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1. Competitive Perimeter Defense of Conical Environments

   https://doi.org/10.1109/CDC51059.2022.9993007  

   Bajaj, Shivam; Torng, Eric; Bopardikar, Shaunak D.; Von Moll, Alexander; Weintraub, Isaac; Garcia, Eloy; Casbeer, David W. (December 2022, Proceedings of the IEEE Conference on Decision and Control)
2. Competitive Perimeter Defense on a Line

   Bajaj, S.; Torng, E.; Bopardikar, S. D. (May 2021, Proceedings of the American Control Conference)

   null (Ed.)
3. Efficient Algorithms for Optimal Perimeter Guarding

   https://doi.org/10.15607/RSS.2019.XV.002  

   Feng, Si Wei; Han, Shuai D.; Goa, Kai; Yu, Jingjin (June 2019, Robotics: science and systems)

   We investigate the problem of optimally assigning a large number of robots (or other types of autonomous agents) to guard the perimeters of closed 2D regions, where the perimeter of each region to be guarded may contain multiple disjoint polygonal chains. Each robot is responsible for guarding a subset of a perimeter and any point on a perimeter must be guarded by some robot. In allocating the robots, the main objective is to minimize the maximum 1D distance to be covered by any robot along the boundary of the regions. For this optimization problem which we call optimal perimeter guarding (OPG), thorough structural analysis is performed, which is then exploited to develop fast exact algorithms that run in guaranteed low polynomial time. In addition to formal analysis and proofs, experimental evaluations and simulations are performed that further validate the correctness and effectiveness of our algorithmic results. 

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4. Characterizing Wildfire Perimeter Polygons from QUIC-Fire

   https://doi.org/10.1007/978-3-031-08751-6_44  

   Tan, L.; de Callafon, R.A.; and Altıntaş, I. (June 2022, International Conference on Computational Science and Technology)
5. Integer Optimal Control with Fractional Perimeter Regularization

   https://doi.org/10.1007/s00245-024-10157-y  

   Antil, Harbir; Manns, Paul (July 2024, Applied Mathematics & Optimization)

   Abstract Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension$$d - 1$$ $d-1$for a domain of dimensiond. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent$$0<\alpha <1$$ $0<\alpha <1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a$$\Gamma $$ $\Gamma$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent$$\alpha $$ $\alpha$tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for$$\alpha \in (0.5,1)$$ $\alpha \in (0.5,1)$under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional. 

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