## Mathematics of Planet Earth Reading’s final-year PhD Student, Cathie Wells has published a news blog on the SIAM website:

## Read the full blog here:

**Cathie has provided this summary:**

In the SIAM Online article we look at different ways to plan flight trajectories across the North Atlantic, between John F Kennedy Airport in New York (JFK) and London Heathrow Airport (LHR), in order to reduce CO_{2} emissions. With commercial aviation currently responsible for 5% of all anthropogenic climate change, it is important that we find immediate ways to reduce aviation fuel use and thus emissions. This is offered by improved trajectory planning, taking advantage of wind conditions and a new satellite communications network.

We consider the problem through the framing of Optimal Control Theory and use two different ways to reach a solution. In the first instance time minimal trajectories at fixed altitude and airspeed are considered, enabling the corresponding optimal control problem to be solved using Pontryagin’s Minimum Principle. This method relies on optimisation of the initial system, before discretisation to allow for an approximate numerical solution. When looking at fuel optimisation for a fixed-time flight, we instead discretise the system first and find the control variables which give the minimum value of a cost functional at each time interval, subject to a non-linear constraint. From these the corresponding states of a fuel minimal trajectory can be retrieved.

Using time minimal optimisation we show that large savings in emissions are possible by changing to trajectory based operations, compared with the current tactic of limiting aircraft to the Organised Track Structure. Research into different formulations of fuel-optimal routing highlights the importance of controlling both airspeed and heading angle, rather than just heading angle.