Geodesic
Dynamical system with boundary control associated with symmetric semi-bounded operator
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 17-39 
M. I. Belishev; M. N. Demchenko. Dynamical system with boundary control associated with symmetric semi-bounded operator. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 17-39. http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a1/
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     author = {M. I. Belishev and M. N. Demchenko},
     title = {Dynamical system with boundary control associated with symmetric semi-bounded operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--39},
     year = {2012},
     volume = {409},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a1/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space $\mathcal H$. It determines a Green system $\{\mathcal H,\mathcal B;L_0,\Gamma_1,\Gamma_2\}$, where $\mathcal B$ is a Hilbert space, and $\Gamma_i\colon\mathcal H\to\mathcal B$ are the operators related through the Green formula $$ (L_0^*u, v)_\mathcal H-(u,L_0^*v)_\mathcal H=(\Gamma_1u,\Gamma_2v)_\mathcal B-(\Gamma_2u,\Gamma_1v)_\mathcal B. $$ The boundary space $\mathcal B$ and boundary operators $\Gamma_i$ are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC) \begin{align*} &u_{tt}+L_0^*u=0,&&u(t)\in\mathcal H,\,\,t>0,\\ &u|_{t=0}=u_t|_{t=0}=0,&&\\ &\Gamma_1u=f,&&f(t)\in\mathcal B,\,\,\,t\geqslant0. \end{align*} We show that this system is controllable if and only if the operator $L_0$ is completely non-self-adjoint. A version of the notion of a wave spectrum of $L_0$ is introduced. It is a topological space determined by $L_0$ and constructed from reachable sets of the DSBC.

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