Interpolation Theory and Its Applications pp 92-108 | Cite as

# Extremal Problems

## Abstract

In this chapter extremal problems of two types are investigated. First we consider the interpolation problems the solution of which *w* (ξ) shall in addition satisfy the extremal condition *w*^{*} (ξ) *w* (ξ) ≤ *ρ*^{2}_{min}, (7.0.1) where *ρ*_{min} is a non-negative matrix. By means of a reformulation we reduce the corresponding interpolation problem to a degenerate one. This degenerate problem has a unique solution which can be found with the help of the results of Chapter 5. It is essential both from the applied and theoretical view-points that the solution of the extremal problem turns out to be a rational matrix function. The case when *ρ*_{min} is scalar matrix was investigated in the works [1], [2] and found its application in the control theory [28]. The transition to the arbitrary non-negative matrix *ρ*_{min} allows to increase considerably the class of the extremal problems which have effective solutions. The second type of the extremal problems considered in this chapter is connected with the maximum jump theorem (A.Sakhnovich [50]). This case is provided with some problems of the canonical differential systems theory, several problems of radio techique and problem connected with the Gauss model (Vladimirov-Volovich problem [63]).

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