4.20 Variational calculus

Problem: Solve the problem of finding the extrema of the functional reducing it to an optimal control problem with restrictions on the phase coordinates and controls: \[ \begin{array}{l} \int_{0}^{2} x d t \rightarrow \text { extr }, \quad|\ddot{x}| \leq 2, \quad x(0)+x(2)=0, \\ \dot{x}(0)=0 . \end{array} \]