Authors

Anthony Hastir, Federico Califano, Hans Zwart

Abstract

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.

Keywords

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

Citation

BibTeX

@article{Hastir_2019,
  title={{Well-posedness of infinite-dimensional linear systems with nonlinear feedback}},
  volume={128},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2019.04.002},
  journal={Systems & Control Letters},
  publisher={Elsevier BV},
  author={Hastir, Anthony and Califano, Federico and Zwart, Hans},
  year={2019},
  pages={19--25}
}

Download the bib file

References

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