On the structure of the optimal thrust for the “intermediate” aircraft model
Authors: Cherkasov O. Yu., Smirnova N.V.
Published in issue: #6(102)/2020
Category: Aviation and Rocket-Space Engineering | Chapter: Aircraft Dynamics, Ballistics, Motion Control
The paper considers a brachistochrone problem modification including in the objective function a fuel consumption penalty apart from the process time. The material point moves in a vertical plane under gravity, viscous nonlinear friction and traction. The trajectory slope angle and thrust are considered as a control variables. The Pontryagin maximum principle allows reducing the optimal control problem to a boundary value problem for a system of two nonlinear differential equations. Qualitative analysis of the resulting system allows studying the key features of extreme trajectories, including their asymptotic behavior. Extreme thrust control is obtained as a function of the velocity and the trajectory slope angle. The structure of extreme thrust is determined, and the number of switches is analytically determined. The results of numerical solving the boundary value problem are presented, illustrating the analytical conclusions.
 Tsien H.S., Evans R.C. Journal of American Rocket Society, 1951, vol. 21, no. 5, pp. 99–107.
 Menon P.K.A., Kelley H.J., Cliff E.M. AIAA Journal of Guidance, 1985, vol. 8, no. 3, pp.312-–319.
 Vratanar B., Saje M. International Journal of Non-Linear Mechanics, 1998, vol. 33, no. 3, pp. 489–505.
 Hayen J.C. International Journal of Non-Linear Mechanics, 2005, vol. 40, no. 8, pp. 1057–1075.
 Salinic S. Acta Mechanica, 2009, no. 208, pp. 97–115.
 Sumbatov A.S. International Journal of Non-Linear Mechanics, 2017, vol. 88, no. 1, pp. 135–141.
 Chen D., Liao G., Wang J. International Journal of Mechanics Research, 2015, vol. 4, no. 4, pp. 76–88.
 Thomas V. The use of variational techniques in the optimization of flight trajectories. Ph.D. thesis. University of Arizona, Parks, E.K., 1963.
 Drummond J.E., Downes G.L. Journal of Optimization Theory and Applications, 1971, vol. 7, no. 6, pp. 444–449.
 Vondrukhov A.S., Golubev Yu.F. Journal of Computer and Systems Sciences International, 2014, vol. 53, no. 6, pp. 824–838.
 Vondrukhov A.S. Journal of Computer and Systems Sciences International, 2015, vol. 54, no. 4, pp. 514–524.
 Zarodnyuk A.V., Cherkasov O.Yu. Journal of Computer and Systems Sciences International, 2017, vol. 56, no. 4, pp. 553560.
 Cherkasov O.Yu, Zarodnyuk A.V. Optimal Controlled Descent in the Atmosphere and the Modified Brachistochrone Problem. Preprints, IFAC CAO 2018. Yekaterinburg, Russia, October 15–19. 2018, pp. 630–635.
 Cherkasov O., Zarodnyuk A., Smirnova N. International Journal of Nonlinear Sciences and Numerical Simulation, 2019, vol. 20, no. 1, pp. 1–6.
 Zarodnyuk A.V., Zakirov A.N., Cherkasov O.Yu. Inzhenernyy zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2018, iss. 4. DOI: 10.18698/2308-6033-2018-4-1758
 Letov A.M. Dinamika poleta i upravleniye [Flight dynamics and control]. Moscow, Nauka Publ., 1969, 360 p.
 Kelley H.J. A Transformation approach to singular subarcs in optimal trajectory and control problems. SIAM Journal of Control, 1965, no. 2, рр. 234–240.