AN IMPORTANT problem in air traffic management (ATM) isnthe design of aircraft trajectories that meet certain arrival timenconstraints at given waypoints, for instance, at the top of descentn(TOD), at the initial approach fix (IAF), or at the runway threshold.nThese are called 4-D trajectories, which are a key element in thentrajectory-based-operations (TBO) concept proposed by NextGennand SESAR for the future ATM system. The time constraints mustnalso be met in certain cases in which the nominal trajectories havento be modified to resolve detected conflicts (e.g., lost of separationnminima) between aircraft; for example, Bilimoria and Lee [1]nanalyze aircraft conflict resolution with an arrival time constraintnat a downstream waypoint.nOn the other hand, the design of fuel-optimal trajectories that leadnto energy-efficient flights is another important problem which hasnbeen treated extensively in the literature, see, for instance, Burrowsn[2], Neuman and Kreindler [3], Menon [4] and the references therein.nFuel-optimal trajectories with fixed arrival times are studied bynSorensen and Waters [5], Burrows [6] and Chakravarty [7], whonanalyze the 4-D fuel-optimization problem as a minimum directoperating-ncost (DOC) problem with free final time, that is, thenproblem is to find the time cost for which the corresponding freenfinal-time DOC-optimal trajectory arrives at the assigned time.nIn this work we analyze the problem of minimum-fuel cruise atnconstant altitude with a fixed arrival time as a singular optimal controlnproblem, building upon the works of Pargett and Ardema [8] andnRivas and Valenzuela [9], who analyze the problem of maximumrangencruise at constant altitude also as a singular optimal controlnproblem; the case of unsteady cruise is considered. The singular arcsnand the corresponding optimal control are obtained as a function ofnthe final time. The optimal paths are obtained as well, which define anvariable-Mach cruise at constant altitude. The influence of cruisenaltitude on the optimal paths is analyzed, and the minimum fuel isncalculated. The final-time constraint may be defined, for example, byna flight delay imposed on the nominal (preferred) cruise trajectoryn(which in our case is the minimum-fuel cruise trajectory with freenfinal time); comparison with a standard constant-Mach procedure tonabsorb delays is made. Results are presented for a model of a Boeingn767-300ER.nProblem FormulationnIt is desired to minimize fuel consumption for a given range andna given final time, that is, to minimize the following performancenindex:nJ u0001nZ tfn0ncTdt (1)nwith final time tfnfixed, subject to the following constraints:n_Vnu0001n1nmnu0002T u0003 Du0004 m_ u0001 u0003cT x_ u0001 V (2)nwhich are the equations of motion for cruise at constant altitude andnconstant heading. In these equations, the drag is a general knownnfunction Du0002V;mu0004, which takes into account the remaining equationnof motion L u0001 mg. The thrust Tu0002Vu0004 is given by T u0001 u0001TMu0002Vu0004, wherenu0001 models the throttle setting u0001m u0005 u0001 u0005 1, and TMu0002Vu0004 is a knownnfunction. The specific fuel consumption cu0002Vu0004 is also a knownnfunction. Thus, this problem has three states (speed V, mass m, andnflight distance x) and one control (u0001). The initial aircraft mass (mi)nand the initial and final flight distances (xi u0001 0 and xf) are given.nTo solve this problem it is convenient to take the distance x asnthe independent variable. Thus, the problem can be formulated asnfollows: minimize the performance index
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