V.I. Berdyshev. An observer and a pair of objects enveloping a set of convex regions ... P. 64-70

In the space $X$ ($X=\mathbb R^2,\mathbb R^3$), there are a family of pairwise disjoint convex closed regions $G_i$ and a shortest trajectory $\cal T$ connecting given initial and finite points and enveloping the regions $G_i$, $\cal T\cap \cup_i \stackrel{\circ} G_i=\varnothing$. Two objects, $t$ and $T$, move under observation along the trajectory $\cal T$ with a constant speed, and the distance $\rho(t,T)$ between the objects along the curve $\cal T$ satisfies the condition $0<\rho(t,T)\le d$ for given $d>0$. We construct a trajectory $\cal T_f$ of the observer's motion and find the observer's speed mode such that the following inequality holds at any time $\tau$ for given $\delta>d$:
$$\min\big\{\|f_{\tau}-t_{\tau}\|,\|f_{\tau}-T_{\tau}\|\big\}=\delta.$$

Keywords: moving object, observer, trajectory, speed mode

Received August 31, 2022

Revised September 19, 2022

Accepted September 26, 2022

Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-874).

Vitalii Ivanovich Berdyshev, RAS Academician, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: bvi@imm.uran.ru

REFERENCES

1.   Berdyshev V.I. Trajectory of the observer tracking object motion around convex obstacles in $\mathbb R^2$ and $\mathbb R^3$. Doklady Mathematics, 2022, vol. 505, pp. 100–104. doi: 10.31857/S268695432204004X (in Russian).

2.   Lyu V. Metody planirovaniya puti v srede s prepyatstviyami (obzor). Matematika i Mat. Modelirovanie, 2018, vol. 1, pp. 15–58 (in Russian). doi: 10.24108/mathm.0118.0000098 

Cite this article as: V.I. Berdyshev. An observer and a pair of objects enveloping a set of convex regions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 64–70