Trajectory Optimization
Minimize 
subject to 
.
This example demonstrates how a variational problem can be discretized to a finite optimization problem efficiently solved by convex methods, such as QuadraticOptimization.
The variational problem will be approximated by discretizing the boundary value problem and using the trapezoidal rule to integrate on a uniformly spaced grid on the interval [0,1], 
with 
.
Let the variable u[i] represent 
and x[i] represent 
for 
.
The differential equation constraint is easily represented using centered second-order difference approximations for 
from 1 to 
.
At the boundary, the zero derivative conditions allow for the use of fictitious points 
and 
. When 
and 
, the second-order difference formula for the first derivative
is zero for 
and 
. Thus, at the boundary, use the following.
The trapezoidal rule for 
is given by the following.
Since the expression from the trapezoidal rule is quadratic and all of the constraints are linear equality constraints, the minimum of the discretized integral can be found using QuadraticOptimization directly.
Approximate functions are constructed with Interpolation.
An exact analytic solution, 
, is known for this problem, so it is possible to plot the error in the discretization.
The asymptotic error is roughly 
, so by doubling 
to 200 and recomputing, the error will be about 1/4 of what is shown here.
The analytic solution can be found by considering a family of curves 
where 
is a parameter. This parametric curve satisfies the prescribed boundary conditions 
. Since 
, one can find an optimal parameter 
that minimizes 
.
The optimal value of 
is at 2, which is the analytic result 
.
