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In this paper, we consider the verification of approximate infinite-step opacity for discrete-time control sys-tems. Relying on finite abstraction techniques for solving this problem requires discretization of the state and input sets, which requires significant computational resources. Here, we propose a discretization-free approach in which we formulate opacity as a safety property over an appropriately constructed augmented system, and seek to verify it by finding suitable barrier certificates. Within our proposed scheme, lack of opacity is also verified by posing it as a reachability property over the augmented system. The main result of this paper offers a discretization-free approach to verify (lack of) infinite-step opacity in discrete-time control systems. We also discuss other notions of opacity, and their relations to one another. We particularly study the conditions under which verifying one form of opacity for a system also implies other forms. Finally, we illustrate our theoretical results on two numerical examples, where we utilize sum-of-squares programming to search for polynomial barrier certificates. In these examples, we verify the infinite-step, and current-step opacity for a vehicle by checking whether its position is concealed from possible outside intruders. 

Abstract Soft, worm-like robots show promise in complex and constrained environments due to their robust, yet simple movement patterns. Although many such robots have been developed, they either rely on tethered power supplies and complex designs or cannot move external loads. To address these issues, we here introduce a novel, maggot-inspired, magnetically driven “mag-bot” that utilizes shape memory alloy-induced, thermoresponsive actuation and surface pattern-induced anisotropic friction to achieve locomotion inspired by fly larvae. This simple, untethered design can carry cargo that weighs up to three times its own weight with only a 17% reduction in speed over unloaded conditions thereby demonstrating, for the first time, how soft, untethered robots may be used to carry loads in controlled environments. Given their small scale and low cost, we expect that these mag-bots may be used in remote, confined spaces for small objects handling or as components in more complex designs.