CONDITIONS FOR THE EQUIVALENCE OF INTERNAL AND EXTERNAL STABILITY FORDISTRIBUTED-PARAMETER SYSTEMS

We give conditions on a controlled, observed distributed parameter sys tem under which external stability, also known as input-output stabili ty, implies internal stability, which is the exponential stability of the underlying semigroup generator. We show that when the system satis fies a general definition of stabilizability and detectability, extern al stability is equivalent to internal stability. Unlike previous defi nitions 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 pla ne. We illustrate this with a simple example.