Dissipativity of the delay semigroup

TL;DR

This paper demonstrates how delay semigroups can be transformed into contraction semigroups through inner product modification, aiding in the stability analysis of delayed differential and stochastic differential equations.

Contribution

It introduces a method to convert delay semigroups into contraction semigroups via inner product modification, facilitating stability studies.

Findings

Delay semigroups can be made contractive with an equivalent inner product.

Application to stability of differential equations with delay.

Application to stochastic differential equations with delay.

Abstract

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.