Dynamical systems with boundary control: Models and characterization of inverse data (Q2747516)

The author uses as basic object the space of boundary values which is a collection \(\{H,G; L,D,N\}\) of Hilbert spaces \(H,G\) and operators \(L:H\to H\), \(D:H\to G\), \(N:H\to G\) connected through the Green formula NEWLINE\[NEWLINE(Lu, v)_H-(u,L_v)_H =(Nu,Dv)_G- (Du,Nv)_G.NEWLINE\]NEWLINE The author considers the dynamical system with boundary control (DSBC) NEWLINE\[NEWLINE\begin{aligned} u_{tt}-Lu & =0\text{ in }H, \quad 0<t<T;\\ u|_{t=0} & =u_t|_{t=0}= 0\text{ in }H;\tag{1}\\ D\bigl[ u(t)\bigr] & =f(t)\text{ in }G,\;0\leq t\leq T,\end{aligned}NEWLINE\]NEWLINE where \(f\in L_2((0,T); G)\) is the boundary control and \(u=u^f(t)\) is the solution.NEWLINENEWLINENEWLINEHe studies the response operator associated with the system (1), \(R^T:F^T\to F^T\), \(\text{Dom} R^T= K^T\), \((R^Tf)(t): =N[u^f(t)]\), \(0\leq t\leq T\), where \(F^T:=L_2((0,T); G)\). Moreover, by introducing some auxiliary operators, he defines a map \(\widetilde R^{2T}: F^{2T}\to F^{2T}\), \(\text{Dom} \widetilde R^{2T}= K^{2T}\), which is called the continued response operator of the system (1). An important class of operators \({\mathcal R}^{2T}\) is identified, so that each \(R\in {\mathcal R}^{2T}\) determines a DSBC in a natural way.NEWLINENEWLINENEWLINEThe main result states that an operator \(R:F^{2T}\to F^{2T}\) is the continued reponse operator of a DSBC if and only if \(R\in{\mathcal R}^{2T}\). Thus, a characterization of the reponse operator of DSBC is given. A set of models (realizations) of DSBCs determined by the reponse operator is presented. As an application, a conditional existence theorem characterizing the dynamical Dirichlet-to-Neumann map of a Riemannian manifold is obtained.