Hyperbolic equations with Dirichlet boundary feedback via position vector: Regularity and almost periodic stabilization-Part I


A hyperbolic equation defined on a bounded domain is considered, with input acting in the Dirichlet boundary condition and expressed as a specified feedback of the position vector only. Two main results are established. First, we prove a well-posedness and regularity result of the feedback solutions. Second, we specialize our equation to the case when the original differential operator with zero boundary conditions is self-adjoint and unstable. Here, under certain natural algebraic conditions based on the finitely many unstable eigenvalues, we establish the existence of boundary vectors, for which the corresponding feedback solutions have the same desirable structural property of a stable free system: They can be expressed as an infinite linear combination of sines and cosines (special case of almost periodicity). A cosine operator approach is employed. © 1981 Springer-Verlag New York Inc.

Publication Title

Applied Mathematics & Optimization