A semigroup approach to boundary feedback systems
In this paper, ``abstract boundary feedback systems'' of the following form are studied: \[ \dot{u}(t) = A_m u(t), \;\dot{x}(t) = Bx(t) + Cu(t), \;Lu(t) = x(t), \;t\geq 0, \;u(0)= f_0, \;x(0)=x_0, \] where \(A_m\) is a linear operator on the state space \(X\), \(B\) is a linear operator on the boundary space \(\partial X\), \(L\) (the boundary operator) is a linear and bounded operator from the domain of \(A_m\) to \(\partial X\), and \(C:X\rightarrow \partial X\) is the feedback operator. In order to solve this abstract boundary feedback system, the authors convert it into an abstract Cauchy problem of the form \(\dot{x}(t) = {\mathcal A}{x}(t) + F(t)\), \(t\geq 0\), \({x}(0)= {x}_0\), where \({\mathcal A}\) is an operator matrix on the product \(X\times \partial X\), and then semigroup methods are used. Further, spectral theory is applied in order to discuss stability under bounded feedback. As an example, the heat equation on a bounded domain \(\Omega\subset \mathbb R^n\) with Dirichlet and Neumann boundary conditions is studied.
scientific article; zbMATH DE number 2018223