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 -