Function Space Controllability of Linear Retarded Systems: A Derivation from Abstract Operator Conditions

Controllability of linear retarded systems is investigated by using the abstract representation of such systems given by x = Ax + Bu, where x belongs to Hilbert space Rⁿ × L₂ ([ - h,0],Rⁿ ) denoted as M₂, and A generates a C₀-semigroup. It is shown that useful, practically verifiable conditions can be obtained by this approach. The following problems are investigated: approximate controllability in the space M₂ and its subspace, L₂, exact Euclidean (Rⁿ ) controllability, spectral controllability, feedback stabilizability and a relation between pointwise degeneracy and function space controllability. Starting from the abstract functional analytic framework, the analysis is carried down to the matrix theory level, through the crucial intermediate role of the theory of entire functions.