Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Input-to-State Stability for PDEs

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_100024-1


This chapter reviews the challenges for the extension of the Input-to-State Stability (ISS) property for systems described by Partial Differential Equations (PDEs). The methodologies that have been used in the literature for the derivation of ISS estimates, are presented. Examples are also provided and possible directions of future research on ISS for PDEs are given.


Input-to-State Stability Partial Differential Equations Infinite-Dimensional Systems 
This is a preview of subscription content, log in to check access.


  1. Bekiaris-Liberis N, Krstic M (2013) Nonlinear control under nonconstant delays. SIAM, PhiladelphiaCrossRefGoogle Scholar
  2. Karafyllis I, Krstic M (2014) On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM Control Optim Calc Var 20: 894–923MathSciNetCrossRefGoogle Scholar
  3. Karafyllis I, Krstic M (2017a) ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J Control Optim 55:1716–1751MathSciNetCrossRefGoogle Scholar
  4. Karafyllis I, Krstic M (2017b) Stability of integral delay equations and stabilization of age-structured models. ESAIM Control Optim Calc Var 23(4):1667–1714MathSciNetCrossRefGoogle Scholar
  5. Karafyllis I, Krstic M (2019a) Input-to-state stability for PDEs. Communications and control engineering. Springer, LondonCrossRefGoogle Scholar
  6. Karafyllis I, Krstic M (2019b) Small-gain-based boundary feedback design for global exponential stabilization of 1-D semilinear parabolic PDEs. SIAM J Control Optim 57(3):2016–2036MathSciNetCrossRefGoogle Scholar
  7. Koga S, Karafyllis I, Krstic M (2018) Input-to-state stability for the control of Stefan problem with respect to heat loss at the interface. In: Proceedings of the 2018 American control conferenceGoogle Scholar
  8. Mironchenko A (2016) Local input-to-state stability: characterizations and counterexamples. Syst Control Lett 87:23–28MathSciNetCrossRefGoogle Scholar
  9. Mironchenko A, Karafyllis I, Krstic M (2019) Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances. SIAM J Control Optim 57:510–532MathSciNetCrossRefGoogle Scholar

Authors and Affiliations

  1. 1.Department of MathematicsNational Technical University of AthensAthensGreece
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

Section editors and affiliations

  • Miroslav Krstic
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA