Boundary feedback stabilizability of parabolic equations

A parabolic equation defined on a bounded domain is considered, with input acting on theboundary expressed as a specifiedfeedback of the solution. Both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated. Algebraic conditions based on the finitely many unstable eigenvalues are given, ensuring the existence ofboundary vectors, for which all the solutions to theboundary feedback parabolic equation decay exponentially to zero ast+ in (essentially) the strongest possible space norm. A semigroup approach is employed.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-77-3338.A preliminary version of this paper has appeared in the Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, held at Delft University, The Netherlands, July 3–6, 1979; pp. 428–433.