Suppose that we wish to achieve
$$\max\int_0^1 (1-x^2-\dot{x}^2)dt, x(0)=0, x(1)\geq 1$$
Two possible ways one can do this is by Euler-Lagrange eqn or maximum principle. Applying the Euler-Lagrange on $F(t,x,\dot{x})=1-x^2-\dot{x}^2$ yields two sorts of solutions, the one is an exponential function, while the other is a trigonometric one. However, how can I proceed if I wish to find the solution through the Pontryagin maximum principle, which I have never -- willingly or knowingly anyway -- used so far?
I am not necessarily asking for an explicit solution, even a direction would be nice, since the Wiki page on this was surprisingly short and lacking on information.
Thanks in advance.