C-半群和李雅普诺夫方程

C-semigroup and Lyapunov equations

考虑线性控制系统x’(f)=Ax(f)+Bu(t)(t>0),x(0)=x0,这里A是Hilbert空间X中指数稳定的C-半群T(t)的无穷小生成元,B是Hilbert空间Y到X的有界算子,且A的豫解集非空,R(C)在X中稠密时,获得了延拓控制映射LB是李雅普诺夫方程的唯一的自伴解.

We consider the linear control systems x'(t) = Ax(t) + Bu(t) (t > 0), x(0) = x0, where A is the generator of an exponentially bounded C-semigroup T(t) in a Hilbert space, B is a bounded operator from Hilbert space 7 to X, the resolvent set of A is not empty and R(C) is dence in X, and obtain that the extended controllability map LB is the unique self-adjoint solution to the Lyapunov equation.