Time-optimal rendezvous transfer trajectory for restricted cone-angle range solar sails

The advantage of solar sails in deep space exploration is that no fuel consumption is required. The heliocentric distance is one factor influencing the solar radiation pressure force exerted on solar sails. In addition, the solar radiation pressure force is also related to the solar sail orientation with respect to the sunlight direction. For an ideal flat solar sail, the cone angle between the sail normal and the sunlight direction determines the magnitude and direction of solar radiation pressure force. In general, the cone angle can change from 0° to 90°. However, in practical applications, a large cone angle may reduce the efficiency of solar radiation pressure force and there is a strict requirement on the attitude control. Usually, the cone angle range is restricted less more than an acute angle （for example, not more than 40°） in engineering practice. In this paper, the time-optimal transfer trajectory is designed over a restricted range of the cone angle, and an indirect method is used to solve the two point boundary value problem associated to the optimal control problem. Relevant numerical examples are provided to compare with the case of an unrestricted case, and the effects of different maximum restricted cone angles are discussed. The results indicate that （1） for the condition of a restricted cone-angle range the transfer time is longer than that for the unrestricted case and （2） the optimal transfer time increases as the maximum restricted cone angle decreases.

On the Ranks of Semigroups of Transformations on a Finite Set with Restricted Range

Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ（X, Y） = {α∈ ΡΥ（X） ： Xα Y}, Υ（X, Y） = ΡΥ（X, Y） ∩Υ（X） and Ι（X, Y） = ΡΥ（X, Y） ∩ Ι（X）. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green＇s relations of Υ（X,Y）. In this paper, we present analogous results for bothΡ Υ（X, Y） and Ι（X, Y）. For a finite set X with ｜x｜≥ 3, the ranks of ΡΥ（X） = ΡΥ（X, X）, Υ（X） = Υ（X, X） and.Ι（X） = Ι（X, X） are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ（X,Y）, Υ（X, Y） and Ι（X, Y） for any proper non-empty subset Y of X.

1-Bit compressive sensing: Reformulation and RRSP-based sign recovery theory

Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption （that has been overlooked in the literature） should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an t0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property （RRSP） of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.