## Singular arcs in the generalized Goddard’s Problem (2007)

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Citations: | 9 - 6 self |

### BibTeX

@MISC{Bonnans07singulararcs,

author = {F. Bonnans and P. Martinon and E. Trélat},

title = {Singular arcs in the generalized Goddard’s Problem },

year = {2007}

}

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### Abstract

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 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 with the problem of nonsmoothness of the optimal control.

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Citation Context ...e meshsize mechanism does not perform well here, which is why we use a fixed meshsize. 4.2.4 Shooting method applied to the original problem (P ) When implementing a shooting method (see for instance =-=[3, 8, 13, 23]-=-), the structure of the trajectory has to be known a priori. The structure of the control must be prescribed here by assigning a fixed number of interior switching times that correspond to junctions b... |

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Citation Context ... such singular arcs, whenever they occur, may be optimal. Their optimal status may be proved using generalized Legendre-Clebsch type conditions or the theory of conjugate points (see [19, 11], or see =-=[1, 4]-=- for a complete second-order optimality theory of singular arcs). 3 Analysis of the optimal control problem With respect to the notations used in the previous section, we set ⎛ ⎞ r x = ⎝ v ⎠ ∈ IR m 3 ... |

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Citation Context ...irect methods In order to validate the solution obtained previously with the shooting algorithm, we next implement a direct method. Although direct methods can be very sophisticated (see for instance =-=[3, 25]-=-), we here use a very rough formulation, since our aim is just to check if the results are consistent with our solution. We discretize the control using piecewise constant functions, 21s1 0.5 Control ... |

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Citation Context ...al dEx0,t0,t1(u) at u is not surjective. In this case, the trajectory x(·, x0, t0, u) is said to be singular on [t0, t1]. Recall the two following standard characterizations of singular controls (see =-=[8, 6]-=-). A control u ∈ Ux0,t0,t1 is singular if and only if the linearized system along the trajectory x(·, x0, t0, u) on [t0, t1] is not controllable. This is also equivalent to the existence of an absolut... |

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Citation Context ...ion of singular arcs. To this aim, we perform a continuation (or homotopic) approach, and regularize the original problem by adding a quadratic (�u� 2 ) term to the objective, as done for instance in =-=[15, 21]-=-. The general meaning of continuation is to solve a difficult problem by starting from the known solution of a somewhat related, but easier problem. By related we mean here that there must exist a cer... |

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Citation Context ...o well known that such singular arcs, whenever they occur, may be optimal. Their optimal status may be proved using generalized Legendre-Clebsch type conditions or the theory of conjugate points (see =-=[19, 11]-=-, or see [1, 4] for a complete second-order optimality theory of singular arcs). 3 Analysis of the optimal control problem With respect to the notations used in the previous section, we set ⎛ ⎞ r x = ... |

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Citation Context ... such singular arcs, whenever they occur, may be optimal. Their optimal status may be proved using generalized Legendre-Clebsch type conditions or the theory of conjugate points (see [19, 11], or see =-=[1, 4]-=- for a complete second-order optimality theory of singular arcs). 3 Analysis of the optimal control problem With respect to the notations used in the previous section, we set ⎛ ⎞ r x = ⎝ v ⎠ ∈ IR m 3 ... |

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Citation Context ...l dEx0,t0,t1 (u) at u is not surjective. In this case, the trajectory x(·, x0, t0, u) is said to be singular on [t0, t1]. Recall the two following standard characterizations of singular controls (see =-=[5, 18]-=-). A control u ∈ Ux0,t0,t1 is singular if and only if the linearized system along the trajectory x(·, x0, t0, u) on [t0, t1] is not controllable. This is also equivalent to the existence of an absolut... |

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Citation Context ...quations, and Equation (21) (similarly, Equation (16)) permits in general to derive an expression for the control u. The vocable ”generic” employed above can now be made more precise: it is proved in =-=[6]-=- that there exists an open and dense (in the sens of Whitney) subset G of the set of couples of smooth vector fields such that, for every control system (18) with (f0, f1) ∈ G, the set where 〈p, [f1, ... |

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Citation Context ...ion of singular arcs. To this aim, we perform a continuation (or homotopic) approach, and regularize the original problem by adding a quadratic (�u� 2 ) term to the objective, as done for instance in =-=[15, 21]-=-. The general meaning of continuation is to solve a difficult problem by starting from the known solution of a somewhat related, but easier problem. By related we mean here that there must exist a cer... |

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Citation Context ...éans cedex 2, France (Emmanuel.Trelat@univ-orleans.fr), and CMAP, UMR CNRS 7641, Ecole Polytechnique, and INRIA Futurs, 91128 Palaiseau, France. 1 1 Introduction The classical Goddard’s problem (see =-=[1, 2, 3, 4, 5]-=-) consists in maximizing the final altitude of a rocket with vertical trajectory, the controls being the norm and direction of the thrust force. Due to nonlinear effects of aerodynamic forces, the opt... |

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Citation Context ...t such singular arcs, whenever they occur, may be optimal. Their optimal status may be proved using generalized Legendre-Clebsch type conditions or the theory of conjugate points (see [9, 10], or see =-=[11, 12]-=- for a complete second-order optimality theory of singular arcs). 3 Analysis of the optimal control problem With respect to the notations used in the previous section, we set x = rv m ∈ IR3 × IR... |

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Citation Context ...ots-clés : Contrôle optimal, problème de Goddard, trajectoires singulières, méthode de tir, homotopie, méthodes directes. Example of RR.sty 3 1 Introduction The classical Goddard’s problem (see =-=[10, 20, 22]-=-) consists in maximizing the final altitude of a rocket with vertical trajectory, the controls being the norm and direction of the thrust force. Due to nonlinear effects of aerodynamic forces, the opt... |

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Citation Context ...léans cedex 2, France (Emmanuel.Trelat@univ-orleans.fr), and CMAP, UMR CNRS 7641, Ecole Polytechnique, and INRIA Futurs, 91128 Palaiseau, France. 1s1 Introduction The classical Goddard’s problem (see =-=[10, 22, 24]-=-) consists in maximizing the final altitude of a rocket with vertical trajectory, the controls being the norm and direction of the thrust force. Due to nonlinear effects of aerodynamic forces, the opt... |