Sobolev’s type optimal topology in the problem of exact observability for Hilbert space dynamical systems connected with Riesz basis of divided differences

This paper considers the problem of exact observability of a general class of distributed parameter systems in Hilbert spaces, connected to Riesz basis properties of some families of exponential functions and the divided differences of those functions. Under some assumptions on asymptotic spectral analysis of the differential operator of the system, the conditions of exact observability are stated in the form of exact observable spaces being the direct sum of some specific Sobolev spaces. The main result consists of proving the optimality of these subspaces of observable states. The result was based on advanced non-harmonic analysis approach connected to the unusual fact that time-space Riesz basis does not consist only of exponential functions but also contains divided differences of these functions.