### On a nonlinear problem of optimal rendezvous

**Published:**09.04.2020

**Authors:** Makieva E.I., Cherkasov O. Yu.

**Published in issue: **#4(100)/2020

**DOI: **10.18698/2308-6033-2020-4-1974

**Category:** Aviation and Rocket-Space Engineering | **Chapter:** Aircraft Dynamics, Ballistics, Motion Control

The paper analyzes a nonlinear problem of optimal rendezvous of two material points in the horizontal plane. The velocity of both participants is constant modulo. The aim of control is to minimize the final distance between participants under given initial conditions. The approach time is fixed. The angle between the line of sight and the velocity vector of the Participant 1 (P1) is used as a control variable. The Participant (P2) uses the proportional-navigation law. This task may be relevant when planning the approach paths of a tanker aircraft to an unmanned aerial vehicle, or in the case of intercepting an attacking unmanned aerial vehicle by a target simulator missile launched from a real target. The principle of maximum procedure allows reducing optimal control problem to the problem of analyzing the phase portrait of a system of two nonlinear differential equations. A qualitative analysis of the system is performed, the characteristic properties of the trajectories of the participants in the horizontal plane are investigated and the results of numerical solution of the boundary value problem are presented.

**References**

**[1]**Kabamba P.T., Girard A.R. Journal of the Society for Industrial and Applied Mathematics, 2015, vol. 57 (4), pp. 611–624.

**[2]**Ben-Asher J.Z., Cliff E.M. Journal of Guidance Control and Dynamics, 1989, vol. 12 (4), pp. 598–600.

**[3]**Turetsky V., Shima T. Hybrid Evasion Strategy against a Missile with Guidance Law of Variable Structure. Proceedings of the American Control Conference, (ACC) July 6–8, 2016. 2016, pp. 3132–3137.

**[4]**Turetsky V., Shima T. Journal of Guidance, Control, and Dynamics, 2016, vol. 39 (10), pp. 2364–2373.

**[5]**Guelman M., Shinar J. Journal of Guidance, Control, and Dynamics, 1984, vol. 7 (4), pp. 471–476.

**[6]**Glizer V.Y. Journal of Optimization Theory and Applications, 1996, vol. 88 (3), pp. 503–539.

**[7]**Pachter M., Yavin Y. Journal of Optimization Theory and Applications, 1986, vol. 51 (1), pp. 129–159.

**[8]**Cherkasov O.Yu., Yakushev A.G. Journal of Optimization Theory and Applications, 2002, vol. 113 (2), pp. 211–226.

**[9]**Makieva E.I., Cherkasov O.Yu. Zadacha optimalnoy vstrechi s presledovatelem, navodyashchimsya meodo proportsionalnoy navigatsii [The problem of optimal meeting with the pursuer, guided by the proportional navigated law]. XII Vserossiyskiy syezd po fundamentalnym problemam teoreticheskoy i prikladnoy mekhaniki: sbornik trudov v 4 tomakh. Tom 1. Obshchaya i prikladnaya mekhanika [Proceedings of XII All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics. In 4 volumes. Vol. 1. General and applied mechanics], 2019, pp. 223–225.

**[10]**Meyer Y., Isaiah P., Shima T. Automatica, 2015, vol. 53, pp. 256–263.

**[11]**Berdyshev Yu.I. Trudy Instituta Matematiki i Mekhaniki UrO RAN — Procee-dings of the Steklov Institute of Mathematics. Supplement: Proceedings of the Institute of Mathematics and Mechanics, 2016, vol. 22 (1), pp. 26–35.

**[12]**Pontryagin, L.S., Boltyansky V.G., Gamkrelidze R.V., Mishchenko E.F. Mate-maticheskaya teoriya optimalnykh protsessov [The Mathematical Theory of Optimal Processes. New York, Interscience Publishers, 1962]. Moscow, Nauka Publ., 1983, 393 p.