Extremal problems generated by operator pencils (Q1071253)

Continuing a series of papers on the same subject [cf. Dokl. Akad. Nauk SSSR 255, 770-780 (1980; Zbl 0484.49010); Usp. Mat. Nauk 36, No.5(221), 161-162 (1981; Zbl 0484.52010); Dokl. Akad. Nauk SSSR 257, 1033-1037 (1981; Zbl 0492.47002) and Optimizatsiya 27(44), 143-152 (1981; Zbl 0529.47006)] the author considers the following optimization problem: given Rayleigh functionals p, \(p_ i\), \(i=1,2,...,n\) on the Hilbert space H, find \(p(V):=\sup \{p(x)|\) \(x\in V\}\), where \(V=\{x\in H\setminus \{0\}| p_ i(x)\leq \alpha_ i\), \(i=1,...,m\), \(p_ i(x)=\alpha_ i\), \(i=m+1,...,n\}.\) In the present case these Rayleigh functionals are generated by operator pencils. The duality theory, as described in the first and second paper loc. cit., is complete for this class of problems. As an application the problem of tuning-out vibrating systems from resonance-endangered areas is discussed.