Problem:
Solve $\int_0^1(x^2+u^2)dt \rightarrow min$, subject to $x'(t) = u(t) + x(t)$ and $x(0) = 0$.
Our approach to the solution
We solve this problem using the Pontryagin Maximum Principle.
We know that $x_1$ is free so we can use the transversality condition, that is, $p(1) = 0$.
$H = p(u+x) - \lambda_0(x^2+u^2)$
$x' = u + x$
$p' = -p + 2\lambda_0x$
$p - 2\lambda_0u = 0 \rightarrow$$\lambda_0$ & $p$ can't both be zero, if $\lambda_0 = 0$ then $p = 0$, hence we take $\lambda_0 =1$.
Then we get $p = 2u \iff u = \frac{1}{2}p$
We now don't know how to solve the differential equations for $p(t)$ & $x(t)$, can anybody please help us!
