A compact formula for the third variation and necessary optimality conditions (Q1874910)

From the text (translated from the Russian): ''For the optimal control problem (1) \(\dot x=f(x,u,t)\), \(x(t_0)=x_0\), \(t\in [t_0,t_1]\), \(u(t)\in U\), \(\phi (x(t_1))\to \min _u\) we construct the third variation of a performance functional of the form \(\phi (\tilde x(t_1))-\phi (x(t_1))\) for the control variation \[ \delta u=\sum ^p_{i=1}\delta u_i,\quad \delta u_i=\begin{cases} c_i&\quad \text{for } t\in [\theta _{i-1},\theta _i),\\ 0&\quad \text{for } t\notin [\theta _{i-1},\theta _i),\end{cases} \] \(\theta _i=\theta +q_i\epsilon\), \(0<q_i\leq 1\), \(q_p=1\), \(\theta \in [t_0,t_1)\), \(i=1,\dots ,p\), where \(\epsilon\) is a small parameter and \(c_i\) is constant with respect to \(t\).''