The Differentiability with Respect to a Parameter of the Solution of a Linear Abstract Cauchy Problem

The differentiability with repect to a parameter of the solution of a linear inhomogeneous abstract Cauchy problem is considered by employing the theory of strongly continuous semigroups. Criteria for differentiability are developed which allow the parameter to appear in unbounded terms of the generator. Sufficient conditions are given for the solvability of an inhomogeneous Cauchy problem with zero initial condition by the variation of constants formula. These conditions are used to show that the derivatives with respect to a parameter satisfy certain sensitivity equations in time. The results are applied to the differentiability of the solution of a linear delay differential equation with respect to its delays.