A Trajectory Minimizing the Exposure of a Moving Object

Autor: V. B. Kostousov, V. I. Berdyshev
Rok vydání: 2021
Předmět:
Zdroj: Proceedings of the Steklov Institute of Mathematics. 313:S21-S32
ISSN: 1531-8605
0081-5438
Popis: A corridor $$Y$$ for the motion of an object is given in the space $$X=\mathbb{R}^{N}$$ ( $$N=2,3$$ ). A finite number of emitters $$s_{i}$$ with fixed convex radiation cones $$K(s_{i})$$ are located outside the corridor. The intensity of radiation $$F(y)$$ , $$y>0$$ , satisfies the condition $$F(y)\geq\lambda F(\lambda y)$$ for $$y>0$$ and $$\lambda>1$$ . It is required to find a trajectory minimizing the value $$J({\mathcal{T}})=\displaystyle\sum_{i}\displaystyle\intop\limits_{0}^{1}F\big{ (}\|s_{i}-t(\tau)\|\big{)}\,d\tau$$ in the class of uniform motion trajectories $${\mathcal{T}}=\big{\{}t(\tau)\colon 0\leq\tau\leq 1,\ t(0)=t_{*},\ t(1)=t^{*} \big{\}}\subset Y$$ , $$t_{*},t^{*}\in\partial Y$$ , $$t_{*}\neq t^{*}$$ . We propose methods for the approximate construction of optimal trajectories in the case where the multiplicity of covering the corridor $$Y$$ with the cones $$K(s_{i})$$ is at most 2.
Databáze: OpenAIRE
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