Projection-difference method in observation and control problems for Schrödinger-type equations (Q1281209)

The problem of restoration of a complex function \(v=v(\vec{x})\in L^{2}(\Omega)\) using an observed information about a corresponding phase trajectory \(E=E(z,\vec{x}; v)\) of the system \[ \partial_{z} E + i\Delta E + i\alpha E=0 \tag{1} \] \[ E_{z=0}=v(\vec{x}), \qquad E_{(0,z)\times (\partial \Omega)}=0 \tag{2} \] is considered. Here \( i^{2}=-1\), \(\vec{x}=(x_{1}, x_{2})\in \Omega=(0, X_{1})\times ( 0, X_{2})\) , \(Q=(0, Z)\times \Omega\), \(\partial \Omega\) is the boundary of the domain \(\Omega\), \(\partial_{z}=\partial/\partial z\), \(\Delta= \partial^{2}/\partial x_{1}^{2} + \partial^{2}/\partial x_{2}^{2}\), \(a=a(z,\vec{x})\) a real function. Let the trajectory \(E( z,\vec{x})\) of the systems (1), (2) be observed only on the part \(Q_{0}\) of the domain \(Q\). The part of the domain is called ``a zone'' and it is of the form \[ Q_0=(z^{0}, Z)\times \Omega_{0},\quad \Omega_{0}=(b_{1}, d_{1})\times (b_{2}, d_{2}),\qquad 0 \leq b_{j} < d_{j}\leq X_{j},\quad j= 1,2, \;0\leq z< Z.\tag{3} \] For any given functional \(\varphi \in L^{2} (\Omega)^{*}=L^{2} (\Omega)\) it is requested to find a method for calculation its values \(\langle \varphi, v\rangle\), using as information the observed signal only \[ g=A^{*} v, \tag{4} \] where \(A^{*}\) is an operator which associates any initial condition \(v(\vec{x})\) from (2) with a narrowing of the solution \(E\) of the problem (1), (2) to the zone \(Q_{0}\). It should be mentioned that there are interesting applications in optics of the considered problem, i.e. the problem of determining initial parameters of light signals, distorted by passing through an optically inhomogeneous medium. The question of solvability of the problem (1)--(4) and the dual problem of the so-called zone control are discussed in the article. The theory of mutually conjugate operators is used as a base of the discussion. Conditions for the convergence of finite-dimensional approximations of the observation and control are given.