Transfer functions for infinite-dimensional systems

In this paper, we study three definitions of the transfer function for an infinite-dimensional system. The first one defines the transfer function as the expression $C(sI-A)^{-1}B+D$. In the second definition, the transfer function is defined as the quotient of the Laplace transform of the output and input, with initial condition zero. In the third definition, we introduce the transfer function as the quotient of the input and output, when the input and output are exponentials. We show that these definitions always agree on the right-half plane bounded to the left by the growth bound of the underlying semigroup, but that they may differ elsewhere.