Renorming for stability and approximation of linear systems: Examples

For the linear Cauchy problem on a Hilbert space, we consider the two issues of exponential stability and preservation of exponential stability under approximation. It is often the case that an appropriate inner product (equivalent to the original energy inner product) can provide insight into these stability issues, and can even be used to construct approximation schemes. We give a heuristic discussion motivating the use of such inner products, and then survey several recent examples in which this idea has been used. Included are examples of damped elastic systems and retarded and neutral delay equations.