On Stability and the Spectrum Determined Growth Condition for Spatially Periodic Systems

Abstract

We consider distributed parameter systems where the underlying dynamics are spatially periodic on the real line. We examine the problem of exponential stability, namely whether the semigroup e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">At</sup> decays exponentially in time. It is known that for distributed systems the condition that the spectrum of A belong to the open left-half plane is, in general, not sufficient for exponential stability. Those systems for which this condition is sufficient are said to satisfy the spectrum determined growth condition (SDGC). In this work we separate A into a spatially invariant operator and a spatially periodic operator. We find conditions for the spatially invariant part to satisfy the SDGC, and show that the SDGC remains satisfied under the addition of a spatially periodic operator if this operator is 'weak' enough relative to the spatially invariant one. A similar method is used to derive conditions which guarantee that A has left-half plane spectrum and thus establish exponential stability