On continuity of solutions for parabolic control systems and input-to-state stability

Birgit Jacob, Felix L. Schwenninger* (Corresponding Author), Hans Zwart

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

28 Citations (Scopus)
177 Downloads (Pure)


We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

Original languageEnglish
Pages (from-to)6284-6306
Number of pages23
JournalJournal of differential equations
Issue number10
Early online date8 Nov 2018
Publication statusPublished - 5 May 2019


  • Abstract parabolic control system
  • Admissible operator
  • Bounded functional calculus
  • H∞ calculus
  • Input-to-state stability
  • Orlicz space


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