Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming

In this paper an optimisation algorithm based on Differential Dynamic Programming is applied to the design of rendezvous and fly-by trajectories to near Earth objects. Differential dynamic programming is a successive approximation technique that computes a feedback control law in correspondence of a fixed number of decision times. In this way the high dimensional problem characteristic of low-thrust optimisation is reduced into a series of small dimensional problems. The proposed method exploits the stage-wise approach to incorporate an adaptive refinement of the discretisation mesh within the optimisation process. A particular interpolation technique was used to preserve the feedback nature of the control law, thus improving robustness against some approximation errors introduced during the adaptation process. The algorithm implements global variations of the control law, which ensure a further increase in robustness. The results presented show how the proposed approach is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York

Quantum magnetism in two dimensions: From semi-classical N\'eel order to magnetic disorder

This is a review of ground-state features of the s=1/2 Heisenberg antiferromagnet on two-dimensional lattices. A central issue is the interplay of lattice topology (e.g. coordination number, non-equivalent nearest-neighbor bonds, geometric frustration) and quantum fluctuations and their impact on possible long-range order. This article presents a unified summary of all 11 two-dimensional uniform Archimedean lattices which include e.g. the square, triangular and kagome lattice. We find that the ground state of the spin-1/2 Heisenberg antiferromagnet is likely to be semi-classically ordered in most cases. However, the interplay of geometric frustration and quantum fluctuations gives rise to a quantum paramagnetic ground state without semi-classical long-range order on two lattices which are precisely those among the 11 uniform Archimedean lattices with a highly degenerate ground state in the classical limit. The first one is the famous kagome lattice where many low-lying singlet excitations are known to arise in the spin gap. The second lattice is called star lattice and has a clear gap to all excitations. Modification of certain bonds leads to quantum phase transitions which are also discussed briefly. Furthermore, we discuss the magnetization process of the Heisenberg antiferromagnet on the 11 Archimedean lattices, focusing on anomalies like plateaus and a magnetization jump just below the saturation field. As an illustration we discuss the two-dimensional Shastry-Sutherland model which is used to describe SrCu2(BO3)2.Comment: This is now the complete 72-page preprint version of the 2004 review article. This version corrects two further typographic errors (three total with respect to the published version), see page 2 for detail

Hydration status affects thirst and salt preference but not energy intake or postprandial ghrelin in healthy adults : a randomised crossover trial

Background Few studies have investigated the effect of hydration status on appetite for food in healthy adults. Prior work suggests hydration status does not alter appetite or energy intake, with mixed findings regarding appetite hormone secretion. However, an extensive investigation into both the psychological and physiological appetitive responses to hydration status has never been conducted. Objective To investigate the effect of hydration status on multiple facets of appetite. Design After 3 days pre-trial standardization, a range of appetite tasks were conducted when hypohydrated (HYPO) and euhydrated (EUHY) in 16 healthy participants (8 men). Hydration status was manipulated via dehydration in a heat tent for 60 min and subsequent fluid restriction (HYPO) or replacement (EUHY). The next day, a food reward computer task was completed followed by an ad libitum pasta meal. Pre- and post-prandial visual analogue scales assessing hunger, fullness, and flavour desires (sweet, salty, savoury and fatty) were additionally completed. Blood samples were taken the previous day before the hydration interventions in a euhydrated state, and in the fasted and post-prandial state during HYPO and EUHY. Results HYPO induced -1.9 ± 1.2% body mass change, compared to -0.2 ± 0.6% , with accompanying changes in markers of hypohydration which were not seen during EUHY. A higher desire for foods was associated with a higher water content but the association was weaker in EUHY compared to HYPO, (β= -0.33 mm/g of food water content, p < 0.001) in the food reward task. Visual analogue scales showed similar hunger and fullness between interventions, but during HYPO there was consistently higher thirst (average range in difference 27–32 mm across all time points) and lower fasted desire for salt (−23, 95% CI −10, −35 mm). Ad libitum energy intake (HYPO 1953 ± 742 kJ, EUHY 2027 ± 926 kJ; p = 0.542) and post-prandial ghrelin concentrations (HYPO 180 ± 65 pg mL−1, EUHY 188 ± 71 pg mL−1; p = 0.736) were similar by hydration status. Conclusions An acute manipulation to hydration status altered desire for salt and foods of differing water contents, but did not influence energy intake at an ad libitum pasta meal. Further research should investigate whether these appetites would alter food choice

A shooting algorithm for problems with singular arcs

In this article we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss-Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system) we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated to the perturbed problem. We present numerical tests that validate our method.Comment: No. RR-7763 (2011); Journal of Optimization, Theory and Applications, published as 'Online first', January 201