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On the reduction of extremal problems to linear equations in a Hilbert space. (English. Russian original) Zbl 0866.49027

Russ. Math. 37, No. 5, 30-36 (1993); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 5 (372), 36-42 (1993).
A problem of infinite-dimensional minimization of a quadratic functional subject to a finite number of linear functional constraints-equalities is considered. In order to simplify and solve the constraints analytically, a special isomorphism is suggested in the paper. With the aid of the isomorphism the initial problem is transformed to unconstrained minimization. For the latter problem necessary conditions of second order are sufficient and give solvability conditions for the initial problem. Here three variants are investigated in dependence on the number of constraints. For every variant, one of the aforesaid conditions is a linear equation in a Hilbert space which corresponds to the first order necessary condition (zero of Fréchet differential), and the other is non-negativeness of the quadratic form corresponding to second derivative operator for the functional obtained. Thus the authors come to solvability characterization. There are some examples.

MSC:

49J99 Existence theories in calculus of variations and optimal control
90C48 Programming in abstract spaces
49R50 Variational methods for eigenvalues of operators (MSC2000)
90C20 Quadratic programming
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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