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

Anthony Hastir, Federico Califano, Hans Zwart

Research output: Contribution to journalArticleAcademicpeer-review

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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.

Original languageEnglish
Pages (from-to)19-25
Number of pages7
JournalSystems and control letters
Volume128
Early online date6 May 2019
DOIs
Publication statusPublished - 1 Jun 2019

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Nonlinear feedback
Closed loop systems
Partial differential equations
Linear systems
Hamiltonians
System theory
Damping

Keywords

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

Cite this

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Well-posedness of infinite-dimensional linear systems with nonlinear feedback. / Hastir, Anthony; Califano, Federico; Zwart, Hans.

In: Systems and control letters, Vol. 128, 01.06.2019, p. 19-25.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Califano, Federico

AU - Zwart, Hans

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N2 - 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.

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KW - Nonlinear feedback

KW - Passive infinite-dimensional systems

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