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 theDirichlet boundary condition and expressed as a specifiedfeedback of theposition 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 withzero boundary conditions is self-adjoint and unstable. Here, under certain natural algebraic conditions based on the finitely many unstable eigenvalues, we establish the existence ofboundary vectors, for which the corresponding feedback solutions have the same desirablestructural property of astable 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.This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (I.L.) and Grant AFOSR-77-3338 (R.T.) through ISU.This research was performed while the author was visiting the Department of Systems Science, University of California, Los Angeles.