zbMATH — the first resource for mathematics

Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems. (English) Zbl 1069.49503
Summary: The dichotomic basis method is further developed for solving completely hyper-sensitive Hamiltonian boundary value problems arising in optimal control. For this class of problems, the solution can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. Constructing the solution in this composite manner alleviates the sensitivity. The method uses a dichotomic basis to decompose the Hamiltonian vector field into its stable and unstable components, thus allowing the missing initial conditions needed to specify the initial and terminal boundary-layer segments to be determined from partial equilibrium conditions. The dichotomic basis reveals the phase-space manifold structure in the neighbourhood of the optimal solution. The challenge is to determine a sufficiently accurate approximation to a dichotomic basis. In this paper we use an approximate dichotomic basis derived from local eigenvectors. An iterative scheme is proposed to handle the approximate nature of the basis. The method is illustrated on an example problem and its general applicability is assessed.

49M30 Other numerical methods in calculus of variations (MSC2010)
65L10 Numerical solution of boundary value problems involving ordinary differential equations
Full Text: DOI
[1] Betts, Journal of Guidance Control and Dynamics 21 pp 193– (1998) · Zbl 1158.49303
[2] Sparse Optimal Control Software, SOCS. Mathematics and Engineering Analysis Library Report, MEA-LR-085, Boeing Information and Support Services, P.O. Box 3797, Seattle, WA, 98124-2297, 15 July 1997.
[3] Betts, Journal of Optimization Theory and Applications 82 (1994) · Zbl 0843.90106
[4] Optimal trajectories by implicit simulation. Technical Report WRDC-TR-90-3056, Wright-Patterson Air Force Base, Boeing Aerospace and Electronics, 1990.
[5] Singular Perturbation Methods in Control: Analysis and Design. Academic Press: San Diego, 1986.
[6] Applied Optimal Control. Hemisphere: New York, 1975.
[7] BNDSCO?A program for the numerical solution of optimal control problems. Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
[8] Rao, Automatica 35 pp 633– (1999) · Zbl 1049.93533
[9] Aircraft maneuver optimization by reduced-order approximation. In Control and Dynamic Systems, vol. 10. (ed.), 1973; 131-178.
[10] An introduction to singular perturbations in non-linear control. In Singular Perturbations in Systems and Control, (ed.), 1983; 1-92.
[11] Anderson, Automatica 23 pp 355– (1987) · Zbl 0623.49010
[12] Lam, Combustion Science and Technology 89 pp 375– (1993)
[13] Lam, International Chemical Kinetics 26 pp 461– (1994)
[14] Geometry of computational singular perturbations. Proceedings of the IFAC Nonlinear Control-Design Symposium, Tahoe City, 1995.
[15] Computational singular perturbation method for optimal control. Proceedings of the American Control Conference, San Diego, 1990.
[16] A new method for solving optimal control problems. Proceedings of the AIAA Guidance, Navigation and Control Conference, Baltimore, 1995; 818-825.
[17] Extension of the computational singular perturbation method to optimal control. PhD Dissertation, Princeton University, 1996.
[18] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer: New York, 1990.
[19] An approach to solving two time-scale trajectory optimization problems. IFAC Workshop on Control Applications of Optimization, Haifa, Israel, December 1995.
[20] Identifying time-scale structure for simplified guidance law development. AIAA Guidance, Navigation, and Control Conference, New Orleans, 1997.
[21] Seywald, Journal of Guidance Control and Dynamics 17 pp 398– (1994)
[22] Seywald, Optimal Control Applications and Methods 18 pp 159– (1997) · Zbl 0873.49026
[23] Ordinary Differential Equations. Springer: Berlin, 1992.
[24] Minimum time-to-climb trajectories using a modified sweep method. Proceedings of the AIAA Guidance, Navigation and Control Conference, Baltimore, 1995; 826-833.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.