Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback

The problem of stabilizability from the boundary $\partial\Omega$ for a parabolic equation given in a bounded domain $\Omega\in\mathbb R^n$, consists in choosing a boundary condition (a control) such that the solution of the resulting mixed boundary-value problem tends as $t\to\infty$ to a given steady-state solution at a prescribed rate $\exp(-\sigma_0t)$. Furthermore, it is required that the control be with feedback, that is, that it react to unpredictable fluctuations of the system by suppressing the results of their action on the stabilizable solution. A new mathematical formulation of the concept of feedback is presented and then used in solving the problem of stabilizability of linear as well as quasi-linear parabolic equations by means of a control with feedback defined on part of the boundary.