An algebraic method of stabilization for a class of boundary control systems of parabolic type (Q5935515)

The boundary control system consists of a parabolic equation \[ {\partial u(t, x) \over \partial t} + {\mathcal L}u(t, x) = 0 \quad (t \geq 0, \;x \in \Omega) \] in a domain \(\Omega\) with boundary \(\Gamma,\) coupled with an ordinary differential system \[ {dv(t) \over dt} + Bv(t) = \sum_{n=1}^N \langle u(t), w_n \rangle_\Gamma \xi_n + \sum_{n=1}^N \langle v(t), \zeta_n\rangle \eta_n \quad (t \geq 0) \] through the boundary condition \[ \tau u(t) = \sum_{n=1}^M \langle v(t), \rho_n \rangle h_n \quad (t \geq 0, \;x \in \Gamma) \] where \(\tau\) is a suitable boundary operator. The \(w_n\) and \(h_n\) are given, the vector \(\{\langle u(t), w_n\rangle\}\) is the output, and the task is to determine the feedback parameters \(B, \xi_n,\) \(\zeta_n,\) \(\eta_n\) and \(\rho_n\) in such a way that \(u(t)\) and \(v(t)\) decay exponentially as \(t \to \infty.\) The author shows that this is the case under smoothness and algebraic conditions on the feedback parameters, generalizing results pertaining to the Dirichlet boundary condition.