### 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 language | English |
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Pages (from-to) | 19-25 |

Number of pages | 7 |

Journal | Systems and control letters |

Volume | 128 |

Early online date | 6 May 2019 |

DOIs | |

Publication status | Published - 1 Jun 2019 |

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### 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.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Well-posedness of infinite-dimensional linear systems with nonlinear feedback

AU - Hastir, Anthony

AU - Califano, Federico

AU - Zwart, Hans

PY - 2019/6/1

Y1 - 2019/6/1

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.

AB - 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.

KW - Boundary feedback

KW - Nonlinear damping

KW - Nonlinear feedback

KW - Passive infinite-dimensional systems

KW - port-Hamiltonian systems

KW - Vibrating string

KW - Well-posedness

UR - http://www.scopus.com/inward/record.url?scp=85065042437&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2019.04.002

DO - 10.1016/j.sysconle.2019.04.002

M3 - Article

VL - 128

SP - 19

EP - 25

JO - Systems and control letters

JF - Systems and control letters

SN - 0167-6911

ER -