ThreeKelvin

Well, yes, \dot{r} is a state, but the problem formulation is continuous, not discrete. There is no last step. Even if the problem formulation was discrete, e.g. x[n+1] = f(x[n], u[n]), you'd need to ensure that there'd be no last step thrusting, i.e., T[N_f - 1] = 0.

Looks nice. You might want some further constraints: T(t_f) = 0 (No thrusting at the last instant in order to cheat the constraint \dot{r}(t_f) = 0) r(t) >= r_{planet} (No shortcut through the planet - It might not be needed though.)

Wether a satellite is tidally locked has nothing to do with the mass of the object. In order to have an object locked it must: 1) Have an atraction attitude, i.e., an attitude that, due to the gravitational gradient, is accelerated towards, and 2) A way to dissipated energy, e.g. a flexible structure that transforms some of the work done on it by the gravity gradient into heat, which is then radiated away from the spacecraft. (Otherwise 1 just gives you a pendulum.) Since no satellite is completely rigid they'll all become tidally locked given enough time. Some satellites even rely on the gravitational gradient for attitude control. (see e.g. http://en.wikipedia.org/wiki/%C3%98rsted_%28satellite%29)

Also, the cheapest part of a rocket is the fuel. The rocket engine and the turbine pump are the horribly expensive parts, but even the tanks alone are more expensive than the fuel they store just in raw materials.

Yeah, it was a great launch! And it does seem like their parachutes could use a bit more work before Peter Madsen goes flying.

Right now the "amateure" rocket group Copenhagen Suborbitals are streaming the test of their guided rocket Sapphire on youtube: They're launching this rocket in order to test a prototype active guidance system, so it's a relatively small rocket, it only has a diameter of 220 mm and is 4.5 m tall. The rocket will be launched from the baltic sea, from their own launch platform, a custom built catamaran. Here's a picture from one of their previous launches, where they launched a larger rocket.

Do you have a good model of the process (aircraft, whaterver) you want this automatic trim on? I did something similar back when I was a student. We designed and implemented a reversing regulator for a model King Hauler truck with two trailers, and it turned out that our angle sensors weren't up to measuring the angles between the parts of the system. We therefore designed and implemented a bias estimator (i.e. the measured angle = actual angle + meas bias), and used the actual angle for feedback instead of the measured angle. (We used the pole placement method for the estimator, nothing fancy.) If you have a good model of the aircraft I think you'd be able to do something along the same lines and construct a trim estimator.