optimal control to minimise a path 
I'm having issues solving this problem. Here is what I have tried so far.
$$ u=\dot {x_1}   + x_1 $$
$$ J= \frac{1}{2} \int_{0}^{t_1}((2x_1)^2+2\dot {x_1} x_1 + \dot {(x_1)^2})dt$$
Can I proceed and say 
$$\frac{d}{dt}(\frac{\partial L}{\partial x_1})-\frac{\partial L}{\partial \dot {x_1}}=0 $$
or would this be incorrect because I don't have $t_1$?
I also think Pontryagin might help here but I'm not sure how to proceed if that's the case. Thanks in advance!
 A: Calling
$$
H(x,u,\lambda) = \frac{1}{2} \left(u(t)^2+x(t)^2\right)+\lambda (t) (u(t)-x(t))
$$
we have
$$
\dot\lambda(t) = -\frac{\partial H}{\partial x} = \lambda(t)+x(t)\\
\frac{\partial H}{\partial u} = \lambda(t) + u(t) = 0
$$
Now solving
$$
\dot\lambda(t) = \lambda(t)+x(t)\\
\lambda(t) + u(t) = 0\\
\dot x(t)+x(t) = u(t)
$$
 we obtain
$$
x(t) = \frac{\left(7 c_1-3 c_2\right) \sinh \left(\sqrt{2} t\right)}{\sqrt{2}}+\left(5 c_1-2 c_2\right) \cosh \left(\sqrt{2} t\right)\\
\lambda(t) = \frac{\left(7 c_2-17 c_1\right) \sinh \left(\sqrt{2} t\right)}{\sqrt{2}}+\left(5 c_2-12 c_1\right) \cosh \left(\sqrt{2} t\right)
$$
Now including the contour conditions $x(0) = 1, x(t_f) = 2$ we obtain
$$
c_1= 2 \sqrt{2} \coth \left(\sqrt{2} t_f\right)-4 \sqrt{2} \text{csch}\left(\sqrt{2} t_f\right)+3\\
c_2= 5\sqrt{2} \coth \left(\sqrt{2} t_f\right)-10 \sqrt{2} \text{csch}\left(\sqrt{2} t_f\right)+7
$$
Now from the conditions
$$
H_{t_f} = 0\\
u(t_f)+\lambda(t_f) = 0
$$
we obtain
$$
\lambda(t_f) = 2\left(-1\pm\sqrt 2\right)
$$
and then follows 
$$
t_f = \frac{\ln 2}{\sqrt 2}
$$
etc.
