Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations

Abstract

The paper gives general necessary and sufficient conditions for completeness of generalized eigenfunctions associated with systems of linear autonomous retarded functional differential equations (FDE), in the Hilbert space R n × L 2([− h, 0], R n ), and also in the space C([− h, 0], R n ). The generalized eigenfunctions are elements of the nullspaces of ( Iλ − A) m , where A is the infinitesimal generator of a C 0-semigroup of bounded linear operators on R n × L 2([− h, 0], R n ) (resp. C([− h, 0], R n )) corresponding to the FDE in question. In addition to the usual notion of completeness, a new concept of F-completeness is introduced and its significance is explained. In particular, it is shown that the F-completeness is related to the absence of solutions of the transposed equation that vanish in finite time. The results are obtained entirely via the C 0-semigroup theory, which results in simplicity of the proofs. As a by-product, some new results on the adjoint semigroup are obtained. The main results are expressed in an operator form. These are translated into conditions expressed in terms of the original system matrices. For systems with one delay, the F-completeness criterion is translated into a verifiable matrix type criterion, in which the concepts of maximal controllability and invariant subspaces of two matrices play a prominent role.