Control subspaces of minimal dimension. Elementary introduction. Discotheca. / Vasyunin, V. I.; Nikol'skii, N. K.
In: Journal of Soviet Mathematics, Vol. 22, No. 6, 01.08.1983, p. 1719-1738.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Control subspaces of minimal dimension. Elementary introduction. Discotheca
AU - Vasyunin, V. I.
AU - Nikol'skii, N. K.
PY - 1983/8/1
Y1 - 1983/8/1
N2 - In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ:[Figure not available: see fulltext.], where Cyc A=R:R is a subspace of Χ, dim R<+∞. Spqn (AnR:n≥0)=χ. Always disc A ≥μA=(the multiplicity of the spectrum of the operator[Figure not available: see fulltext.] (dim R:R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ≤ disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties of disc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H2); disc S=2, (but μS=i); disc Sn*=n (but μ=1), where Sn=S⊕. ⊕S; disc S=2, (but μS=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μN=1 {mapping} (the operator N is reductive). There are other results also, and also a list of unsolved problems.
AB - In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ:[Figure not available: see fulltext.], where Cyc A=R:R is a subspace of Χ, dim R<+∞. Spqn (AnR:n≥0)=χ. Always disc A ≥μA=(the multiplicity of the spectrum of the operator[Figure not available: see fulltext.] (dim R:R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ≤ disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties of disc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H2); disc S=2, (but μS=i); disc Sn*=n (but μ=1), where Sn=S⊕. ⊕S; disc S=2, (but μS=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μN=1 {mapping} (the operator N is reductive). There are other results also, and also a list of unsolved problems.
UR - http://www.scopus.com/inward/record.url?scp=34250145129&partnerID=8YFLogxK
U2 - 10.1007/BF01882576
DO - 10.1007/BF01882576
M3 - Article
AN - SCOPUS:34250145129
VL - 22
SP - 1719
EP - 1738
JO - Journal of Mathematical Sciences
JF - Journal of Mathematical Sciences
SN - 1072-3374
IS - 6
ER -
ID: 49881069