How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

Weiss, George; Tucsnak, Marius

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 247-273.

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 to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded $n$-dimensional domain with boundary control and boundary observation on part of the boundary.

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