An observation problem for the Bessel differential operator

Abstract

In this paper, the parabolic partial differential equation u, = u rr + {\/r)u r -(v 2 /r 2 )u, where v S= 0 is a parameter, with Dirichlet, Neumann, and mixed boundary conditions is considered. The final state observability for such problems is investigated.

Boundary value controllability and observability problems for the wave and heat equation

“…Thus, this final state observability result agrees with those obtained in [6], [7], [14], [19] and [21].…”

“…On this basis, a duality theorem proved in [4] can then be used to obtain the controllability result, where the control / again enters in the mixed boundary condition. In [2], [3], [14], [19], [21], results concerning various observability problems for the heat equation in one space dimension or higher dimensions are established. For more details concerning controllability results see [6], [7].…”

Boundary value control problems involving the bessel differential operator

“…The proof of Theorem 1.1 and the proofs of results concerning necessary conditions for controllability depend on special properties of the eigenvalues and eigenfunctions of B{ 01) and the spectral representation of the energy spaces H B . Some of the preparatory results reported in Section 3 (see also [20]) may also be of interest by themselves.…”