Abstract
This paper investigates a periodic search problem for a parabolic spiral moving target in the plane in which any information of the target position is not always available to the searcher. We have one searcher that starts searching for the target from (0, 0). The problem is mathematically formulated as an allocation of searching effort which is the expected value of the first meeting time between the searcher and the target. We study the same method which is detailed on the real line by Mohamed et al. (Appl. Math. Model. 35:4127–4139, 2011) in the case of the plane to find the conditions from which the expected value is finite.
Similar content being viewed by others
References
Polites, M.E.: An assessment of the technology of automated rendezvous and capture in space. Technical Report TP-208528, NASA MSFC (1970)
Zimpfer, D., Tuohy, S.: Autonomous rendezvous, capture, and in-space assembly: past, present, and future. AIAA Paper 05-2523, 1st Space Exploration Conference: Continuing the Voyage of Discovery, Orlando, FL (2005)
Ianni, J.D., Graves, J.D.: The human’s dismissing role in future on-orbit servicing missions. AIAA Paper 2001-4539, AIAA Space Conference and Exposition, Albuquerque, NM (2001)
Matsumoto, S., Oda, M., Kawano, I.: Attitude dynamics and control of space cargo for reusable orbital logistic supply servicing. AIAA Paper 2002-4785, AIAA Guidance, Navigation, and Control Conference, Monterey, CA (2002)
Defense Advanced Research Projects Agency (DARPA): Orbital Express space operation architecture. http://www.darpa.mil/tto/programs/oe.html
Air Force Research Laboratory (AFRL): XSS-11 Microsatellite. http://www.vs.afrl.af.mil/FactSheets/XSS11-MicroSatellite.pdf
Anonymous: Executive summary (introduction-CEV): NASA exploration system architecture study final report (DRAFT) (2005). http://www.spaceref.com/news/viewsr.html?pid=1967
Bailey, J.W.: NASA JSC Solicitation: commercial orbital transportation services (COTS) space flight demonstrations. NASA JSC Solicitation on spaceref.com. http://www.spaceref.com/news/viewsr.html?pid=18511
Dittmar, M.: Commercial avenues for space utilization. AIAA Paper 2003-6234, AIAA Space Conference and Exposition, Long Beach, CA (2003)
Long, A., Hastings, D.: Catching the wave: a unique opportunity for the development of an on-orbit satellite servicing infrastructure. AIAA Paper 04-6051, AIAA Space Conference and Exhibition, San Diego, CA (2004)
Mohamed A., Kassem M., El-Hadidy M.: Multiplicative linear search for a brownian target motion. Appl. Math. Model. 35, 4127–4139 (2011)
Mathematische Basteleien, Spirals. http://www.mathematische-basteleien.de/spiral.htm
Stone L.: Theory of Optimal Search. Academic Press, New York (1975)
Lukka, M.: On the optimal searching tracks for a stationary target. Publications of the Institute for Applied Mathematics, No. 4, University of Turku (Finland) (1974)
Bourgault F., Furukawa T., Durrant-Whyte H.: Coordinated decentralized search for a lost target in a bayesian world. Proc. IEEE/RSJ Int. Conf. Intell. Robot. Sys. 1, 48–53 (2003)
Bourgault F., Furukawa T., Durrant-Whyte H.: Optimal search for a lost target in a bayesian world. Field Serv. Robot. 24, 209–222 (2006)
Miller A., Moskowitz I.: Generalizations of the Carlton-Kimball distribution for a target’s future location. Comput. Math. Appl. 31, 61–68 (1996)
Mohamed A.: The generalized search for one dimensional random walker. Int. J. Pure Appl. Math. 19, 375–387 (2005)
El-Rayes A., Mohamed A., Abou-Gabal H.: Linear search for a brownian target motion. Acta Math. Sci. J. 23(B), 321–327 (2003)
Beck A., Beck M.: The Revenge of the linear search problem. SIAM J. Control Optim. 30, 112–122 (1992)
Reyniers D.: Coordinated search for an object on the line. Eur. J. Oper. Res. 95, 663–670 (1996)
Reyniers D.: Coordinated two searchers for an object hidden on an interval. J. Oper. Res. Soc. 46, 1386–1392 (1995)
Balkhi Z.: The generalized linear search problem, existence of optimal search paths. J. Oper. Res. Soc. Jpn. 30, 399–420 (1987)
Balkhi Z.: Generalized optimal search paths for continuous univariate randomvariable. Oper. Res. 23, 67–96 (1987)
Franck W.: An optimal Search Problem. SIAM Rev. 7, 503–512 (1965)
Mohamed A., Abou-Gabal H., El-Hadidy M.: Coordinated search for a randomly located target on the plane. Eur. J. Pure Appl. Math. 2, 97–111 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mohamed, A.EM.A., El-Hadidy, M.A.A. Existence of a periodic search strategy for a parabolic spiral target motion in the plane. Afr. Mat. 24, 145–160 (2013). https://doi.org/10.1007/s13370-011-0049-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-011-0049-3