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Published in:
2019 | OriginalPaper | Chapter
5. ISS in Spatial L2 and H1 Norms
Authors : Iasson Karafyllis, Miroslav Krstic
Published in: Input-to-State Stability for PDEs
Publisher: Springer International Publishing
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Abstract
The chapter provides ISS estimates of the solutions of 1-D, parabolic PDEs with respect to boundary and distributed disturbances, which are expressed in the spatial \( L^{2} \) and \( H^{1} \) norms of the state. Two different methodologies for the derivation of ISS estimates are developed. The first methodology involves an eigenfunction expansion of the solution. The second methodology involves the use of an ISS Lyapunov functional and deals with special cases for the boundary conditions where boundary disturbances appear only in boundary conditions of Robin or Neumann type. The derivation of ISS estimates is illustrated in two different applications: the study of the sensitivity of the temperature distribution in a solid bar subject to variations of the air temperature and the study of the sensitivity of R–A–D PDEs subject to inlet disturbances.