The differential equations are shown on
the right. In the lanuage of functionals, it is a minimum
final time problem. There are three state variables. Two control
variables. There is also a non-differential constraint on the controls
to keep it bounded. Again, as in typical trajectory optimization
problem, the boundary conditions are specified at two points.
The problem has an analytical solution
Fig. 1. The System Dynamics
Three solutions are compared.
The analytical solution, a numerical solution and , the Bezier solution.
A Cubic Bezier curve is used to model the problem. The Bezier
curve indicates a small error in the middle portion of the curve.
Figure 2. The Brachistochrone solution