Bibliography

M. Althoff. On Computing the Minkowski Difference of Zonotopes. CoRR abs/1512.02794 (2015), arXiv:1512.02794.

M. Althoff. Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In: Hybrid Systems: Computation and Control (HSCC), edited by C. Belta and F. Ivancic (ACM, 2013); pp. 173–182.

M. Althoff and B. H. Krogh. Avoiding geometric intersection operations in reachability analysis of hybrid systems. In: Hybrid Systems: Computation and Control (HSCC), edited by T. Dang and I. M. Mitchell (ACM, 2012); pp. 45–54.

M. Althoff, O. Stursberg and M. Buss. Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: Conference on Decision and Control (CDC) (IEEE, 2008); pp. 4042–4048.

M. Althoff, O. Stursberg and M. Buss. Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes. Nonlinear Analysis: Hybrid Systems 4, 233–249 (2010).

S. Bak and P. S. Duggirala. Simulation-Equivalent Reachability of Large Linear Systems with Inputs. In: Computer Aided Verification (CAV), Vol. 10426 of LNCS, edited by R. Majumdar and V. Kuncak (Springer, 2017); pp. 401–420.

M. Behroozi. Largest Inscribed Rectangles in Geometric Convex Sets. CoRR abs/1905.13246 (2019), arXiv:1905.13246.

C. Combastel. A state bounding observer based on zonotopes. In: European Control Conference (ECC) (IEEE, 2003); pp. 2589–2594.

T. Dreossi, T. Dang and C. Piazza. Reachability computation for polynomial dynamical systems. Formal Methods in System Design 50, 1–38 (2017).

T. Dreossi, T. Dang and C. Piazza. Parallelotope Bundles for Polynomial Reachability. In: Hybrid Systems: Computation and Control (HSCC), edited by A. Abate and G. Fainekos (ACM, 2016); pp. 297–306.

P. S. Duggirala and M. Viswanathan. Parsimonious, Simulation Based Verification of Linear Systems. In: Computer Aided Verification (CAV), Vol. 9779 of LNCS, edited by S. Chaudhuri and A. Farzan (Springer, 2016); pp. 477–494.

G. Frehse, C. L. Guernic, A. Donzé, S. Cotton, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang and O. Maler. SpaceEx: Scalable Verification of Hybrid Systems. In: Computer Aided Verification (CAV), Vol. 6806 of LNCS, edited by G. Gopalakrishnan and S. Qadeer (Springer, 2011); pp. 379–395.

G. Frehse and R. Ray. Flowpipe-Guard Intersection for Reachability Computations with Support Functions. In: Analysis and Design of Hybrid Systems (ADHS), Vol. 45 no. 9 of IFAC Proceedings Volumes, edited by M. Heemels and B. D. Schutter (Elsevier, 2012); pp. 94–101.

K. Ghorbal, E. Goubault and S. Putot. The Zonotope Abstract Domain Taylor1+. In: Computer Aided Verification (CAV), Vol. 5643 of LNCS, edited by A. Bouajjani and O. Maler (Springer, 2009); pp. 627–633.

A. Girard. Reachability of Uncertain Linear Systems Using Zonotopes. In: Hybrid Systems: Computation and Control (HSCC), Vol. 3414 of LNCS, edited by M. Morari and L. Thiele (Springer, 2005); pp. 291–305.

R. Goldman. Intersection of Two Lines in Three-Space. In: Graphics Gems, edited by A. S. Glassner (Academic Press, 1990); pp. 304–304.

C. L. Guernic. Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics. Ph.D. Thesis, Joseph Fourier University, Grenoble, France (2009).

L. J. Guibas, A. T. Nguyen and L. Zhang. Zonotopes as bounding volumes. In: Symposium on Discrete Algorithms (SODA) (ACM/SIAM, 2003); pp. 803–812.

G. K. Kamenev. An algorithm for approximating polyhedra. Computational Mathematics and Mathematical Physics 36, 533–544 (1996).

N. Kochdumper. Extensions of Polynomial Zonotopes and their Application to Verification of Cyber-Physical Systems. Ph.D. Thesis, Technical University of Munich, Germany (2022).

N. Kochdumper and M. Althoff. Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis. Transactions on Automatic Control 66, 4043–4058 (2021).

I. Kolmanovsky and E. G. Gilbert. Theory and computation of disturbance invariant sets for discrete-time linear systems. Mathematical Problems in Engineering 4, 317–367 (1998).

A.-K. Kopetzki, B. Schürmann and M. Althoff. Methods for order reduction of zonotopes. In: Conference on Decision and Control (CDC) (IEEE, 2017); pp. 5626–5633.

M. Kvasnica. Minkowski addition of convex polytopes (2005).

A. V. Lotov and A. I. Pospelov. The modified method of refined bounds for polyhedral approximation of convex polytopes. Computational Mathematics and Mathematical Physics 48, 933–941 (2008).

O. L. Mangasarian. Nonlinear programming (SIAM, 1994).

M. Maı̈ga, N. Ramdani, L. Travé-Massuyès and C. Combastel. A CSP Versus a Zonotope-Based Method for Solving Guard Set Intersection in Nonlinear Hybrid Reachability. Mathematics in Computer Science 8, 407–423 (2014).

I. M. Mitchell, J. Budzis and A. Bolyachevets. Invariant, viability and discriminating kernel under-approximation via zonotope scaling: poster abstract. In: Hybrid Systems: Computation and Control (HSCC), edited by N. Ozay and P. Prabhakar (ACM, 2019); pp. 268–269.

M. E. Muller. A Note on a Method for Generating Points Uniformly on N-Dimensional Spheres. Communications of the ACM 2, 19–20 (1959).

J. O’Rourke. Computational geometry in C (Cambridge University Press, 1998).

R. T. Rockafellar and R. J.-B. Wets. Variational Analysis. Vol. 317 of Grundlehren der mathematischen Wissenschaften (Springer, 1998).

D. B. Rubin. The Bayesian bootstrap. The annals of statistics, 130–134 (1981).

G. Singh, T. Gehr, M. Mirman, M. Püschel and M. T. Vechev. Fast and Effective Robustness Certification. In: Advances in Neural Information Processing Systems (NeurIPS), edited by S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett (2018); pp. 10825–10836.

P. Valtr. Probability that n Random Points are in Convex Position. Discrete & Computational Geometry 13, 637–643 (1995).

M. Wetzlinger, N. Kochdumper, S. Bak and M. Althoff. Fully Automated Verification of Linear Systems Using Inner and Outer Approximations of Reachable Sets. Transactions on Automatic Control 68, 7771–7786 (2023).

X. Yang and J. K. Scott. A comparison of zonotope order reduction techniques. Automatica 95, 378–384 (2018).