Conditions for the equivalence of internal and external stability for distributed parameter systems

The author gives conditions under which external stability, also known as input-output stability, implies internal stability, which is the exponential stability of the underlying semigroup. In particular, he gives a very general definition of stabilizability and detectability and shows that, when the system satisfies both of these, external stability is equivalent to internal stability. Therefore, in order to check internal stability, it is often sufficient to check the boundedness of the transfer function in the right half complex plane. The author applies this to a simple example, and also obtains a robustness result