In distributed optimization and iterative consensus literature, a standard problem is for N agents to minimize a function f over a subset of Euclidean space, where the cost function is expressed as a sum fi. In this paper, we study the private distributed optimization problem (PDOP) with the additional requirement that the cost function of the individual agents should remain differentially private. The adversary attempts to infer information about the private cost functions from the messages that the agents exchange. Achieving differential privacy requires that any change of an individual's cost function only results in unsubstantial changes in the statistics of the messages. We propose a class of iterative algorithms for solving PDOP, which achieves differential privacy and convergence to a common value. Our analysis reveals the dependence of the achieved accuracy and the privacy levels on the the parameters of the algorithm. We observe that to achieve -differential privacy the accuracy of the algorithm has the order of O( 1 2 ).

The iterative consensus problem requires a set of processes or agents with different initial values, to interact and update their states to eventually converge to a common value. Protocols solving iterative consensus serve as building blocks in a variety of systems where distributed coordination is required for load balancing, data aggregation, sensor fusion, filtering, clock synchronization and platooning of autonomous vehicles. In this paper, we introduce the private iterative consensus problem where agents are required to converge while protecting the privacy of their initial values from honest but curious adversaries. Protecting the initial states, in many applications, suffice to protect all subsequent states of the individual participants.First, we adapt the notion of differential privacy in this setting of iterative computation. Next, we present a server-based and a completely distributed randomized mechanism for solving private iterative consensus with adversaries who can observe the messages as well as the internal states of the server and a subset of the clients. Finally, we establish the tradeoff between privacy and the accuracy of the proposed randomized mechanism.

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Mathworks' Stateflow is a predominant environment for modeling embedded and cyber-physical systems where control software interacts with physical processes. We present Compare-Execute-Check-Engine (C2E2)-a verification tool for continuous and hybrid Stateflow models. It checks bounded time invariant properties of models with nonlinear dynamics, and discrete transitions with guards and resets. C2E2 transforms the model, generates simulations using a validated numerical solver, and then computes reachtube over-approximations with increasing precision. For this last step it uses annotations that have to be added to the model. These annotations are extensions of proof certificates studied in Control Theory and can be automatically obtained for linear dynamics. The C2E2 algorithm is sound and it is guaranteed to terminate if the system is robustly safe (or unsafe) with respect to perturbations of guards and invariants of the model. We present the architecture of C2E2, its workflow, and examples illustrating its potential role in model-based design, verification, and validation. 1 Introduction Cyber-physical systems (CPS) are systems that involve the close interaction between a software controller and a physical plant. The state of the physical plant evolves continuously with time and is often modeled using ordinary differential equations (ODE). The software controller, on the other hand, evolves through discrete steps and these steps influence the evolution of the physical process. This results in a "hybrid" behavior of discrete and continuous steps that makes the formal analysis of these models particularly challenging, so much so, that even models that are mathematically extremely simple are computationally intractable. In addition, many physical plants have complicated continuous dynamics that are described by nonlinear differential equations. Such plants, even without any interaction with a controlling software, are often unamenable to automated analysis. On the other hand, the widespread deployment of CPS in safety critical scenarios like automotives, avionics, and medical devices, have made formal, automated analysis of such systems necessary. This is evident from the extensive activity in the research community [20,19,7]. Given the challenges of formally verifying CPS, the sole analysis technique that is commonly used to analyze nonlinear systems is numerical simulation. However, given the large, uncountable space of behaviors, using numerical simulations

Simulation-based verification algorithms can provide formal safety guarantees for nonlinear and hybrid systems. The previous algorithms rely on user provided model annotations called discrepancy function, which are crucial for computing reachtubes from simulations. In this paper, we eliminate this requirement by presenting an algorithm for computing piece-wise exponential discrepancy functions. The algorithm relies on computing local convergence or divergence rates of trajectories along a simulation using a coarse over-approximation of the reach set and bounding the maximal eigenvalue of the Jacobian over this over-approximation. The resulting discrepancy function preserves the soundness and the relative completeness of the verification algorithm. We also provide a coordinate transformation method to improve the local estimates for the convergence or divergence rates in practical examples. We extend the method to get the input-to-state discrepancy of nonlinear dynamical systems which can be used for compositional analysis. Our experiments show that the approach is effective in terms of running time for several benchmark problems, scales reasonably to larger dimensional systems, and compares favorably with respect to available tools for nonlinear models.

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