Higher derivatives problem

Simplest problem can be posed not only in 1-time continuously differentiable functions, but also in n-times continuously differentiable functions space:

$I(x) = \int_{t_0}^{t_1} L(t, x(t), \dot x(t), \ddot x(t), ..., x^{(n)}(t)) dt \to extr$
$x(t_0) = x_00, x(t_1) = x_01$
$\dot x(t_0) = x_10, \dot x(t_1) = x_11$
...
$x^{(n-1)}(t_0) = x_{(n-1)0}, x^{(n-1)}(t_1) = x_{(n-1)1}$

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

$I(x): C^n([t_0, t_1], \mathbb{R}) \to \mathbb{R}$ - functional to maximize/minimize,
$[t_0, t_1]: t_0 < t_1$ - fixed closed line segment,
$x(t) \in C^n([t_0, t_1], \mathbb{R})$ - n-times continuously differentiable function,
$x_{kj} \in \mathbb{R}$ - boundary conditions.

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Higher derivatives problem

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