's ability to verify quantified properties of hybrid systems and dL models of switched systems as looping hybrid programs whose stability can be formally specified and proven by finding appropriate loop invariants, i.e., properties that are preserved across each loop iteration. This blend of ideas enables a trustworthy implementation of switched system stability verification in the KeYmaera X prover based on dL. For standard classes of switching mechanisms, the implementation provides fully automated stability proofs, including searching for suitable Lyapunov functions. Moreover, the generality of the deductive approach also enables verification of switching control laws that require non-standard stability arguments through the design of loop invariants that suitably express specific intuitions behind those control laws. This flexibility is demonstrated on three case studies: a model for longitudinal flight control by Branicky, an automatic cruise controller, and Brockett's nonholonomic integrator. 

Continuous invariants are an important component in deductive verification of hybrid and continuous systems. Just like discrete invariants are used to reason about correctness in discrete systems without having to unroll their loops, continuous invariants are used to reason about differential equations without having to solve them. Automatic generation of continuous invariants remains one of the biggest practical challenges to the automation of formal proofs of safety for hybrid systems. There are at present many disparate methods available for generating continuous invariants; however, this wealth of diverse techniques presents a number of challenges, with different methods having different strengths and weaknesses. To address some of these challenges, we develop Pegasus : an automatic continuous invariant generator which allows for combinations of various methods, and integrate it with the KeYmaera X theorem prover for hybrid systems. We describe some of the architectural aspects of this integration, comment on its methods and challenges, and present an experimental evaluation on a suite of benchmarks. 

Continuous invariants are an important ingredient for deductive verification of hybrid and continuous systems. Just like discrete invariants are used to reason about correctness in discrete systems without unrolling their loops forever, continuous invariants are used to reason about differential equations without having to solve them. Automatic generation of continuous invariants remains one of the biggest practical challenges for automating formal proofs of safety for hybrid systems. There are at present many disparate methods available for generating continuous invariants; however, this wealth of diverse techniques presents a number of challenges, with different methods having different strengths and weaknesses. To address some of these challenges, we develop Pegasus: an automatic continuous invariant generator which allows for combinations of various methods, and integrate it with the KeYmaera X theorem prover for hybrid systems. We describe some of the architectural aspects of this integration, comment on its methods and challenges, and present an experimental evaluation on a suite of benchmarks.