The use of conditional approximation in optimization problems (Q1335820)

This is an ``engineer's mathematical paper'' in which the reasoning is not very clear. It proposes a ``conditional approximation in direct methods for the variational calculus''. Suppose that \(H(l)\) is a solution of a variational problem which satisfies the following conditions: \[ H^{(p_ m)}(l_ m)- C_ m= 0,\qquad m= 1,\dots, M,\leqno(*) \] and has an approximate expansion: \(H(a, l)= \sum^ n_{i= 1} a_ i F_ i(l)\), where \(F_ i(l)\) are known. The author wants to determine the deviations \(da_ i\) by minimizing the mean square deviation of the unperturbed function \(H(a_ 1, \dots, a_ n, l)\) from the perturbed function \[ H(a_ 1+ da_ 1,\dots, a_ n+ da_ n, l) \] for a given deviation \(\delta a_ j\), \(j= 1,\dots, n\), when conditions \((*)\) hold; and reduces it into \(n\) problems as follows: \[ \begin{cases} \min{1\over 2} \int^{t_ 2}_{t_ 1} [\sum^ n_{i= 1} da_ i F_ i(l)]^ 2 dl,\\ da_ j= \delta a_ j,\qquad & j= 1,\dots, n,\\ \sum^ n_{i= 1} da_ i F^{(p_ m)}_ i(l_ m)= 0,\qquad & m= 1,\dots, M.\end{cases} \] \vskip3mm An application to inverse dynamical problems about a plane model of the motion of an aircraft is examined.