Stability for semidiscrete Galerkin approximation of neutral delay equations

We consider the issue of stability for semidiscrete Galerkin approximations of neutral delay-differential equations. We recall recent results which show how a reforming of the energy state space can be used to obtain a dissipative inequality which implies exponential stability of the solution semigroup associated with the delay differential equation. We then show in detail how the norm is used to construct finite dimensional semidiscrete Galerliin approximations which preserve the stability behavior of the original neutral equation. In applications to optimal control problems, it is important that semi-discrete approximation schemes have this property.