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The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts. 

Prior work on automatic control synthesis for cyberphysical systems under logical constraints has primarily focused on environmental disturbances or modeling uncertainties, however, the impact of deliberate and malicious attacks has been less studied. In this paper, we consider a discrete-time dynamical system with a linear temporal logic (LTL) constraint in the presence of an adversary, which is modeled as a stochastic game. We assume that the adversary observes the control policy before choosing an attack strategy. We investigate two problems. In the first problem, we synthesize a robust control policy for the stochastic game that maximizes the probability of satisfying the LTL constraint. A value iteration based algorithm is proposed to compute the optimal control policy. In the second problem, we focus on a subclass of LTL constraints, which consist of an arbitrary LTL formula and an invariant constraint. We then investigate the problem of computing a control policy that minimizes the expected number of invariant constraint violations while maximizing the probability of satisfying the arbitrary LTL constraint. We characterize the optimality condition for the desired control policy. A policy iteration based algorithm is proposed to compute the control policy. We illustrate the proposed approaches using two numerical case studies. 

In this work, we consider an LTI system with a Kalman filter, detector, and Linear Quadratic Gaussian (LQG) controller under false data injection attack. The interaction between the controller and adversary is captured by a Stackelberg game, in which the controller is the leader and the adversary is the follower. We propose a framework under which the system chooses time-varying detection thresholds to reduce the effectiveness of the attack and enhance the control performance. We model the impact of the detector as a switching signal, resulting in a switched linear system. A closed form solution for the optimal attack is first computed using the proposed framework, as the best response to any detection threshold. We then present a convex program to compute the optimal detection threshold. Our approach is evaluated using a numerical case study.