Generalized backstepping and Gramian methods for spectral assignability in infinite dimension

In this article, we explore connections between two stabilization methods: Gramian stabilization and backstepping (and more generally F -equivalence). These methods are related to the problem of spectral assignability (also known as the pole placement or pole shifting problem), which consists in finding a linear feedback such that the resulting closed-loop system has a prescribed spectrum. We provide a constructive solution to the spectral assignability problem for a broad class of infinitedimensional controllable systems and target spectra, with unbounded but admissible control operators. We construct a feedback using a generalization of the backstepping method, and provide an extensive study of the resulting closed-loop operator. We prove that it has a Riesz basis of eigenvectors, which we explicitly describe, along with its biorthonormal basis. This allows us to introduce general Gramian operators, and give an alternative expression of the feedback using these operators.