A note on algebraic aspects of boundary feedback control systems of parabolic type (Q1969187)

The control system is \[ {\partial u \over \partial t} + {\mathcal L} u = 0 \quad (t \geq 0, x \in \Omega) \] in a \(n\)-dimensional domain \(\Omega\) with boundary \(\Gamma,\) subject to finite dimensional boundary feedback \[ \tau u = \sum_{k=1}^N \langle u, w_k \rangle_\Omega h_k \quad (t \geq 0, x \in \Gamma) . \] Here, \(\mathcal L\) is (the negative of) a second order uniformly elliptic differential operator in \(\Omega\) and \(\tau\) is a Dirichlet or Robin boundary operator. The system is reduced to the abstract differential equation \(u'(t) + Mu(t) = 0\) in \(L^2(\Omega),\) where \(M = {\mathcal L}\) with the feedback boundary condition. The problem is to fix the feedback parameters in such a way that \(\|e^{-Mt}\|\leq ce^{-\mu t}\) for \(t > 0.\) This was previously done by the author using fractional powers \((L + cI)^\omega,\) \(L = {\mathcal L}\) with the homogeneous boundary condition. The author proposes an alternative algebraic approach to stabilization that does not require fractional powers of \(L + cI.\)