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Let $A_0$ be a possibly unbounded positive operator on the Hilbert space $H$, which is boundedly invertible. Let $C_0$ be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space $U$. We prove that the system of equations $$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$ $$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input $u$ and output $y$. The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$ where $X$ is the state space. Moreover, we have the energy identity $${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$ We show that the system described above is isomorphic...

Let be a possibly unbounded positive operator on the Hilbert space , which is boundedly invertible. Let be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space . We prove that the system of equations $$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$ $$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input and output . The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$ where is the state space. Moreover, we have the energy identity $${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$ We show that the system described above is isomorphic to its dual, so that a similar...

We survey the literature on well-posed linear systems, which has been an area of rapid development in recent years. We examine the particular subclass of conservative systems and its connections to scattering theory. We study some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators. We describe a simple way to generate conservative systems via a second-order differential equation in a Hilbert space....