Minimization of quadratic functionals with constraints in the form of equalities
It is known that there is a wide class of the variational problems for which there are no methods of the classical variational calculus to solve them. To solve such problems one has to elaborate the method of reduction of these problems to the minimization problem of some functional in some Hilbert space [see \textit{A. A. Gruzdev}, Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 5(372), 46-42 (1993; Zbl 0866.49027) and \textit{A. A. Gruzdev} and \textit{S. A. Gusarenko}, ibid. 1994, No. 6(385), 39-49 (1994; Zbl 0846.49006)]. The author presents a general approach to such reduction, which is based on the direct examination of the extremal problems. The approach is illustrated by the investigation, among others, of the following problems: \[ \textstyle{{1\over 2}} \langle x,Ux\rangle- \langle x,f\rangle\to \inf,\;\ell x= a, \] where \(U: D\to D\) (\(D\) is a Hilbert space) is a linear, self-conjugate and bounded operator, the components of the vector-valued functional \(\ell\) are linearly independent, \(a\in R^k\) and \(f\in D\).
scientific article; zbMATH DE number 1541612