Simplest problem

$I(x) = \int_{t_0}^{t_1} L(t, x(t), \dot x(t)) dt \to extr$
$x(t_0) = x_0$
$x(t_1) = x_1$

This is the task of finding an extrema in continuously differentiable functions space $C^1([t_0, t_1], \mathbb{R})$ , where:

$I(x): C^1([t_0, t_1], \mathbb{R}) \to \mathbb{R}$ - functional to maximize/minimize,
$[t_0, t_1]: t_0 < t_1$ - fixed closed line segment,
$x(t) \in C^1([t_0, t_1], \mathbb{R})$ - continuously differentiable function,
$\dot x(t) = \frac {dx}{dt})$ - function derivative,
$x_0, x_1 \in \mathbb{R}$ - boundary conditions.

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Simplest problem

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