Feedback stabilization of a typical unstable heat equation by means of an auxiliary functional observer is studied. The control system is described by \[ (\partial /\partial t)u(t,x)=\sigma u(t,x)=(\Delta - q(x))u(t,x),\quad u(0,x)=u_ 0(x),\quad x\in \Omega, \]
\[ \tau u(t,\xi)=\alpha(\xi)u(t,\xi)+(1-\alpha(\xi))(\partial /\partial n)u(t,\xi)=\sum^{M}_{k=1}f_ k(t)h_ k(\xi),\quad \xi \in \Gamma. \] Here, \(\Omega\) denotes a connected bounded domain in \({\mathbb{R}}^ m\) with a boundary \(\Gamma\), \(f_ k(t)\) inputs, and \(h_ k(\xi)\) controllers. The outputs are a finite number of observations on \(\Gamma\) given by \((u(t,\cdot),w_ k)_{\Gamma},\quad 1\leq k\leq N.\) Let \(\{\lambda_ i,\phi_{ij};\quad i\geq 1,\quad 1\leq j\leq m_ i(<\infty)\}\) be the eigenpairs such that \(\sigma \phi_{ij}=-\lambda_ i\phi_{ij},\quad \tau \phi_{ij}=0,\) and \(\lambda_ 1<\lambda_ 2<...\to \infty.\) Our goal is as follows: Given \(h_ k\) and \(w_ k\) on \(\Gamma\), design \(f_ k(t)\) as a suitable feedback of the outputs in order to stabilize the evolution u(t,\(\cdot)\). The feedback is implemented via a functional observer, which is first a conceptional one. The main result is as follows: Under some conditions we can find suitable \(\alpha_ k,\quad \rho_ k,\quad 1\leq k\leq M,\) such that the state u and the observer v satisfy \[ \| u(t,\cdot)\| \quad and\quad \| v(t)\|_ H\leq const e^{-\kappa t}\{\| u_ 0\| +\| v_ 0\|_ H\},\quad t\geq 0. \] The observer is reduced to purely finite- dimensional one by adding a small perturbation. The system finally obtained has a state \(\{u(t,\cdot),v_ 1(t)\}\), where \(v_ 1(t)\) belongs to \({\mathbb{R}}^ S\), S being finite. The final estimate is the same as above with \(\| v\|_ H\) replaced by \(| v_ 1|_{{\mathbb{R}}^ S}\).