On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract diff...On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.展开更多
Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. I...Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. In the condition that the resolvent set A is not empty and the range of C is dense in X, we obtain that the extended controllability map is the unique self-adjoint solution to the Lyapunov equation. Moreover, sufficient conditions for asymptotically stability of C-semigroup are given.展开更多
文摘On the assumption that the Cauchy problem for incomplete second order abstract differential equation (u″(t)=Au(t), -∞ <t <∞) is well posed and the Cauchy problem for complete second order abstract differential equation ( u″(t)+A 1u′(t)+A 0u(t)=0, t≥0 ) is strongly well posed, the necessary conditions for their solutions to be pseudo almost periodic are derived.
文摘Consider the linear control systems x′(t)=Ax(t)+Bu(t)(t>0), x(0)=x_0 , where A is the generator of an exponentially stable C-semigroup on a Hilbert space X, B is a bounded operator from the Hilbert space Y to X. In the condition that the resolvent set A is not empty and the range of C is dense in X, we obtain that the extended controllability map is the unique self-adjoint solution to the Lyapunov equation. Moreover, sufficient conditions for asymptotically stability of C-semigroup are given.
