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

Conditions are given on a controlled, observed distributed parameter system under which external stability, also known as input-output stability, implies internal stability, which is the exponential stability of the underlying semigroup generator. It is shown that when the system satisfies a general definition of stabilizability and detectability, external stability is equivalent to internal stability. Unlike the case previous definitions of stabilizability and detectability, a system which satisfies these does not necessarily have a spectrum decomposition. 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. This is illustrated with a simple example