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Báo cáo toán học: "Control subspaces of minimal dimension, unitary and model operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Kiểm soát subspaces kích thước tối thiểu, mô hình và điều hành đơn nhất. | J. OPERATOR THEORY 10 1983 307-330 Copyright by INCREST 1983 CONTROL SUBSPACES OF MINIMAL DIMENSION UNITARY AND MODEL OPERATORS N. K. NIKOLSKII and V. I. VASJUNIN Let us consider a linear dynamical system x t Ax t Bu t r 0 where A XX B U - X are bounded linear operators X the state space and u the control or input space are some normed linear spaces. The general problem of control is to describe operator pairs A B admitting an appropriate input signal w - such that the system starting from a fixed initial state x 0 eventually reaches a prescribed neighbourhood of X X 6 X. If the pair A B has this property one says that the system is controllable. It is well known see e.g. 1 2 that for x 0 0 the system is controllable iff the subspace BU is cyclic for A i.e. X span zt j3t k 0 where span . is the closed linear hull of the set . . In the paper 3 see also 4 6 the following characteristic of an operator A was introduced disc A sup min dim7 R c R R eCycT Recycx where Cyc 1 is the family of all finite dimensional cyclic subspaces i.e. of all such R s R c X dim J oo that ER X where def Er ER span 4 R n ỳ 0 . Here disc stands for Dimension of the Input Subspace of Control . For a transfer operator A of a controllable system the quantity discX shows to what extent it is possible to minimize the dimension of the control subspace of our system 308 N. K. NIKOLSKII and V. I. VASJUN IN without loss of controllability. The reader can find in 3 4 some discussions of properties of disc 1 and of its connection with the spectral multiplicity fiA min dim7 R 6 CycA . The knowledge of LatA the lattice of invariant subspaces of A in some cases enables US to compute disc A this quantity depending on Lat 1 only. This paper yields a collection of such cases unitary and semiunitary operators C o-con-tractions and sums of these operators. 1. MAIN RESULT. Let u be a unitary operator in a separable Hilbert space Hv . We can think that Hv . tf t dv i T r f uyw dv i - p v dv 0. T T where V is the