On reduction of variational problems to extremal problems without constraints (Q1897949)

For a wide class of variational problems on \(W_2^{(l)}\) \[ \int^b_a f(t, (T_1 x) (t), \dots, (T_m x) (t)) dt\to \min;\quad \varphi_i (x) =0, \quad i=1, \dots, k, \tag \(*\) \] with \(T_i: W_2^{(l)}\to L_2\) linear and bounded, \(\varphi\) twice Fréchet-differentiable and \(f\) twice continuously differentiable, a method is introduced, which allows to reduce \((*)\) to a free extremal problem on \(L_2\). Applications to some important special cases of the problem \((*)\) are discussed. For certain variational problems, for instance quadratic problems with linear constraints, effective necessary and sufficient conditions are obtained.