Stabilization of Neumann boundary feedback of parabolic equations: The case of trace in the feedback loop
A parabolic equation defined on a bounded domain is considered, with input acting in the Neumann (or mixed) boundary condition, and expressed as a specified finite dimensional, nondynamical feedback of the Dirichlet trace of the solution ("boundary observation"). The free system is assumed unstable. Conditions are given at the unstable eigenvalues, under which one can select boundary vectors of the feedback operator, so that the corresponding feedback solutions decay exponentially to zero, in the uniform operator norm as t → + ∞. These conditions consist of (i) verifiable algebraic (full rank) conditions, plus (ii) an Invertibility Condition. The latter depends crucially on properties of the Dirichlet traces of the ('free system') eigenfunctions, whose direct knowledge is available only in special cases. We then specialize-in the appendix-to canonical situations, involving the Laplacian (translated) on spheres and parallelepipeds. In these cases, we indicate how to construct (in infinitely many ways, in fact) boundary vectors of the feedback operator, which satisfy both the algebraic conditions and the Invertibility Condition, thereby yielding stabilization. © 1983 Springer-Verlag New York Inc.
Applied Mathematics & Optimization
Lasiecka, I., & Triggiani, R. (1983). Stabilization of Neumann boundary feedback of parabolic equations: The case of trace in the feedback loop. Applied Mathematics & Optimization, 10 (1), 307-350. https://doi.org/10.1007/BF01448392