Singular Arcs in the Generalized Goddard’s Problem
We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal traj...
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| Veröffentlicht in: | Journal of optimization theory and applications 2008-11, Vol.139 (2), p.439-461 |
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| container_title | Journal of optimization theory and applications |
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| creator | Bonnans, F. Martinon, P. Trélat, E. |
| description | We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control. |
| doi_str_mv | 10.1007/s10957-008-9387-1 |
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The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. 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The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. 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| subjects | Aerodynamics Applications of Mathematics Applied sciences Calculus of Variations and Optimal Control Optimization Computer science control theory systems Computer simulation Control system synthesis Control theory. Systems Energy consumption Engineering Exact sciences and technology Fundamental areas of phenomenology (including applications) Mathematics Mathematics and Statistics Maximum principle Methods Norms Operations Research/Decision Theory Optimal control Optimization Optimization and Control Physics Regularization Research methodology Simulation Solid dynamics (ballistics, collision, multibody system, stabilization...) Solid mechanics Strategy Studies Theory of Computation Thrust Trajectories Variance analysis Velocity |
| title | Singular Arcs in the Generalized Goddard’s Problem |
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