The maximum principle is due to Lev Pontryagin. It is remarkable that despite being blind he was one of the greatest mathematicians of his generation. The key thing to grasp is that the PMP provides necessary conditions. We use the fact that an adjoint trajectory $\lambda$ exists to deduce properties of, or completely determine, the optimal control and optimally controlled trajectory. To my thinking, the PMP is notoriously badly explained in most books that it appears. I hope I have been able to make it seem more intuitive. Lagrange multipliers give a helpful interpretation, as does differentiation of the infinitestimal version of the optimality equation.
The rocket car example is a celebrated problem that is nicely solved using the PMP. Whittle calls it the "Bush problem", but does not explain why. There are two plausible answers. This may be a reference to the famous engineer Vannevar Bush, but I have not been able to track down any definitive reference. Whittle talks about bringing the rollers of a rolling mill to rest in a standard position in minimum time. So perhaps he is thinking about a roller chain or "bush roller chain", which (according to Wikipedia) is the type of chain drive most commonly used for transmission of mechanical power on many kinds of domestic, industrial and agricultural machinery, including conveyors, wire- and tube-drawing machines, printing presses, cars, motorcycles, and bicycles. I will ask Peter Whittle if he remembers.
I think I have finally guessed the answer to the question posed above. According to J. T. Olympio, A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory
Problems. J Optim Theory Appl, 2013,
"The double integrator problem (also called the Feldbaum–Bushaw problem) is a time minimization problem, where a frictionless particle moving along a line with an initial velocity and position should be put to rest."
This is exactly our rocket car problem. Apparently, it was first solved by D.W. Bushaw, Differential Equations with a Discontinuous Forcing Term, PhD Thesis, Princeton, 1952.
In the obituary of Donald W. Bushaw (1926-2012) it is stated that "Don’s PhD thesis is recognized as the beginning of modern optimal control theory."
There is a nice interactive demo of the solution to the rocket car parking problem that you can try.
The rocket car example is a celebrated problem that is nicely solved using the PMP. Whittle calls it the "Bush problem", but does not explain why. There are two plausible answers. This may be a reference to the famous engineer Vannevar Bush, but I have not been able to track down any definitive reference. Whittle talks about bringing the rollers of a rolling mill to rest in a standard position in minimum time. So perhaps he is thinking about a roller chain or "bush roller chain", which (according to Wikipedia) is the type of chain drive most commonly used for transmission of mechanical power on many kinds of domestic, industrial and agricultural machinery, including conveyors, wire- and tube-drawing machines, printing presses, cars, motorcycles, and bicycles. I will ask Peter Whittle if he remembers.
I think I have finally guessed the answer to the question posed above. According to J. T. Olympio, A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory
Problems. J Optim Theory Appl, 2013,
"The double integrator problem (also called the Feldbaum–Bushaw problem) is a time minimization problem, where a frictionless particle moving along a line with an initial velocity and position should be put to rest."
This is exactly our rocket car problem. Apparently, it was first solved by D.W. Bushaw, Differential Equations with a Discontinuous Forcing Term, PhD Thesis, Princeton, 1952.
In the obituary of Donald W. Bushaw (1926-2012) it is stated that "Don’s PhD thesis is recognized as the beginning of modern optimal control theory."
There is a nice interactive demo of the solution to the rocket car parking problem that you can try.
