完全可控线性系统的谱分布

<正> 其中x(·)∈R_n,u(·)∈R_r,A,B分别为n×n,n×r阶复常数阵,R_n,R_r分别为n,r维酉空间,有如下熟知结果:任给n个复数λ_1,…,λ_n,恒存在r×n阶复数阵C,使阵的谱集σ(A+BC)={λ_1,…,λ_n}的充要条件为

ON SPECTRUM DISTRIBUTION OF COMPLETELY CONTROLLABLE LINEAR SYSTEMS

Let be a Hilbert space. A be a linear operator satisfying the condition (F) given as follows:(F_1) A is an unbounded spectral operator with spectral resolution where dim E(λ_k)= 1; in general, we assume that E(λ_k) (k) is self-adjoint, and φ_k(k ≥ 1) is the normal eigenveetor in E(λ_k);In this note, we proved the followingTheorem 1.1. Let A be a linear operator satisfying condition (F). Then, for a given b ∈and any given sequence A {ν_1,ν_2,…, ν_n,…} of complex numbers, there exists at least an element g ∈such that σ(A + <., g>b)= σ_p(A + <., g>b) = A, if and only ifTheorem 1.2. Let A boa linear operator satisfying condition (F), and J a set of indexes such that Then the necessary and sufficient condition for the stabilization of linear system dx/dt=Ax + bu(t) by the feedback law u(t) = 〈x(t), k〉 with some pair of g, b (∈), is