Reduced Transversality Conditions in Optimal Space Trajectories

2013 ◽  
Vol 36 (5) ◽  
pp. 1289-1300 ◽  
Author(s):  
Binfeng Pan ◽  
Zheng Chen ◽  
Ping Lu ◽  
Bo Gao
Keyword(s):  
Space Trajectories ◽  
Transversality Conditions ◽  
Optimal Space Trajectories ◽  
Optimal Space

Optimal Space Trajectories—A Review of Published Work

1968 ◽  
Vol 72 (686) ◽  
pp. 141-146 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Gravitational Field ◽  
General Theory ◽  
Optimal Trajectory ◽  
Space Vehicle ◽  
Two Dimensions ◽  
Performance Criteria ◽  
Space Vehicles ◽  
Optimal Space Trajectories ◽  
End Conditions ◽  
Optimal Space

The problem of transferring a space vehicle between two points in a given gravitational field such that the minimum amount of fuel is used has been called the fundamental navigational problem of astronautics. In such a problem it may be required to find the optimum thrust magnitude and thrust direction which yields a minimum fuel trajectory. Furthermore, certain end conditions may be specified which the optimal trajectory must satisfy. In a large number of published papers the velocity of the vehicle is supposed known both at the beginning of the transfer and at the end whereas the time taken to complete the manoeuvre may or may not be given. Also, other performance criteria have been chosen besides minimum fuel. For example, minimum time of transit or maximum orbital altitude at perigee. Papers mentioned in this review deal mainly with flight in two dimensions apart from those sections on general theory. Furthermore, all space vehicles considered are assumed to have a fixed exhaust velocity unless otherwise stated.

scholarly journals Optimal Space Trajectories with Multiple Coast Arcs Using Modified Equinoctial Elements

2021 ◽  
Author(s):  
Mauro Pontani
Keyword(s):  
Optimization Problem ◽  
Space Mission ◽  
Low Thrust ◽  
Optimal Space Trajectories ◽  
Sequential Solution ◽  
Necessary Conditions For Optimality ◽  
Single Arc ◽  
Finite Thrust ◽  
Solution Methodology ◽  
Optimal Space

AbstractThe detection of optimal trajectories with multiple coast arcs represents a significant and challenging problem of practical relevance in space mission analysis. Two such types of optimal paths are analyzed in this study: (a) minimum-time low-thrust trajectories with eclipse intervals and (b) minimum-fuel finite-thrust paths. Modified equinoctial elements are used to describe the orbit dynamics. Problem (a) is formulated as a multiple-arc optimization problem, and additional, specific multipoint necessary conditions for optimality are derived. These yield the jump conditions for the costate variables at the transitions from light to shadow (and vice versa). A sequential solution methodology capable of enforcing all the multipoint conditions is proposed and successfully applied in an illustrative numerical example. Unlike several preceding researches, no regularization or averaging is required to make tractable and solve the problem. Moreover, this work revisits problem (b), formulated as a single-arc optimization problem, while emphasizing the substantial analytical differences between minimum-fuel paths and problem (a). This study also proves the existence and provides the derivation of the closed-form expressions for the costate variables (associated with equinoctial elements) along optimal coast arcs.

scholarly journals Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Kyungchan Lee ◽  
Gunnar F. Lange ◽  
Lin-Lin Wang ◽  
Brinda Kuthanazhi ◽  
Thaís V. Trevisan ◽  
...  
Keyword(s):  
Time Reversal ◽  
Topological Insulators ◽  
Dirac Point ◽  
Saddle Points ◽  
Surface States ◽  
Van Hove Singularity ◽  
Space Group ◽  
Topological Materials ◽  
Topological Surface ◽  
Optimal Space

AbstractTime reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P$$\bar{1}$$ 1 ¯ , which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.

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