A paper entitled

Flight Trajectory Design in the Presence of Contrails: Application of a Multiphase Mixed-Integer Optimal Control Approach.

by  M. Soler, B. Zou, M. Hansen.

has been published in Transportation Research Part C: Emerging Technologies. Volume 48, November 2014, Pages 172–194 DOI 10.1016/j.trc.2014.08.009

The article can be downloaded under free access (valid for 50 days, until November 11, 2014)



In this paper we study the 4D trajectory planning problem in a contrail sensitive environment. We identify the control inputs that steer the aircraft from 5: “Cuts” the and sells ounce quantities (30-40% pure) to the pushers for $800-$1,200 per ounce, realizing a profit of $10,000 from the pound he originally purchased. the initial fix to the final fix following a horizontal route of waypoints while performing step climbs and descents, in order to minimize the overall flying cost of fuel consumption, $CO_2$ emissions, passenger travel time, and persistent contrail formation. The optimal trajectory design problem is formulated as a multiphase mixed integer optimal control problem, which is converted into a mixed integer non-linear program by first making the unknown switching times part of the state, then applying a Hermite-Simpson direct collocation method, and finally introducing binary variables to model the decision making. We solve the mixed-integer nonlinear program using a branch-and-bound algorithm. The numerical results show the effectiveness of the approach.


Flight 4D trajectory design; Persistent contrails; Climate impact; Mixed-integer optimal control; Mixed-integer non-linear program.