I've been struggling to solve calculus of variation problems with terminal conditions. The current textbook i'm using for my course seems to only tangentially touch upon the methodology. Here is one problem for example:
min $\int_{0}^{1}(10-16x-8x^2-\frac{1}{2}\dot{x}^2)dt$
with the initial condition $x(0) = 0$ and in each case of the terminal condition:
- $x(1) \in [0,1]$
- $x(1) \leq 0$
- $x(1) \geq 1$
How would I go about solving a problem like this? After I get the ODE and isolate for the constant A, i'm not sure how to continue
Choose $x(t) = \begin{cases}Kt,& t\in [0, {1 \over 2}] \\ K(1-t), & t \in [{1 \over 2},1]\end{cases} $, then $x(0) = x(1) = 0$, and the cost is $J(x) = -{1 \over 3}(5 K^2+12K-30)$. By choosing $K$ arbitrarily large we can make the cost as small as we want.
It is straightforward to see that we can make small adjustments to $x$ at the midpoint so that $x$ is smooth, however this complicates the analysis and obscures the point.
It follows that 1., 2. have no solution. (Assuming that my computations were correct, a big assumption.)