Well-posedness of infinite-dimensional linear systems with nonlinear feedback

Anthony HASTIR, Federico Califano, Hans Zwart

Research output: Contribution to journalArticlepeer-review


We study existence of solutions, and in particular well-posedness, for a class of inhomogeneous, nonlinear partial differential equations (PDE's). The main idea is to use system theory to write the nonlinear PDE as a well-posed infinite-dimensional linear system interconnected with a static nonlinearity. By a simple example, it is shown that in general well-posedness of the closed-loop system is not guaranteed. We show that well-posedness of the closed-loop system is guaranteed for linear systems whose input to output map is coercive for small times interconnected to monotone nonlinearities. This work generalizes the results presented in [1], where only globally Lipschitz continuous nonlinearities were considered. Furthermore, it is shown that a general class of linear port-Hamiltonian systems satisfies the conditions asked on the open-loop system. The result is applied to show well-posedness of a system consisting of a vibrating string with nonlinear damping at the boundary.

Original languageEnglish
Pages (from-to)19-25
Number of pages7
JournalSystems and Control Letters
Publication statusPublished - Jun 2019


  • Boundary feedback
  • Nonlinear damping
  • Nonlinear feedback
  • Passive infinite-dimensional systems
  • port-Hamiltonian systems
  • Vibrating string
  • Well-posedness


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