Solutions and constrained null-controllability for a differential-difference equation

Abstract

Abstract A formula for the solution of a differential-difference equation is given, and then the corresponding semigroup theory is developed, with a detailed description of the infinitesimal generator and some of its spectral properties. Results in Sections 2 and 3 are used in particular to show how the equation can be rewritten as a classical Cauchy problem. This, in Section 4, allows to study its exact null-controllability, by using some techniques from Functional Analysis and Operator Theory. The control u is constrained to lie in a separable weakly compact subset Ω of an arbitrary Banach space U.